1,1,102,0,0.1015123,"\int \sin ^3(e+f x) (a+a \sin (e+f x))^2 \, dx","Int[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2,x]","-\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a^2 \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a^2 x}{4}","-\frac{a^2 \cos ^5(e+f x)}{5 f}+\frac{a^2 \cos ^3(e+f x)}{f}-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a^2 \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a^2 x}{4}",1,"(3*a^2*x)/4 - (2*a^2*Cos[e + f*x])/f + (a^2*Cos[e + f*x]^3)/f - (a^2*Cos[e + f*x]^5)/(5*f) - (3*a^2*Cos[e + f*x]*Sin[e + f*x])/(4*f) - (a^2*Cos[e + f*x]*Sin[e + f*x]^3)/(2*f)","A",9,4,21,0.1905,1,"{2757, 2633, 2635, 8}"
2,1,129,0,0.1448261,"\int \sin ^3(e+f x) (a+a \sin (e+f x))^3 \, dx","Int[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^3,x]","-\frac{3 a^3 \cos ^5(e+f x)}{5 f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{a^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{23 a^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{23 a^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{23 a^3 x}{16}","-\frac{3 a^3 \cos ^5(e+f x)}{5 f}+\frac{7 a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{a^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{23 a^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{23 a^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{23 a^3 x}{16}",1,"(23*a^3*x)/16 - (4*a^3*Cos[e + f*x])/f + (7*a^3*Cos[e + f*x]^3)/(3*f) - (3*a^3*Cos[e + f*x]^5)/(5*f) - (23*a^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (23*a^3*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (a^3*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f)","A",13,4,21,0.1905,1,"{2757, 2633, 2635, 8}"
3,1,53,0,0.069486,"\int \frac{\sin ^4(x)}{a+a \sin (x)} \, dx","Int[Sin[x]^4/(a + a*Sin[x]),x]","-\frac{3 x}{2 a}+\frac{4 \cos ^3(x)}{3 a}-\frac{4 \cos (x)}{a}+\frac{\sin ^3(x) \cos (x)}{a \sin (x)+a}+\frac{3 \sin (x) \cos (x)}{2 a}","-\frac{3 x}{2 a}+\frac{4 \cos ^3(x)}{3 a}-\frac{4 \cos (x)}{a}+\frac{\sin ^3(x) \cos (x)}{a \sin (x)+a}+\frac{3 \sin (x) \cos (x)}{2 a}",1,"(-3*x)/(2*a) - (4*Cos[x])/a + (4*Cos[x]^3)/(3*a) + (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^3)/(a + a*Sin[x])","A",6,5,13,0.3846,1,"{2767, 2748, 2635, 8, 2633}"
4,1,42,0,0.048874,"\int \frac{\sin ^3(x)}{a+a \sin (x)} \, dx","Int[Sin[x]^3/(a + a*Sin[x]),x]","\frac{3 x}{2 a}+\frac{2 \cos (x)}{a}+\frac{\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac{3 \sin (x) \cos (x)}{2 a}","\frac{3 x}{2 a}+\frac{2 \cos (x)}{a}+\frac{\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac{3 \sin (x) \cos (x)}{2 a}",1,"(3*x)/(2*a) + (2*Cos[x])/a - (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^2)/(a + a*Sin[x])","A",2,2,13,0.1538,1,"{2767, 2734}"
5,1,27,0,0.0664069,"\int \frac{\sin ^2(x)}{a+a \sin (x)} \, dx","Int[Sin[x]^2/(a + a*Sin[x]),x]","-\frac{x}{a}-\frac{\cos (x)}{a}-\frac{\cos (x)}{a (\sin (x)+1)}","-\frac{x}{a}-\frac{\cos (x)}{a}-\frac{\cos (x)}{a (\sin (x)+1)}",1,"-(x/a) - Cos[x]/a - Cos[x]/(a*(1 + Sin[x]))","A",4,4,13,0.3077,1,"{2746, 12, 2735, 2648}"
6,1,17,0,0.0285508,"\int \frac{\sin (x)}{a+a \sin (x)} \, dx","Int[Sin[x]/(a + a*Sin[x]),x]","\frac{x}{a}+\frac{\cos (x)}{a \sin (x)+a}","\frac{x}{a}+\frac{\cos (x)}{a \sin (x)+a}",1,"x/a + Cos[x]/(a + a*Sin[x])","A",2,2,11,0.1818,1,"{2735, 2648}"
7,1,12,0,0.0102863,"\int \frac{1}{a+a \sin (x)} \, dx","Int[(a + a*Sin[x])^(-1),x]","-\frac{\cos (x)}{a \sin (x)+a}","-\frac{\cos (x)}{a \sin (x)+a}",1,"-(Cos[x]/(a + a*Sin[x]))","A",1,1,8,0.1250,1,"{2648}"
8,1,20,0,0.0379892,"\int \frac{\csc (x)}{a+a \sin (x)} \, dx","Int[Csc[x]/(a + a*Sin[x]),x]","\frac{\cos (x)}{a \sin (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a}","\frac{\cos (x)}{a \sin (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"-(ArcTanh[Cos[x]]/a) + Cos[x]/(a + a*Sin[x])","A",3,3,11,0.2727,1,"{2747, 3770, 2648}"
9,1,26,0,0.0602264,"\int \frac{\csc ^2(x)}{a+a \sin (x)} \, dx","Int[Csc[x]^2/(a + a*Sin[x]),x]","-\frac{2 \cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}+\frac{\cot (x)}{a \sin (x)+a}","-\frac{2 \cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}+\frac{\cot (x)}{a \sin (x)+a}",1,"ArcTanh[Cos[x]]/a - (2*Cot[x])/a + Cot[x]/(a + a*Sin[x])","A",5,5,13,0.3846,1,"{2768, 2748, 3767, 8, 3770}"
10,1,42,0,0.0663447,"\int \frac{\csc ^3(x)}{a+a \sin (x)} \, dx","Int[Csc[x]^3/(a + a*Sin[x]),x]","\frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a \sin (x)+a}","\frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a \sin (x)+a}",1,"(-3*ArcTanh[Cos[x]])/(2*a) + (2*Cot[x])/a - (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x])/(a + a*Sin[x])","A",6,6,13,0.4615,1,"{2768, 2748, 3768, 3770, 3767, 8}"
11,1,55,0,0.0696497,"\int \frac{\csc ^4(x)}{a+a \sin (x)} \, dx","Int[Csc[x]^4/(a + a*Sin[x]),x]","-\frac{4 \cot ^3(x)}{3 a}-\frac{4 \cot (x)}{a}+\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc ^2(x)}{a \sin (x)+a}","-\frac{4 \cot ^3(x)}{3 a}-\frac{4 \cot (x)}{a}+\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc ^2(x)}{a \sin (x)+a}",1,"(3*ArcTanh[Cos[x]])/(2*a) - (4*Cot[x])/a - (4*Cot[x]^3)/(3*a) + (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x]^2)/(a + a*Sin[x])","A",6,5,13,0.3846,1,"{2768, 2748, 3767, 3768, 3770}"
12,1,66,0,0.1207506,"\int \frac{\sin ^4(x)}{(a+a \sin (x))^2} \, dx","Int[Sin[x]^4/(a + a*Sin[x])^2,x]","\frac{7 x}{2 a^2}+\frac{16 \cos (x)}{3 a^2}+\frac{8 \sin ^2(x) \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{7 \sin (x) \cos (x)}{2 a^2}+\frac{\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}","\frac{7 x}{2 a^2}+\frac{16 \cos (x)}{3 a^2}+\frac{8 \sin ^2(x) \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{7 \sin (x) \cos (x)}{2 a^2}+\frac{\sin ^3(x) \cos (x)}{3 (a \sin (x)+a)^2}",1,"(7*x)/(2*a^2) + (16*Cos[x])/(3*a^2) - (7*Cos[x]*Sin[x])/(2*a^2) + (8*Cos[x]*Sin[x]^2)/(3*a^2*(1 + Sin[x])) + (Cos[x]*Sin[x]^3)/(3*(a + a*Sin[x])^2)","A",3,3,13,0.2308,1,"{2765, 2977, 2734}"
13,1,47,0,0.1431517,"\int \frac{\sin ^3(x)}{(a+a \sin (x))^2} \, dx","Int[Sin[x]^3/(a + a*Sin[x])^2,x]","-\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}-\frac{2 \cos (x)}{a^2 (\sin (x)+1)}+\frac{\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2}","-\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}-\frac{2 \cos (x)}{a^2 (\sin (x)+1)}+\frac{\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2}",1,"(-2*x)/a^2 - (4*Cos[x])/(3*a^2) - (2*Cos[x])/(a^2*(1 + Sin[x])) + (Cos[x]*Sin[x]^2)/(3*(a + a*Sin[x])^2)","A",6,6,13,0.4615,1,"{2765, 2968, 3023, 12, 2735, 2648}"
14,1,35,0,0.0725763,"\int \frac{\sin ^2(x)}{(a+a \sin (x))^2} \, dx","Int[Sin[x]^2/(a + a*Sin[x])^2,x]","\frac{x}{a^2}+\frac{5 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\cos (x)}{3 (a \sin (x)+a)^2}","\frac{x}{a^2}+\frac{5 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\cos (x)}{3 (a \sin (x)+a)^2}",1,"x/a^2 + (5*Cos[x])/(3*a^2*(1 + Sin[x])) - Cos[x]/(3*(a + a*Sin[x])^2)","A",3,3,13,0.2308,1,"{2758, 2735, 2648}"
15,1,33,0,0.0314165,"\int \frac{\sin (x)}{(a+a \sin (x))^2} \, dx","Int[Sin[x]/(a + a*Sin[x])^2,x]","\frac{\cos (x)}{3 (a \sin (x)+a)^2}-\frac{2 \cos (x)}{3 \left(a^2 \sin (x)+a^2\right)}","\frac{\cos (x)}{3 (a \sin (x)+a)^2}-\frac{2 \cos (x)}{3 \left(a^2 \sin (x)+a^2\right)}",1,"Cos[x]/(3*(a + a*Sin[x])^2) - (2*Cos[x])/(3*(a^2 + a^2*Sin[x]))","A",2,2,11,0.1818,1,"{2750, 2648}"
16,1,33,0,0.0211535,"\int \frac{1}{(a+a \sin (x))^2} \, dx","Int[(a + a*Sin[x])^(-2),x]","-\frac{\cos (x)}{3 \left(a^2 \sin (x)+a^2\right)}-\frac{\cos (x)}{3 (a \sin (x)+a)^2}","-\frac{\cos (x)}{3 \left(a^2 \sin (x)+a^2\right)}-\frac{\cos (x)}{3 (a \sin (x)+a)^2}",1,"-Cos[x]/(3*(a + a*Sin[x])^2) - Cos[x]/(3*(a^2 + a^2*Sin[x]))","A",2,2,8,0.2500,1,"{2650, 2648}"
17,1,38,0,0.0882768,"\int \frac{\csc (x)}{(a+a \sin (x))^2} \, dx","Int[Csc[x]/(a + a*Sin[x])^2,x]","\frac{4 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{\cos (x)}{3 (a \sin (x)+a)^2}","\frac{4 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{\cos (x)}{3 (a \sin (x)+a)^2}",1,"-(ArcTanh[Cos[x]]/a^2) + (4*Cos[x])/(3*a^2*(1 + Sin[x])) + Cos[x]/(3*(a + a*Sin[x])^2)","A",4,4,11,0.3636,1,"{2766, 2978, 12, 3770}"
18,1,45,0,0.1345028,"\int \frac{\csc ^2(x)}{(a+a \sin (x))^2} \, dx","Int[Csc[x]^2/(a + a*Sin[x])^2,x]","-\frac{10 \cot (x)}{3 a^2}+\frac{2 \tanh ^{-1}(\cos (x))}{a^2}+\frac{2 \cot (x)}{a^2 (\sin (x)+1)}+\frac{\cot (x)}{3 (a \sin (x)+a)^2}","-\frac{10 \cot (x)}{3 a^2}+\frac{2 \tanh ^{-1}(\cos (x))}{a^2}+\frac{2 \cot (x)}{a^2 (\sin (x)+1)}+\frac{\cot (x)}{3 (a \sin (x)+a)^2}",1,"(2*ArcTanh[Cos[x]])/a^2 - (10*Cot[x])/(3*a^2) + (2*Cot[x])/(a^2*(1 + Sin[x])) + Cot[x]/(3*(a + a*Sin[x])^2)","A",6,6,13,0.4615,1,"{2766, 2978, 2748, 3767, 8, 3770}"
19,1,64,0,0.1451653,"\int \frac{\csc ^3(x)}{(a+a \sin (x))^2} \, dx","Int[Csc[x]^3/(a + a*Sin[x])^2,x]","\frac{16 \cot (x)}{3 a^2}-\frac{7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac{\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}","\frac{16 \cot (x)}{3 a^2}-\frac{7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac{7 \cot (x) \csc (x)}{2 a^2}+\frac{8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac{\cot (x) \csc (x)}{3 (a \sin (x)+a)^2}",1,"(-7*ArcTanh[Cos[x]])/(2*a^2) + (16*Cot[x])/(3*a^2) - (7*Cot[x]*Csc[x])/(2*a^2) + (8*Cot[x]*Csc[x])/(3*a^2*(1 + Sin[x])) + (Cot[x]*Csc[x])/(3*(a + a*Sin[x])^2)","A",7,7,13,0.5385,1,"{2766, 2978, 2748, 3768, 3770, 3767, 8}"
20,1,71,0,0.1501088,"\int \frac{\csc ^4(x)}{(a+a \sin (x))^2} \, dx","Int[Csc[x]^4/(a + a*Sin[x])^2,x]","-\frac{4 \cot ^3(x)}{a^2}-\frac{12 \cot (x)}{a^2}+\frac{5 \tanh ^{-1}(\cos (x))}{a^2}+\frac{5 \cot (x) \csc (x)}{a^2}+\frac{10 \cot (x) \csc ^2(x)}{3 a^2 (\sin (x)+1)}+\frac{\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2}","-\frac{\cot ^3(x)}{3 a^2}-\frac{4 \cot (x)}{a^2}-\frac{13 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac{\cos (x)}{3 a^2 (\sin (x)+1)^2}+\frac{5 \tanh ^{-1}(\cos (x))}{a^2}+\frac{\cot (x) \csc (x)}{a^2}",1,"(5*ArcTanh[Cos[x]])/a^2 - (12*Cot[x])/a^2 - (4*Cot[x]^3)/a^2 + (5*Cot[x]*Csc[x])/a^2 + (10*Cot[x]*Csc[x]^2)/(3*a^2*(1 + Sin[x])) + (Cot[x]*Csc[x]^2)/(3*(a + a*Sin[x])^2)","A",7,6,13,0.4615,1,"{2766, 2978, 2748, 3767, 3768, 3770}"
21,1,101,0,0.2260267,"\int \frac{\sin ^6(x)}{(a+a \sin (x))^3} \, dx","Int[Sin[x]^6/(a + a*Sin[x])^3,x]","-\frac{23 x}{2 a^3}+\frac{136 \cos ^3(x)}{15 a^3}-\frac{136 \cos (x)}{5 a^3}+\frac{23 \sin ^3(x) \cos (x)}{3 \left(a^3 \sin (x)+a^3\right)}+\frac{23 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{13 \sin ^4(x) \cos (x)}{15 a (a \sin (x)+a)^2}","-\frac{23 x}{2 a^3}+\frac{136 \cos ^3(x)}{15 a^3}-\frac{136 \cos (x)}{5 a^3}+\frac{23 \sin ^3(x) \cos (x)}{3 \left(a^3 \sin (x)+a^3\right)}+\frac{23 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^5(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{13 \sin ^4(x) \cos (x)}{15 a (a \sin (x)+a)^2}",1,"(-23*x)/(2*a^3) - (136*Cos[x])/(5*a^3) + (136*Cos[x]^3)/(15*a^3) + (23*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]^5)/(5*(a + a*Sin[x])^3) + (13*Cos[x]*Sin[x]^4)/(15*a*(a + a*Sin[x])^2) + (23*Cos[x]*Sin[x]^3)/(3*(a^3 + a^3*Sin[x]))","A",8,6,13,0.4615,1,"{2765, 2977, 2748, 2635, 8, 2633}"
22,1,90,0,0.2080003,"\int \frac{\sin ^5(x)}{(a+a \sin (x))^3} \, dx","Int[Sin[x]^5/(a + a*Sin[x])^3,x]","\frac{13 x}{2 a^3}+\frac{152 \cos (x)}{15 a^3}+\frac{76 \sin ^2(x) \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{13 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2}","\frac{13 x}{2 a^3}+\frac{152 \cos (x)}{15 a^3}+\frac{76 \sin ^2(x) \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{13 \sin (x) \cos (x)}{2 a^3}+\frac{\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2}",1,"(13*x)/(2*a^3) + (152*Cos[x])/(15*a^3) - (13*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]^4)/(5*(a + a*Sin[x])^3) + (11*Cos[x]*Sin[x]^3)/(15*a*(a + a*Sin[x])^2) + (76*Cos[x]*Sin[x]^2)/(15*(a^3 + a^3*Sin[x]))","A",4,3,13,0.2308,1,"{2765, 2977, 2734}"
23,1,71,0,0.2210077,"\int \frac{\sin ^4(x)}{(a+a \sin (x))^3} \, dx","Int[Sin[x]^4/(a + a*Sin[x])^3,x]","-\frac{3 x}{a^3}-\frac{9 \cos (x)}{5 a^3}-\frac{3 \cos (x)}{a^3 \sin (x)+a^3}+\frac{\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{3 \sin ^2(x) \cos (x)}{5 a (a \sin (x)+a)^2}","-\frac{3 x}{a^3}-\frac{9 \cos (x)}{5 a^3}-\frac{3 \cos (x)}{a^3 \sin (x)+a^3}+\frac{\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac{3 \sin ^2(x) \cos (x)}{5 a (a \sin (x)+a)^2}",1,"(-3*x)/a^3 - (9*Cos[x])/(5*a^3) + (Cos[x]*Sin[x]^3)/(5*(a + a*Sin[x])^3) + (3*Cos[x]*Sin[x]^2)/(5*a*(a + a*Sin[x])^2) - (3*Cos[x])/(a^3 + a^3*Sin[x])","A",7,7,13,0.5385,1,"{2765, 2977, 2968, 3023, 12, 2735, 2648}"
24,1,59,0,0.1569793,"\int \frac{\sin ^3(x)}{(a+a \sin (x))^3} \, dx","Int[Sin[x]^3/(a + a*Sin[x])^3,x]","\frac{x}{a^3}+\frac{29 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}","\frac{x}{a^3}+\frac{29 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}",1,"x/a^3 + (Cos[x]*Sin[x]^2)/(5*(a + a*Sin[x])^3) - (7*Cos[x])/(15*a*(a + a*Sin[x])^2) + (29*Cos[x])/(15*(a^3 + a^3*Sin[x]))","A",5,5,13,0.3846,1,"{2765, 2968, 3019, 2735, 2648}"
25,1,50,0,0.0760257,"\int \frac{\sin ^2(x)}{(a+a \sin (x))^3} \, dx","Int[Sin[x]^2/(a + a*Sin[x])^3,x]","-\frac{7 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3}","-\frac{7 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{8 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"-Cos[x]/(5*(a + a*Sin[x])^3) + (8*Cos[x])/(15*a*(a + a*Sin[x])^2) - (7*Cos[x])/(15*(a^3 + a^3*Sin[x]))","A",3,3,13,0.2308,1,"{2758, 2750, 2648}"
26,1,50,0,0.0460519,"\int \frac{\sin (x)}{(a+a \sin (x))^3} \, dx","Int[Sin[x]/(a + a*Sin[x])^3,x]","-\frac{\cos (x)}{5 \left(a^3 \sin (x)+a^3\right)}-\frac{\cos (x)}{5 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3}","-\frac{\cos (x)}{5 \left(a^3 \sin (x)+a^3\right)}-\frac{\cos (x)}{5 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"Cos[x]/(5*(a + a*Sin[x])^3) - Cos[x]/(5*a*(a + a*Sin[x])^2) - Cos[x]/(5*(a^3 + a^3*Sin[x]))","A",3,3,11,0.2727,1,"{2750, 2650, 2648}"
27,1,50,0,0.0350515,"\int \frac{1}{(a+a \sin (x))^3} \, dx","Int[(a + a*Sin[x])^(-3),x]","-\frac{2 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{2 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3}","-\frac{2 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{2 \cos (x)}{15 a (a \sin (x)+a)^2}-\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"-Cos[x]/(5*(a + a*Sin[x])^3) - (2*Cos[x])/(15*a*(a + a*Sin[x])^2) - (2*Cos[x])/(15*(a^3 + a^3*Sin[x]))","A",3,2,8,0.2500,1,"{2650, 2648}"
28,1,58,0,0.1605285,"\int \frac{\csc (x)}{(a+a \sin (x))^3} \, dx","Int[Csc[x]/(a + a*Sin[x])^3,x]","\frac{22 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{\tanh ^{-1}(\cos (x))}{a^3}+\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3}","\frac{22 \cos (x)}{15 \left(a^3 \sin (x)+a^3\right)}-\frac{\tanh ^{-1}(\cos (x))}{a^3}+\frac{7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac{\cos (x)}{5 (a \sin (x)+a)^3}",1,"-(ArcTanh[Cos[x]]/a^3) + Cos[x]/(5*(a + a*Sin[x])^3) + (7*Cos[x])/(15*a*(a + a*Sin[x])^2) + (22*Cos[x])/(15*(a^3 + a^3*Sin[x]))","A",5,4,11,0.3636,1,"{2766, 2978, 12, 3770}"
29,1,65,0,0.2287113,"\int \frac{\csc ^2(x)}{(a+a \sin (x))^3} \, dx","Int[Csc[x]^2/(a + a*Sin[x])^3,x]","-\frac{24 \cot (x)}{5 a^3}+\frac{3 \tanh ^{-1}(\cos (x))}{a^3}+\frac{3 \cot (x)}{a^3 \sin (x)+a^3}+\frac{3 \cot (x)}{5 a (a \sin (x)+a)^2}+\frac{\cot (x)}{5 (a \sin (x)+a)^3}","-\frac{24 \cot (x)}{5 a^3}+\frac{3 \tanh ^{-1}(\cos (x))}{a^3}+\frac{3 \cot (x)}{a^3 \sin (x)+a^3}+\frac{3 \cot (x)}{5 a (a \sin (x)+a)^2}+\frac{\cot (x)}{5 (a \sin (x)+a)^3}",1,"(3*ArcTanh[Cos[x]])/a^3 - (24*Cot[x])/(5*a^3) + Cot[x]/(5*(a + a*Sin[x])^3) + (3*Cot[x])/(5*a*(a + a*Sin[x])^2) + (3*Cot[x])/(a^3 + a^3*Sin[x])","A",7,6,13,0.4615,1,"{2766, 2978, 2748, 3767, 8, 3770}"
30,1,86,0,0.2350195,"\int \frac{\csc ^3(x)}{(a+a \sin (x))^3} \, dx","Int[Csc[x]^3/(a + a*Sin[x])^3,x]","\frac{152 \cot (x)}{15 a^3}-\frac{13 \tanh ^{-1}(\cos (x))}{2 a^3}-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{76 \cot (x) \csc (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{11 \cot (x) \csc (x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}","\frac{152 \cot (x)}{15 a^3}-\frac{13 \tanh ^{-1}(\cos (x))}{2 a^3}-\frac{13 \cot (x) \csc (x)}{2 a^3}+\frac{76 \cot (x) \csc (x)}{15 \left(a^3 \sin (x)+a^3\right)}+\frac{11 \cot (x) \csc (x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc (x)}{5 (a \sin (x)+a)^3}",1,"(-13*ArcTanh[Cos[x]])/(2*a^3) + (152*Cot[x])/(15*a^3) - (13*Cot[x]*Csc[x])/(2*a^3) + (Cot[x]*Csc[x])/(5*(a + a*Sin[x])^3) + (11*Cot[x]*Csc[x])/(15*a*(a + a*Sin[x])^2) + (76*Cot[x]*Csc[x])/(15*(a^3 + a^3*Sin[x]))","A",8,7,13,0.5385,1,"{2766, 2978, 2748, 3768, 3770, 3767, 8}"
31,1,103,0,0.2448446,"\int \frac{\csc ^4(x)}{(a+a \sin (x))^3} \, dx","Int[Csc[x]^4/(a + a*Sin[x])^3,x]","-\frac{136 \cot ^3(x)}{15 a^3}-\frac{136 \cot (x)}{5 a^3}+\frac{23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{23 \cot (x) \csc (x)}{2 a^3}+\frac{23 \cot (x) \csc ^2(x)}{3 \left(a^3 \sin (x)+a^3\right)}+\frac{13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}","-\frac{136 \cot ^3(x)}{15 a^3}-\frac{136 \cot (x)}{5 a^3}+\frac{23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{23 \cot (x) \csc (x)}{2 a^3}+\frac{23 \cot (x) \csc ^2(x)}{3 \left(a^3 \sin (x)+a^3\right)}+\frac{13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac{\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3}",1,"(23*ArcTanh[Cos[x]])/(2*a^3) - (136*Cot[x])/(5*a^3) - (136*Cot[x]^3)/(15*a^3) + (23*Cot[x]*Csc[x])/(2*a^3) + (Cot[x]*Csc[x]^2)/(5*(a + a*Sin[x])^3) + (13*Cot[x]*Csc[x]^2)/(15*a*(a + a*Sin[x])^2) + (23*Cot[x]*Csc[x]^2)/(3*(a^3 + a^3*Sin[x]))","A",8,6,13,0.4615,1,"{2766, 2978, 2748, 3767, 3768, 3770}"
32,1,158,0,0.2301148,"\int \sin ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{32 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 a d}+\frac{64 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}-\frac{32 a \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 a \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{32 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 a d}+\frac{64 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}-\frac{32 a \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}",1,"(-32*a*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]]) - (16*a*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (64*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (32*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*a*d)","A",5,4,23,0.1739,1,"{2770, 2759, 2751, 2646}"
33,1,122,0,0.1693197,"\int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{4 a \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{4 a \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}",1,"(-4*a*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) - (12*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a*d)","A",4,4,23,0.1739,1,"{2770, 2759, 2751, 2646}"
34,1,86,0,0.1107937,"\int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{14 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{14 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}",1,"(-14*a*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a*d)","A",3,3,23,0.1304,1,"{2759, 2751, 2646}"
35,1,56,0,0.0451658,"\int \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}",1,"(-2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",2,2,21,0.09524,1,"{2751, 2646}"
36,1,26,0,0.0131947,"\int \sqrt{a+a \sin (c+d x)} \, dx","Int[Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}","-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"(-2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])","A",1,1,14,0.07143,1,"{2646}"
37,1,37,0,0.0548745,"\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d","A",2,2,21,0.09524,1,"{2773, 206}"
38,1,64,0,0.1034915,"\int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a \cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{a \cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])","A",3,3,23,0.1304,1,"{2772, 2773, 206}"
39,1,102,0,0.156279,"\int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}","-\frac{3 a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(-3*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (3*a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])","A",4,3,23,0.1304,1,"{2772, 2773, 206}"
40,1,138,0,0.2126973,"\int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{5 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","-\frac{5 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(-5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (5*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])","A",5,3,23,0.1304,1,"{2772, 2773, 206}"
41,1,38,0,0.0510491,"\int \csc (c+d x) \sqrt{a-a \sin (c+d x)} \, dx","Int[Csc[c + d*x]*Sqrt[a - a*Sin[c + d*x]],x]","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a-a \sin (c+d x)}}\right)}{d}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a-a \sin (c+d x)}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a - a*Sin[c + d*x]]])/d","A",2,2,22,0.09091,1,"{2773, 206}"
42,1,39,0,0.049979,"\int \csc (c+d x) \sqrt{-a+a \sin (c+d x)} \, dx","Int[Csc[c + d*x]*Sqrt[-a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)-a}}\right)}{d}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)-a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Cos[c + d*x])/Sqrt[-a + a*Sin[c + d*x]]])/d","A",2,2,23,0.08696,1,"{2773, 204}"
43,1,40,0,0.0525074,"\int \csc (c+d x) \sqrt{-a-a \sin (c+d x)} \, dx","Int[Csc[c + d*x]*Sqrt[-a - a*Sin[c + d*x]],x]","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a (-\sin (c+d x))-a}}\right)}{d}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a (-\sin (c+d x))-a}}\right)}{d}",1,"(2*Sqrt[a]*ArcTan[(Sqrt[a]*Cos[c + d*x])/Sqrt[-a - a*Sin[c + d*x]]])/d","A",2,2,24,0.08333,1,"{2773, 204}"
44,1,162,0,0.2413491,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{68 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{68 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{136 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{68 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{68 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{136 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}",1,"(-68*a^2*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]]) - (34*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (136*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (68*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d)","A",6,6,23,0.2609,1,"{2763, 21, 2770, 2759, 2751, 2646}"
45,1,116,0,0.1353778,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{152 a^2 \cos (c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{38 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 d}","-\frac{152 a^2 \cos (c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{38 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 d}",1,"(-152*a^2*Cos[c + d*x])/(105*d*Sqrt[a + a*Sin[c + d*x]]) - (38*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*d) + (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2647, 2646}"
46,1,86,0,0.062433,"\int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}","-\frac{8 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}",1,"(-8*a^2*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)","A",3,3,21,0.1429,1,"{2751, 2647, 2646}"
47,1,59,0,0.0281834,"\int (a+a \sin (c+d x))^{3/2} \, dx","Int[(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}","-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}",1,"(-8*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",2,2,14,0.1429,1,"{2647, 2646}"
48,1,66,0,0.0978839,"\int \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{2 a^2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (2*a^2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,21,0.1905,1,"{2763, 21, 2773, 206}"
49,1,66,0,0.1100399,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a^2 \cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{a^2 \cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (a^2*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,23,0.1739,1,"{2762, 21, 2773, 206}"
50,1,106,0,0.165678,"\int \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{7 a^2 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}","-\frac{7 a^2 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(-7*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (7*a^2*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])","A",5,5,23,0.2174,1,"{2762, 21, 2772, 2773, 206}"
51,1,144,0,0.2312383,"\int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{11 a^2 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","-\frac{11 a^2 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(-11*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (11*a^2*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (11*a^2*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,23,0.2174,1,"{2762, 21, 2772, 2773, 206}"
52,1,203,0,0.3519708,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}-\frac{46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}+\frac{568 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 d}-\frac{284 a^3 \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d}","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}-\frac{46 a^3 \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{710 a^3 \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}+\frac{568 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 d}-\frac{284 a^3 \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{284 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{231 d}",1,"(-284*a^3*Cos[c + d*x])/(99*d*Sqrt[a + a*Sin[c + d*x]]) - (710*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(693*d*Sqrt[a + a*Sin[c + d*x]]) - (46*a^3*Cos[c + d*x]*Sin[c + d*x]^4)/(99*d*Sqrt[a + a*Sin[c + d*x]]) + (568*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(693*d) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) - (284*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(231*d)","A",6,6,23,0.2609,1,"{2763, 2981, 2770, 2759, 2751, 2646}"
53,1,146,0,0.1546805,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{208 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}-\frac{832 a^3 \cos (c+d x)}{315 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{7/2}}{9 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{63 d}-\frac{26 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}","-\frac{208 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}-\frac{832 a^3 \cos (c+d x)}{315 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{7/2}}{9 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{63 d}-\frac{26 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}",1,"(-832*a^3*Cos[c + d*x])/(315*d*Sqrt[a + a*Sin[c + d*x]]) - (208*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (26*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d) + (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(63*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/2))/(9*a*d)","A",5,4,23,0.1739,1,"{2759, 2751, 2647, 2646}"
54,1,116,0,0.0829704,"\int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{64 a^3 \cos (c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}","-\frac{64 a^3 \cos (c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}",1,"(-64*a^3*Cos[c + d*x])/(21*d*Sqrt[a + a*Sin[c + d*x]]) - (16*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(7*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*d)","A",4,3,21,0.1429,1,"{2751, 2647, 2646}"
55,1,89,0,0.0461202,"\int (a+a \sin (c+d x))^{5/2} \, dx","Int[(a + a*Sin[c + d*x])^(5/2),x]","-\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}","-\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}",1,"(-64*a^3*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (16*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (2*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)","A",3,2,14,0.1429,1,"{2647, 2646}"
56,1,98,0,0.1923387,"\int \csc (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Csc[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{14 a^3 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-2*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (14*a^3*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",4,4,21,0.1905,1,"{2763, 2981, 2773, 206}"
57,1,94,0,0.1934348,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a^3 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{a^3 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-5*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (a^3*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d","A",4,4,23,0.1739,1,"{2762, 2981, 2773, 206}"
58,1,106,0,0.2184767,"\int \csc ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{19 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}","-\frac{9 a^3 \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{19 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{a^2 \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"(-19*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (9*a^3*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)","A",4,4,23,0.1739,1,"{2762, 2980, 2773, 206}"
59,1,144,0,0.2751324,"\int \csc ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{25 a^3 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{25 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","-\frac{25 a^3 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{25 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(-25*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (25*a^3*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (13*a^3*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",5,5,23,0.2174,1,"{2762, 2980, 2772, 2773, 206}"
60,1,182,0,0.3353209,"\int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{163 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}","-\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{163 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}",1,"(-163*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) - (163*a^3*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) - (163*a^3*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*d)","A",6,5,23,0.2174,1,"{2762, 2980, 2772, 2773, 206}"
61,1,139,0,0.2321213,"\int \frac{\sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}-\frac{28 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}-\frac{28 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (28*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d)","A",6,6,23,0.2609,1,"{2778, 2968, 3023, 2751, 2649, 206}"
62,1,105,0,0.1193864,"\int \frac{\sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d)) + (4*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2649, 206}"
63,1,72,0,0.0483382,"\int \frac{\sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sin[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])","A",3,3,21,0.1429,1,"{2751, 2649, 206}"
64,1,47,0,0.0200594,"\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[1/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d))","A",2,2,14,0.1429,1,"{2649, 206}"
65,1,84,0,0.1139394,"\int \frac{\csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Csc[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d)","A",5,4,21,0.1905,1,"{2780, 2649, 206, 2773}"
66,1,109,0,0.2039459,"\int \frac{\csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,23,0.2174,1,"{2779, 2985, 2649, 206, 2773}"
67,1,146,0,0.3434683,"\int \frac{\csc ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}","\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(-7*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*Sqrt[a]*d) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])","A",7,6,23,0.2609,1,"{2779, 2984, 2985, 2649, 206, 2773}"
68,1,183,0,0.3821267,"\int \frac{\sin ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","\frac{13 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{10 a^2 d}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sin ^2(c+d x) \cos (c+d x)}{10 a d \sqrt{a \sin (c+d x)+a}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}","\frac{13 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{10 a^2 d}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sin ^2(c+d x) \cos (c+d x)}{10 a d \sqrt{a \sin (c+d x)+a}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}",1,"(15*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (31*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (9*Cos[c + d*x]*Sin[c + d*x]^2)/(10*a*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(10*a^2*d)","A",7,7,23,0.3043,1,"{2765, 2983, 2968, 3023, 2751, 2649, 206}"
69,1,145,0,0.2527759,"\int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{7 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{6 a^2 d}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}","-\frac{7 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{6 a^2 d}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}",1,"(-11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(2*d*(a + a*Sin[c + d*x])^(3/2)) + (13*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) - (7*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(6*a^2*d)","A",6,6,23,0.2609,1,"{2765, 2968, 3023, 2751, 2649, 206}"
70,1,105,0,0.1280162,"\int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"(7*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x])/(a*d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,23,0.1739,1,"{2758, 2751, 2649, 206}"
71,1,77,0,0.0569845,"\int \frac{\sin (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sin[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}","\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))","A",3,3,21,0.1429,1,"{2750, 2649, 206}"
72,1,77,0,0.0386647,"\int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^(-3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"-ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) - Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))","A",3,3,14,0.2143,1,"{2650, 2649, 206}"
73,1,114,0,0.2134514,"\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Csc[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) + (5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))","A",6,5,21,0.2381,1,"{2766, 2985, 2649, 206, 2773}"
74,1,144,0,0.3576786,"\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \cot (c+d x)}{2 a d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \cot (c+d x)}{2 a d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"(3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - (9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Cot[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (3*Cot[c + d*x])/(2*a*d*Sqrt[a + a*Sin[c + d*x]])","A",7,6,23,0.2609,1,"{2766, 2984, 2985, 2649, 206, 2773}"
75,1,186,0,0.4864768,"\int \frac{\csc ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"(-19*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d) + (13*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*d*(a + a*Sin[c + d*x])^(3/2)) + (7*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(a*d*Sqrt[a + a*Sin[c + d*x]])","A",8,6,23,0.2609,1,"{2766, 2984, 2985, 2649, 206, 2773}"
76,1,221,0,0.5211027,"\int \frac{\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sin[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{157 \sin ^2(c+d x) \cos (c+d x)}{80 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{787 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{240 a^3 d}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{283 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{21 \sin ^3(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{157 \sin ^2(c+d x) \cos (c+d x)}{80 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{787 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{240 a^3 d}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{283 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{21 \sin ^3(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}",1,"(283*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (21*Cos[c + d*x]*Sin[c + d*x]^3)/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (1729*Cos[c + d*x])/(120*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (157*Cos[c + d*x]*Sin[c + d*x]^2)/(80*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (787*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(240*a^3*d)","A",8,8,23,0.3478,1,"{2765, 2977, 2983, 2968, 3023, 2751, 2649, 206}"
77,1,183,0,0.3854964,"\int \frac{\sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{95 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{48 a^3 d}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{163 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{17 \sin ^2(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{95 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{48 a^3 d}+\frac{197 \cos (c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{163 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{17 \sin ^2(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-163*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (17*Cos[c + d*x]*Sin[c + d*x]^2)/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) + (197*Cos[c + d*x])/(24*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (95*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(48*a^3*d)","A",7,7,23,0.3043,1,"{2765, 2977, 2968, 3023, 2751, 2649, 206}"
78,1,145,0,0.2678278,"\int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{9 \cos (c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{75 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac{13 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{9 \cos (c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{75 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}-\frac{13 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}",1,"(75*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (13*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (9*Cos[c + d*x])/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",6,6,23,0.2609,1,"{2765, 2968, 3019, 2751, 2649, 206}"
79,1,107,0,0.1284787,"\int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{13 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{13 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"(-19*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (13*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{2758, 2750, 2649, 206}"
80,1,107,0,0.0738224,"\int \frac{\sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sin[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"(-5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (5*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))","A",4,4,21,0.1905,1,"{2750, 2650, 2649, 206}"
81,1,107,0,0.0578148,"\int \frac{1}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^(-5/2),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{3 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{3 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (3*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))","A",4,3,14,0.2143,1,"{2650, 2649, 206}"
82,1,144,0,0.3253421,"\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Csc[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{43 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{11 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{43 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{11 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d) + (43*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (11*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))","A",7,6,21,0.2857,1,"{2766, 2978, 2985, 2649, 206, 2773}"
83,1,174,0,0.5068485,"\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{35 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{115 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{15 \cot (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}","-\frac{35 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{115 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{15 \cot (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"(5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d) - (115*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Cot[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (15*Cot[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (35*Cot[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",8,7,23,0.3043,1,"{2766, 2978, 2984, 2985, 2649, 206, 2773}"
84,1,224,0,0.6601263,"\int \frac{\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{39 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{5/2} d}+\frac{219 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}","\frac{63 \cot (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{39 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{5/2} d}+\frac{219 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac{\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"(-39*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(5/2)*d) + (219*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (19*Cot[c + d*x]*Csc[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) + (63*Cot[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (31*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",9,7,23,0.3043,1,"{2766, 2978, 2984, 2985, 2649, 206, 2773}"
85,1,37,0,0.0555735,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{\sin (e+f x)}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/Sqrt[Sin[e + f*x]],x]","-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}","-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right)}{f}",1,"(-2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f","A",2,2,25,0.08000,1,"{2774, 216}"
86,1,38,0,0.0701887,"\int \frac{\sqrt{a-a \sin (e+f x)}}{\sqrt{-\sin (e+f x)}} \, dx","Int[Sqrt[a - a*Sin[e + f*x]]/Sqrt[-Sin[e + f*x]],x]","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-a \sin (e+f x)}}\right)}{f}","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-a \sin (e+f x)}}\right)}{f}",1,"(2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a - a*Sin[e + f*x]]])/f","A",2,2,28,0.07143,1,"{2774, 216}"
87,1,17,0,0.0390379,"\int \frac{1}{\sqrt{\sin (x)} \sqrt{1+\sin (x)}} \, dx","Int[1/(Sqrt[Sin[x]]*Sqrt[1 + Sin[x]]),x]","-\sqrt{2} \sin ^{-1}\left(\frac{\cos (x)}{\sin (x)+1}\right)","-\sqrt{2} \sin ^{-1}\left(\frac{\cos (x)}{\sin (x)+1}\right)",1,"-(Sqrt[2]*ArcSin[Cos[x]/(1 + Sin[x])])","A",2,2,15,0.1333,1,"{2781, 216}"
88,1,42,0,0.0574363,"\int \frac{1}{\sqrt{\sin (x)} \sqrt{a+a \sin (x)}} \, dx","Int[1/(Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]]),x]","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a \sin (x)+a}}\right)}{\sqrt{a}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a \sin (x)+a}}\right)}{\sqrt{a}}",1,"-((Sqrt[2]*ArcTan[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]])])/Sqrt[a])","A",2,2,17,0.1176,1,"{2782, 205}"
89,1,31,0,0.0459595,"\int \frac{1}{\sqrt{1-\sin (x)} \sqrt{\sin (x)}} \, dx","Int[1/(Sqrt[1 - Sin[x]]*Sqrt[Sin[x]]),x]","\sqrt{2} \tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{2} \sqrt{1-\sin (x)} \sqrt{\sin (x)}}\right)","\sqrt{2} \tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{2} \sqrt{1-\sin (x)} \sqrt{\sin (x)}}\right)",1,"Sqrt[2]*ArcTanh[Cos[x]/(Sqrt[2]*Sqrt[1 - Sin[x]]*Sqrt[Sin[x]])]","A",2,2,17,0.1176,1,"{2782, 206}"
90,1,42,0,0.064463,"\int \frac{1}{\sqrt{\sin (x)} \sqrt{a-a \sin (x)}} \, dx","Int[1/(Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]]),x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a-a \sin (x)}}\right)}{\sqrt{a}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (x)}{\sqrt{2} \sqrt{\sin (x)} \sqrt{a-a \sin (x)}}\right)}{\sqrt{a}}",1,"(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]])])/Sqrt[a]","A",2,2,18,0.1111,1,"{2782, 208}"
91,1,184,0,0.213211,"\int \frac{\sqrt[3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx","Int[Sin[c + d*x]^(1/3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\sin ^{\frac{4}{3}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(c+d x)\right)}{36 a^2 d \sqrt{\cos ^2(c+d x)}}+\frac{4 \sqrt[3]{\sin (c+d x)} \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right)}{9 a^2 d \sqrt{\cos ^2(c+d x)}}-\frac{\sqrt[3]{\sin (c+d x)} \cos (c+d x)}{9 a^2 d (\sin (c+d x)+1)}-\frac{\sqrt[3]{\sin (c+d x)} \cos (c+d x)}{3 d (a \sin (c+d x)+a)^2}","-\frac{\sin ^{\frac{4}{3}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(c+d x)\right)}{36 a^2 d \sqrt{\cos ^2(c+d x)}}+\frac{4 \sqrt[3]{\sin (c+d x)} \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right)}{9 a^2 d \sqrt{\cos ^2(c+d x)}}-\frac{\sqrt[3]{\sin (c+d x)} \cos (c+d x)}{9 a^2 d (\sin (c+d x)+1)}-\frac{\sqrt[3]{\sin (c+d x)} \cos (c+d x)}{3 d (a \sin (c+d x)+a)^2}",1,"(4*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[c + d*x]^2]*Sin[c + d*x]^(1/3))/(9*a^2*d*Sqrt[Cos[c + d*x]^2]) - (Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[c + d*x]^2]*Sin[c + d*x]^(4/3))/(36*a^2*d*Sqrt[Cos[c + d*x]^2]) - (Cos[c + d*x]*Sin[c + d*x]^(1/3))/(9*a^2*d*(1 + Sin[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x]^(1/3))/(3*d*(a + a*Sin[c + d*x])^2)","A",5,4,23,0.1739,1,"{2764, 2978, 2748, 2643}"
92,1,161,0,0.2780301,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Int[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{67 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{55\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{11 d}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{44 a d}-\frac{63 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{220 d}","-\frac{67 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{55\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{11 d}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{44 a d}-\frac{63 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{220 d}",1,"(-63*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(220*d) - (3*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3))/(11*d) - (67*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(2/3))/(55*2^(5/6)*d*(1 + Sin[c + d*x])^(7/6)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/3))/(44*a*d)","A",6,6,23,0.2609,1,"{2783, 2968, 3023, 2751, 2652, 2651}"
93,1,126,0,0.1454961,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d}","-\frac{19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d}",1,"(9*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(40*d) - (19*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(2/3))/(10*2^(5/6)*d*(1 + Sin[c + d*x])^(7/6)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/3))/(8*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2652, 2651}"
94,1,96,0,0.0682446,"\int \sin (c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{4 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}","-\frac{4 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d (\sin (c+d x)+1)^{7/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 d}",1,"(-3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(5*d) - (4*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(2/3))/(5*d*(1 + Sin[c + d*x])^(7/6))","A",3,3,21,0.1429,1,"{2751, 2652, 2651}"
95,1,66,0,0.0312848,"\int (a+a \sin (c+d x))^{2/3} \, dx","Int[(a + a*Sin[c + d*x])^(2/3),x]","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"(-2*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6))","A",2,2,14,0.1429,1,"{2652, 2651}"
96,1,77,0,0.1108118,"\int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Int[Csc[c + d*x]*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};1,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};1,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"(-2*2^(1/6)*AppellF1[1/2, 1, -1/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6))","A",4,4,21,0.1905,1,"{2787, 2785, 130, 429}"
97,1,77,0,0.1314902,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx","Int[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3),x]","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};2,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};2,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"(-2*2^(1/6)*AppellF1[1/2, 2, -1/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6))","A",4,4,23,0.1739,1,"{2787, 2785, 130, 429}"
98,1,162,0,0.2787082,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Int[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(4/3),x]","-\frac{388\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{455 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{13 d}-\frac{6 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{65 a d}-\frac{72 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{455 d}","-\frac{388\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{455 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{13 d}-\frac{6 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{65 a d}-\frac{72 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{455 d}",1,"(-388*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-5/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1/3))/(455*d*(1 + Sin[c + d*x])^(5/6)) - (72*Cos[c + d*x]*(a + a*Sin[c + d*x])^(4/3))/(455*d) - (3*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3))/(13*d) - (6*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/3))/(65*a*d)","A",6,6,23,0.2609,1,"{2783, 2968, 3023, 2751, 2652, 2651}"
99,1,127,0,0.1425966,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3),x]","-\frac{37\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{35 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{10 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{70 d}","-\frac{37\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{35 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{10 a d}+\frac{9 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{70 d}",1,"(-37*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-5/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1/3))/(35*d*(1 + Sin[c + d*x])^(5/6)) + (9*Cos[c + d*x]*(a + a*Sin[c + d*x])^(4/3))/(70*d) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/3))/(10*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2652, 2651}"
100,1,97,0,0.0671871,"\int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^(4/3),x]","-\frac{8\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{7 d}","-\frac{8\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d (\sin (c+d x)+1)^{5/6}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{7 d}",1,"(-8*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-5/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1/3))/(7*d*(1 + Sin[c + d*x])^(5/6)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(4/3))/(7*d)","A",3,3,21,0.1429,1,"{2751, 2652, 2651}"
101,1,67,0,0.0300104,"\int (a+a \sin (c+d x))^{4/3} \, dx","Int[(a + a*Sin[c + d*x])^(4/3),x]","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"(-2*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-5/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6))","A",2,2,14,0.1429,1,"{2652, 2651}"
102,1,78,0,0.1136897,"\int \csc (c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Int[Csc[c + d*x]*(a + a*Sin[c + d*x])^(4/3),x]","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} F_1\left(\frac{1}{2};1,-\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} F_1\left(\frac{1}{2};1,-\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"(-2*2^(5/6)*a*AppellF1[1/2, 1, -5/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6))","A",4,4,21,0.1905,1,"{2787, 2785, 130, 429}"
103,1,78,0,0.1299666,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^{4/3} \, dx","Int[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3),x]","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} F_1\left(\frac{1}{2};2,-\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} F_1\left(\frac{1}{2};2,-\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"(-2*2^(5/6)*a*AppellF1[1/2, 2, -5/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6))","A",4,4,23,0.1739,1,"{2787, 2785, 130, 429}"
104,1,161,0,0.2528839,"\int \frac{\sin ^3(c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Int[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(1/3),x]","\frac{37 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{40\ 2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt[3]{a \sin (c+d x)+a}}+\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac{99 \cos (c+d x)}{80 d \sqrt[3]{a \sin (c+d x)+a}}","\frac{37 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{40\ 2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt[3]{a \sin (c+d x)+a}}+\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac{99 \cos (c+d x)}{80 d \sqrt[3]{a \sin (c+d x)+a}}",1,"(-99*Cos[c + d*x])/(80*d*(a + a*Sin[c + d*x])^(1/3)) - (3*Cos[c + d*x]*Sin[c + d*x]^2)/(8*d*(a + a*Sin[c + d*x])^(1/3)) + (37*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(40*2^(5/6)*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)) + (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(40*a*d)","A",6,6,23,0.2609,1,"{2783, 2968, 3023, 2751, 2652, 2651}"
105,1,126,0,0.1316738,"\int \frac{\sin ^2(c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Int[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3),x]","-\frac{7 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 a d}+\frac{9 \cos (c+d x)}{10 d \sqrt[3]{a \sin (c+d x)+a}}","-\frac{7 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{5 a d}+\frac{9 \cos (c+d x)}{10 d \sqrt[3]{a \sin (c+d x)+a}}",1,"(9*Cos[c + d*x])/(10*d*(a + a*Sin[c + d*x])^(1/3)) - (7*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(5*2^(5/6)*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(5*a*d)","A",4,4,23,0.1739,1,"{2759, 2751, 2652, 2651}"
106,1,93,0,0.0626866,"\int \frac{\sin (c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Int[Sin[c + d*x]/(a + a*Sin[c + d*x])^(1/3),x]","\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{2 d \sqrt[3]{a \sin (c+d x)+a}}","\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{2 d \sqrt[3]{a \sin (c+d x)+a}}",1,"(-3*Cos[c + d*x])/(2*d*(a + a*Sin[c + d*x])^(1/3)) + (Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(2^(5/6)*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))","A",3,3,21,0.1429,1,"{2751, 2652, 2651}"
107,1,66,0,0.0309078,"\int \frac{1}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^(-1/3),x]","-\frac{\sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","-\frac{\sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"-((2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))","A",2,2,14,0.1429,1,"{2652, 2651}"
108,1,77,0,0.106019,"\int \frac{\csc (c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Int[Csc[c + d*x]/(a + a*Sin[c + d*x])^(1/3),x]","-\frac{\sqrt[6]{2} \cos (c+d x) F_1\left(\frac{1}{2};1,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","-\frac{\sqrt[6]{2} \cos (c+d x) F_1\left(\frac{1}{2};1,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"-((2^(1/6)*AppellF1[1/2, 1, 5/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))","A",4,4,21,0.1905,1,"{2787, 2785, 130, 429}"
109,1,77,0,0.1193382,"\int \frac{\csc ^2(c+d x)}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Int[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3),x]","-\frac{\sqrt[6]{2} \cos (c+d x) F_1\left(\frac{1}{2};2,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","-\frac{\sqrt[6]{2} \cos (c+d x) F_1\left(\frac{1}{2};2,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"-((2^(1/6)*AppellF1[1/2, 2, 5/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))","A",4,4,23,0.1739,1,"{2787, 2785, 130, 429}"
110,1,162,0,0.2696804,"\int \frac{\sin ^3(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Int[Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(4/3),x]","-\frac{2 \sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt[3]{a \sin (c+d x)+a}}+\frac{6 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}","-\frac{2 \sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}+\frac{6 \cos (c+d x)}{5 a d \sqrt[3]{a \sin (c+d x)+a}}+\frac{6 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}",1,"(6*Cos[c + d*x])/(5*d*(a + a*Sin[c + d*x])^(4/3)) - (3*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*(a + a*Sin[c + d*x])^(4/3)) + (6*Cos[c + d*x])/(5*a*d*(a + a*Sin[c + d*x])^(1/3)) - (2*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))","A",6,6,23,0.2609,1,"{2783, 2968, 3019, 2751, 2652, 2651}"
111,1,129,0,0.136466,"\int \frac{\sin ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Int[Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3),x]","\frac{13 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{2 a d \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}","\frac{13 \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{2 a d \sqrt[3]{a \sin (c+d x)+a}}-\frac{3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}",1,"(-3*Cos[c + d*x])/(5*d*(a + a*Sin[c + d*x])^(4/3)) - (3*Cos[c + d*x])/(2*a*d*(a + a*Sin[c + d*x])^(1/3)) + (13*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(5*2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))","A",4,4,23,0.1739,1,"{2758, 2751, 2652, 2651}"
112,1,99,0,0.0789303,"\int \frac{\sin (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Int[Sin[c + d*x]/(a + a*Sin[c + d*x])^(4/3),x]","\frac{3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}-\frac{4 \sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","\frac{3 \cos (c+d x)}{5 d (a \sin (c+d x)+a)^{4/3}}-\frac{4 \sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(3*Cos[c + d*x])/(5*d*(a + a*Sin[c + d*x])^(4/3)) - (4*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sin[c + d*x])/2])/(5*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))","A",3,3,21,0.1429,1,"{2750, 2652, 2651}"
113,1,69,0,0.0307922,"\int \frac{1}{(a+a \sin (c+d x))^{4/3}} \, dx","Int[(a + a*Sin[c + d*x])^(-4/3),x]","-\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","-\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"-((Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 3/2, (1 - Sin[c + d*x])/2])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))","A",2,2,14,0.1429,1,"{2652, 2651}"
114,1,80,0,0.1171593,"\int \frac{\csc (c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Int[Csc[c + d*x]/(a + a*Sin[c + d*x])^(4/3),x]","-\frac{\cos (c+d x) F_1\left(\frac{1}{2};1,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","-\frac{\cos (c+d x) F_1\left(\frac{1}{2};1,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"-((AppellF1[1/2, 1, 11/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))","A",4,4,21,0.1905,1,"{2787, 2785, 130, 429}"
115,1,80,0,0.1370829,"\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx","Int[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3),x]","-\frac{\cos (c+d x) F_1\left(\frac{1}{2};2,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}","-\frac{\cos (c+d x) F_1\left(\frac{1}{2};2,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"-((AppellF1[1/2, 2, 11/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))","A",4,4,23,0.1739,1,"{2787, 2785, 130, 429}"
116,1,96,0,0.1105499,"\int \sin ^n(e+f x) (1+\sin (e+f x))^{3/2} \, dx","Int[Sin[e + f*x]^n*(1 + Sin[e + f*x])^(3/2),x]","-\frac{2 (4 n+5) \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{\sin (e+f x)+1}}","-\frac{2 (4 n+5) \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{\sin (e+f x)+1}}",1,"(-2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*(3 + 2*n)*Sqrt[1 + Sin[e + f*x]]) - (2*Cos[e + f*x]*Sin[e + f*x]^(1 + n))/(f*(3 + 2*n)*Sqrt[1 + Sin[e + f*x]])","A",4,4,21,0.1905,1,"{2763, 21, 2776, 65}"
117,1,43,0,0.047783,"\int \sin ^n(e+f x) \sqrt{1+\sin (e+f x)} \, dx","Int[Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]],x]","-\frac{2 \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{2 \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(-2*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[1 + Sin[e + f*x]])","A",2,2,21,0.09524,1,"{2776, 65}"
118,1,58,0,0.0676877,"\int \frac{\sin ^n(e+f x)}{\sqrt{1+\sin (e+f x)}} \, dx","Int[Sin[e + f*x]^n/Sqrt[1 + Sin[e + f*x]],x]","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}",1,"-((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]))","A",3,3,21,0.1429,1,"{2785, 130, 429}"
119,1,60,0,0.0722906,"\int \frac{\sin ^n(e+f x)}{(1+\sin (e+f x))^{3/2}} \, dx","Int[Sin[e + f*x]^n/(1 + Sin[e + f*x])^(3/2),x]","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{2 f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{2 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x])/(2*f*Sqrt[1 + Sin[e + f*x]])","A",3,3,21,0.1429,1,"{2785, 130, 429}"
120,1,106,0,0.1402694,"\int \sin ^n(e+f x) (a+a \sin (e+f x))^{3/2} \, dx","Int[Sin[e + f*x]^n*(a + a*Sin[e + f*x])^(3/2),x]","-\frac{2 a^2 (4 n+5) \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}","-\frac{2 a^2 (4 n+5) \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"(-2*a^2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sin[e + f*x]^(1 + n))/(f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])","A",4,4,23,0.1739,1,"{2763, 21, 2776, 65}"
121,1,46,0,0.0615039,"\int \sin ^n(e+f x) \sqrt{a+a \sin (e+f x)} \, dx","Int[Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]],x]","-\frac{2 a \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{a \sin (e+f x)+a}}","-\frac{2 a \cos (e+f x) \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"(-2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])","A",2,2,23,0.08696,1,"{2776, 65}"
122,1,60,0,0.1242682,"\int \frac{\sin ^n(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Sin[e + f*x]^n/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"-((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))","A",4,4,23,0.1739,1,"{2787, 2785, 130, 429}"
123,1,65,0,0.1360248,"\int \frac{\sin ^n(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Sin[e + f*x]^n/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{2 a f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"-(AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,23,0.1739,1,"{2787, 2785, 130, 429}"
124,1,130,0,0.1408993,"\int (d \sin (e+f x))^n (1+\sin (e+f x))^{3/2} \, dx","Int[(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(3/2),x]","\frac{(4 n+5) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},n+1;n+2;\sin (e+f x)\right)}{d f (n+1) (2 n+3) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{\sin (e+f x)+1}}","\frac{(4 n+5) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},n+1;n+2;\sin (e+f x)\right)}{d f (n+1) (2 n+3) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{\sin (e+f x)+1}}",1,"(-2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[1 + Sin[e + f*x]]) + ((5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + n, 2 + n, Sin[e + f*x]]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(3 + 2*n)*Sqrt[1 - Sin[e + f*x]]*Sqrt[1 + Sin[e + f*x]])","A",4,4,23,0.1739,1,"{2763, 21, 2776, 64}"
125,1,72,0,0.055852,"\int (d \sin (e+f x))^n \sqrt{1+\sin (e+f x)} \, dx","Int[(d*Sin[e + f*x])^n*Sqrt[1 + Sin[e + f*x]],x]","\frac{\cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},n+1;n+2;\sin (e+f x)\right)}{d f (n+1) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}","\frac{\cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},n+1;n+2;\sin (e+f x)\right)}{d f (n+1) \sqrt{1-\sin (e+f x)} \sqrt{\sin (e+f x)+1}}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + n, 2 + n, Sin[e + f*x]]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[1 - Sin[e + f*x]]*Sqrt[1 + Sin[e + f*x]])","A",2,2,23,0.08696,1,"{2776, 64}"
126,1,78,0,0.1183931,"\int \frac{(d \sin (e+f x))^n}{\sqrt{1+\sin (e+f x)}} \, dx","Int[(d*Sin[e + f*x])^n/Sqrt[1 + Sin[e + f*x]],x]","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{f \sqrt{\sin (e+f x)+1}}",1,"-((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]]))","A",4,4,23,0.1739,1,"{2786, 2785, 130, 429}"
127,1,80,0,0.1352208,"\int \frac{(d \sin (e+f x))^n}{(1+\sin (e+f x))^{3/2}} \, dx","Int[(d*Sin[e + f*x])^n/(1 + Sin[e + f*x])^(3/2),x]","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{2 f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{2 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(2*f*Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]])","A",4,4,23,0.1739,1,"{2786, 2785, 130, 429}"
128,1,131,0,0.1700915,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^{3/2} \, dx","Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(3/2),x]","-\frac{2 a^2 (4 n+5) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}","-\frac{2 a^2 (4 n+5) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"(-2*a^2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(f*(3 + 2*n)*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])","A",5,5,25,0.2000,1,"{2763, 21, 2776, 67, 65}"
129,1,66,0,0.0745947,"\int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \, dx","Int[(d*Sin[e + f*x])^n*Sqrt[a + a*Sin[e + f*x]],x]","-\frac{2 a \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{a \sin (e+f x)+a}}","-\frac{2 a \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"(-2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]])","A",3,3,25,0.1200,1,"{2776, 67, 65}"
130,1,80,0,0.1804798,"\int \frac{(d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(d*Sin[e + f*x])^n/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,1;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{f \sqrt{a \sin (e+f x)+a}}",1,"-((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]]))","A",5,5,25,0.2000,1,"{2787, 2786, 2785, 130, 429}"
131,1,85,0,0.2022228,"\int \frac{(d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{2 a f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sin ^{-n}(e+f x) F_1\left(\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) (d \sin (e+f x))^n}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"-(AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(2*a*f*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]])","A",5,5,25,0.2000,1,"{2787, 2786, 2785, 130, 429}"
132,1,71,0,0.0599366,"\int \sin ^n(e+f x) (1+\sin (e+f x))^m \, dx","Int[Sin[e + f*x]^n*(1 + Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}",1,"-((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]))","A",2,2,19,0.1053,1,"{2785, 133}"
133,1,68,0,0.0607976,"\int (1-\sin (e+f x))^m (-\sin (e+f x))^n \, dx","Int[(1 - Sin[e + f*x])^m*(-Sin[e + f*x])^n,x]","\frac{2^{m+\frac{1}{2}} \cos (e+f x) F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f \sqrt{1-\sin (e+f x)}}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f \sqrt{1-\sin (e+f x)}}",1,"(2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1 + Sin[e + f*x])/2]*Cos[e + f*x])/(f*Sqrt[1 - Sin[e + f*x]])","A",2,2,23,0.08696,1,"{2785, 133}"
134,1,91,0,0.0997954,"\int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx","Int[(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}",1,"-((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqrt[1 + Sin[e + f*x]]))","A",3,3,21,0.1429,1,"{2786, 2785, 133}"
135,1,90,0,0.1095567,"\int (1-\sin (e+f x))^m (d \sin (e+f x))^n \, dx","Int[(1 - Sin[e + f*x])^m*(d*Sin[e + f*x])^n,x]","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f \sqrt{1-\sin (e+f x)}}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (-\sin (e+f x))^{-n} (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f \sqrt{1-\sin (e+f x)}}",1,"(2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1 + Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(f*Sqrt[1 - Sin[e + f*x]]*(-Sin[e + f*x])^n)","A",3,3,23,0.1304,1,"{2786, 2785, 133}"
136,1,87,0,0.0981528,"\int \sin ^n(e+f x) (a+a \sin (e+f x))^m \, dx","Int[Sin[e + f*x]^n*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"-((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)","A",3,3,21,0.1429,1,"{2787, 2785, 133}"
137,1,85,0,0.109815,"\int (-\sin (e+f x))^n (a-a \sin (e+f x))^m \, dx","Int[(-Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{-m-\frac{1}{2}} (a-a \sin (e+f x))^m F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{-m-\frac{1}{2}} (a-a \sin (e+f x))^m F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f}",1,"(2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1 + Sin[e + f*x])/2]*Cos[e + f*x]*(1 - Sin[e + f*x])^(-1/2 - m)*(a - a*Sin[e + f*x])^m)/f","A",3,3,24,0.1250,1,"{2787, 2785, 133}"
138,1,107,0,0.1499008,"\int (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx","Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} \sin ^{-n}(e+f x) (a \sin (e+f x)+a)^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"-((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^n))","A",4,4,23,0.1739,1,"{2787, 2786, 2785, 133}"
139,1,107,0,0.1597564,"\int (d \sin (e+f x))^n (a-a \sin (e+f x))^m \, dx","Int[(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{-m-\frac{1}{2}} (-\sin (e+f x))^{-n} (a-a \sin (e+f x))^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f}","\frac{2^{m+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{-m-\frac{1}{2}} (-\sin (e+f x))^{-n} (a-a \sin (e+f x))^m (d \sin (e+f x))^n F_1\left(\frac{1}{2};-n,\frac{1}{2}-m;\frac{3}{2};\sin (e+f x)+1,\frac{1}{2} (\sin (e+f x)+1)\right)}{f}",1,"(2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1 + Sin[e + f*x])/2]*Cos[e + f*x]*(1 - Sin[e + f*x])^(-1/2 - m)*(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m)/(f*(-Sin[e + f*x])^n)","A",4,4,24,0.1667,1,"{2787, 2786, 2785, 133}"
140,1,294,0,0.5097314,"\int \sin ^4(c+d x) (a+a \sin (c+d x))^n \, dx","Int[Sin[c + d*x]^4*(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} \left(n^4+6 n^3+17 n^2+12 n+9\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2) (n+3) (n+4)}+\frac{\left(-n^2-n+9\right) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3) (n+4)}-\frac{\left(n^2+3 n+9\right) \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2) (n+3) (n+4)}-\frac{\sin ^3(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+4)}-\frac{n \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3) (n+4)}","-\frac{2^{n+\frac{1}{2}} \left(n^4+6 n^3+17 n^2+12 n+9\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2) (n+3) (n+4)}+\frac{\left(-n^2-n+9\right) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3) (n+4)}-\frac{\left(n^2+3 n+9\right) \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2) (n+3) (n+4)}-\frac{\sin ^3(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+4)}-\frac{n \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3) (n+4)}",1,"((9 - n - n^2)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n)*(4 + n)) - (n*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n)/(d*(3 + n)*(4 + n)) - (Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n)/(d*(4 + n)) - (2^(1/2 + n)*(9 + 12*n + 17*n^2 + 6*n^3 + n^4)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n)*(4 + n)) - ((9 + 3*n + n^2)*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(2 + n)*(3 + n)*(4 + n))","A",7,7,21,0.3333,1,"{2783, 2983, 2968, 3023, 2751, 2652, 2651}"
141,1,215,0,0.2940569,"\int \sin ^3(c+d x) (a+a \sin (c+d x))^n \, dx","Int[Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} n \left(n^2+3 n+5\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2) (n+3)}-\frac{n \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d \left(n^2+5 n+6\right)}-\frac{\sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3)}-\frac{(n+4) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3)}","-\frac{2^{n+\frac{1}{2}} n \left(n^2+3 n+5\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2) (n+3)}-\frac{n \cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d \left(n^2+5 n+6\right)}-\frac{\sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+3)}-\frac{(n+4) \cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1) (n+2) (n+3)}",1,"-(((4 + n)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n))) - (Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n)/(d*(3 + n)) - (2^(1/2 + n)*n*(5 + 3*n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n)) - (n*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(6 + 5*n + n^2))","A",6,6,21,0.2857,1,"{2783, 2968, 3023, 2751, 2652, 2651}"
142,1,156,0,0.1429276,"\int \sin ^2(c+d x) (a+a \sin (c+d x))^n \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} \left(n^2+n+1\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2)}+\frac{\cos (c+d x) (a \sin (c+d x)+a)^n}{d \left(n^2+3 n+2\right)}-\frac{\cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2)}","-\frac{2^{n+\frac{1}{2}} \left(n^2+n+1\right) \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1) (n+2)}+\frac{\cos (c+d x) (a \sin (c+d x)+a)^n}{d \left(n^2+3 n+2\right)}-\frac{\cos (c+d x) (a \sin (c+d x)+a)^{n+1}}{a d (n+2)}",1,"(Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(2 + 3*n + n^2)) - (2^(1/2 + n)*(1 + n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)) - (Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(2 + n))","A",4,4,21,0.1905,1,"{2759, 2751, 2652, 2651}"
143,1,109,0,0.0638162,"\int \sin (c+d x) (a+a \sin (c+d x))^n \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} n \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1)}-\frac{\cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1)}","-\frac{2^{n+\frac{1}{2}} n \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (n+1)}-\frac{\cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1)}",1,"-((Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n))) - (2^(1/2 + n)*n*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n))","A",3,3,19,0.1579,1,"{2751, 2652, 2651}"
144,1,74,0,0.0303929,"\int (a+a \sin (c+d x))^n \, dx","Int[(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"-((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/d)","A",2,2,12,0.1667,1,"{2652, 2651}"
145,1,85,0,0.1108086,"\int \csc (c+d x) (a+a \sin (c+d x))^n \, dx","Int[Csc[c + d*x]*(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n F_1\left(\frac{1}{2};1,\frac{1}{2}-n;\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n F_1\left(\frac{1}{2};1,\frac{1}{2}-n;\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"-((2^(1/2 + n)*AppellF1[1/2, 1, 1/2 - n, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/d)","A",4,4,19,0.2105,1,"{2787, 2785, 130, 429}"
146,1,85,0,0.1152228,"\int \csc ^2(c+d x) (a+a \sin (c+d x))^n \, dx","Int[Csc[c + d*x]^2*(a + a*Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n F_1\left(\frac{1}{2};2,\frac{1}{2}-n;\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n F_1\left(\frac{1}{2};2,\frac{1}{2}-n;\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"-((2^(1/2 + n)*AppellF1[1/2, 2, 1/2 - n, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(1 + Sin[c + d*x])^(-1/2 - n)*(a + a*Sin[c + d*x])^n)/d)","A",4,4,21,0.1905,1,"{2787, 2785, 130, 429}"
147,1,58,0,0.0139806,"\int (1+\sin (c+d x))^n \, dx","Int[(1 + Sin[c + d*x])^n,x]","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt{\sin (c+d x)+1}}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt{\sin (c+d x)+1}}",1,"-((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 - Sin[c + d*x])/2])/(d*Sqrt[1 + Sin[c + d*x]]))","A",1,1,10,0.1000,1,"{2651}"
148,1,57,0,0.0182336,"\int (1-\sin (c+d x))^n \, dx","Int[(1 - Sin[c + d*x])^n,x]","\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}","\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"(2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1 + Sin[c + d*x])/2])/(d*Sqrt[1 - Sin[c + d*x]])","A",1,1,12,0.08333,1,"{2651}"
149,1,77,0,0.0583029,"\int \sin ^3(e+f x) (a+b \sin (e+f x)) \, dx","Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x]),x]","\frac{a \cos ^3(e+f x)}{3 f}-\frac{a \cos (e+f x)}{f}-\frac{b \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{3 b \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3 b x}{8}","\frac{a \cos ^3(e+f x)}{3 f}-\frac{a \cos (e+f x)}{f}-\frac{b \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{3 b \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3 b x}{8}",1,"(3*b*x)/8 - (a*Cos[e + f*x])/f + (a*Cos[e + f*x]^3)/(3*f) - (3*b*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)","A",6,4,19,0.2105,1,"{2748, 2633, 2635, 8}"
150,1,55,0,0.044637,"\int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx","Int[Sin[e + f*x]^2*(a + b*Sin[e + f*x]),x]","-\frac{a \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a x}{2}+\frac{b \cos ^3(e+f x)}{3 f}-\frac{b \cos (e+f x)}{f}","-\frac{a \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a x}{2}+\frac{b \cos ^3(e+f x)}{3 f}-\frac{b \cos (e+f x)}{f}",1,"(a*x)/2 - (b*Cos[e + f*x])/f + (b*Cos[e + f*x]^3)/(3*f) - (a*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",5,4,19,0.2105,1,"{2748, 2635, 8, 2633}"
151,1,39,0,0.0140097,"\int \sin (e+f x) (a+b \sin (e+f x)) \, dx","Int[Sin[e + f*x]*(a + b*Sin[e + f*x]),x]","-\frac{a \cos (e+f x)}{f}-\frac{b \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b x}{2}","-\frac{a \cos (e+f x)}{f}-\frac{b \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b x}{2}",1,"(b*x)/2 - (a*Cos[e + f*x])/f - (b*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,17,0.05882,1,"{2734}"
152,1,16,0,0.0081813,"\int (a+b \sin (e+f x)) \, dx","Int[a + b*Sin[e + f*x],x]","a x-\frac{b \cos (e+f x)}{f}","a x-\frac{b \cos (e+f x)}{f}",1,"a*x - (b*Cos[e + f*x])/f","A",2,1,10,0.1000,1,"{2638}"
153,1,17,0,0.0215194,"\int \csc (e+f x) (a+b \sin (e+f x)) \, dx","Int[Csc[e + f*x]*(a + b*Sin[e + f*x]),x]","b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}","b x-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}",1,"b*x - (a*ArcTanh[Cos[e + f*x]])/f","A",2,2,17,0.1176,1,"{2735, 3770}"
154,1,26,0,0.0370161,"\int \csc ^2(e+f x) (a+b \sin (e+f x)) \, dx","Int[Csc[e + f*x]^2*(a + b*Sin[e + f*x]),x]","-\frac{a \cot (e+f x)}{f}-\frac{b \tanh ^{-1}(\cos (e+f x))}{f}","-\frac{a \cot (e+f x)}{f}-\frac{b \tanh ^{-1}(\cos (e+f x))}{f}",1,"-((b*ArcTanh[Cos[e + f*x]])/f) - (a*Cot[e + f*x])/f","A",4,4,19,0.2105,1,"{2748, 3767, 8, 3770}"
155,1,48,0,0.0470495,"\int \csc ^3(e+f x) (a+b \sin (e+f x)) \, dx","Int[Csc[e + f*x]^3*(a + b*Sin[e + f*x]),x]","-\frac{a \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}-\frac{b \cot (e+f x)}{f}","-\frac{a \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}-\frac{b \cot (e+f x)}{f}",1,"-(a*ArcTanh[Cos[e + f*x]])/(2*f) - (b*Cot[e + f*x])/f - (a*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",5,5,19,0.2632,1,"{2748, 3768, 3770, 3767, 8}"
156,1,64,0,0.050834,"\int \csc ^4(e+f x) (a+b \sin (e+f x)) \, dx","Int[Csc[e + f*x]^4*(a + b*Sin[e + f*x]),x]","-\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-\frac{b \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{b \cot (e+f x) \csc (e+f x)}{2 f}","-\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-\frac{b \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{b \cot (e+f x) \csc (e+f x)}{2 f}",1,"-(b*ArcTanh[Cos[e + f*x]])/(2*f) - (a*Cot[e + f*x])/f - (a*Cot[e + f*x]^3)/(3*f) - (b*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",5,4,19,0.2105,1,"{2748, 3767, 3768, 3770}"
157,1,112,0,0.1044279,"\int \sin ^3(e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x])^2,x]","\frac{\left(a^2+2 b^2\right) \cos ^3(e+f x)}{3 f}-\frac{\left(a^2+b^2\right) \cos (e+f x)}{f}-\frac{a b \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a b \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a b x}{4}-\frac{b^2 \cos ^5(e+f x)}{5 f}","\frac{\left(a^2+2 b^2\right) \cos ^3(e+f x)}{3 f}-\frac{\left(a^2+b^2\right) \cos (e+f x)}{f}-\frac{a b \sin ^3(e+f x) \cos (e+f x)}{2 f}-\frac{3 a b \sin (e+f x) \cos (e+f x)}{4 f}+\frac{3 a b x}{4}-\frac{b^2 \cos ^5(e+f x)}{5 f}",1,"(3*a*b*x)/4 - ((a^2 + b^2)*Cos[e + f*x])/f + ((a^2 + 2*b^2)*Cos[e + f*x]^3)/(3*f) - (b^2*Cos[e + f*x]^5)/(5*f) - (3*a*b*Cos[e + f*x]*Sin[e + f*x])/(4*f) - (a*b*Cos[e + f*x]*Sin[e + f*x]^3)/(2*f)","A",7,5,21,0.2381,1,"{2789, 2635, 8, 3013, 373}"
158,1,101,0,0.0890224,"\int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Sin[e + f*x]^2*(a + b*Sin[e + f*x])^2,x]","-\frac{\left(4 a^2+3 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x \left(4 a^2+3 b^2\right)+\frac{2 a b \cos ^3(e+f x)}{3 f}-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f}","-\frac{\left(4 a^2+3 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x \left(4 a^2+3 b^2\right)+\frac{2 a b \cos ^3(e+f x)}{3 f}-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f}",1,"((4*a^2 + 3*b^2)*x)/8 - (2*a*b*Cos[e + f*x])/f + (2*a*b*Cos[e + f*x]^3)/(3*f) - ((4*a^2 + 3*b^2)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)","A",6,5,21,0.2381,1,"{2789, 2633, 3014, 2635, 8}"
159,1,71,0,0.048996,"\int \sin (e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Sin[e + f*x]*(a + b*Sin[e + f*x])^2,x]","-\frac{2 \left(a^2+b^2\right) \cos (e+f x)}{3 f}-\frac{\cos (e+f x) (a+b \sin (e+f x))^2}{3 f}-\frac{a b \sin (e+f x) \cos (e+f x)}{3 f}+a b x","-\frac{2 \left(a^2+b^2\right) \cos (e+f x)}{3 f}-\frac{\cos (e+f x) (a+b \sin (e+f x))^2}{3 f}-\frac{a b \sin (e+f x) \cos (e+f x)}{3 f}+a b x",1,"a*b*x - (2*(a^2 + b^2)*Cos[e + f*x])/(3*f) - (a*b*Cos[e + f*x]*Sin[e + f*x])/(3*f) - (Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)","A",2,2,19,0.1053,1,"{2753, 2734}"
160,1,50,0,0.0149389,"\int (a+b \sin (e+f x))^2 \, dx","Int[(a + b*Sin[e + f*x])^2,x]","\frac{1}{2} x \left(2 a^2+b^2\right)-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}","\frac{1}{2} x \left(2 a^2+b^2\right)-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}",1,"((2*a^2 + b^2)*x)/2 - (2*a*b*Cos[e + f*x])/f - (b^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,12,0.08333,1,"{2644}"
161,1,35,0,0.0589407,"\int \csc (e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Csc[e + f*x]*(a + b*Sin[e + f*x])^2,x]","-\frac{a^2 \tanh ^{-1}(\cos (e+f x))}{f}+2 a b x-\frac{b^2 \cos (e+f x)}{f}","-\frac{a^2 \tanh ^{-1}(\cos (e+f x))}{f}+2 a b x-\frac{b^2 \cos (e+f x)}{f}",1,"2*a*b*x - (a^2*ArcTanh[Cos[e + f*x]])/f - (b^2*Cos[e + f*x])/f","A",3,3,19,0.1579,1,"{2746, 2735, 3770}"
162,1,34,0,0.0655697,"\int \csc ^2(e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Csc[e + f*x]^2*(a + b*Sin[e + f*x])^2,x]","-\frac{a^2 \cot (e+f x)}{f}-\frac{2 a b \tanh ^{-1}(\cos (e+f x))}{f}+b^2 x","-\frac{a^2 \cot (e+f x)}{f}-\frac{2 a b \tanh ^{-1}(\cos (e+f x))}{f}+b^2 x",1,"b^2*x - (2*a*b*ArcTanh[Cos[e + f*x]])/f - (a^2*Cot[e + f*x])/f","A",4,4,21,0.1905,1,"{2789, 3770, 3012, 8}"
163,1,59,0,0.0756538,"\int \csc ^3(e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Csc[e + f*x]^3*(a + b*Sin[e + f*x])^2,x]","-\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 \cot (e+f x) \csc (e+f x)}{2 f}-\frac{2 a b \cot (e+f x)}{f}","-\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a^2 \cot (e+f x) \csc (e+f x)}{2 f}-\frac{2 a b \cot (e+f x)}{f}",1,"-((a^2 + 2*b^2)*ArcTanh[Cos[e + f*x]])/(2*f) - (2*a*b*Cot[e + f*x])/f - (a^2*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",5,5,21,0.2381,1,"{2789, 3767, 8, 3012, 3770}"
164,1,82,0,0.0872102,"\int \csc ^4(e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Csc[e + f*x]^4*(a + b*Sin[e + f*x])^2,x]","-\frac{\left(2 a^2+3 b^2\right) \cot (e+f x)}{3 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac{a b \tanh ^{-1}(\cos (e+f x))}{f}-\frac{a b \cot (e+f x) \csc (e+f x)}{f}","-\frac{\left(2 a^2+3 b^2\right) \cot (e+f x)}{3 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac{a b \tanh ^{-1}(\cos (e+f x))}{f}-\frac{a b \cot (e+f x) \csc (e+f x)}{f}",1,"-((a*b*ArcTanh[Cos[e + f*x]])/f) - ((2*a^2 + 3*b^2)*Cot[e + f*x])/(3*f) - (a*b*Cot[e + f*x]*Csc[e + f*x])/f - (a^2*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f)","A",6,6,21,0.2857,1,"{2789, 3768, 3770, 3012, 3767, 8}"
165,1,110,0,0.0939383,"\int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx","Int[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^2,x]","-\frac{\left(3 a^2+4 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{\left(3 a^2+4 b^2\right) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{2 a b \cot ^3(e+f x)}{3 f}-\frac{2 a b \cot (e+f x)}{f}","-\frac{\left(3 a^2+4 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{\left(3 a^2+4 b^2\right) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{a^2 \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{2 a b \cot ^3(e+f x)}{3 f}-\frac{2 a b \cot (e+f x)}{f}",1,"-((3*a^2 + 4*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - (2*a*b*Cot[e + f*x])/f - (2*a*b*Cot[e + f*x]^3)/(3*f) - ((3*a^2 + 4*b^2)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)","A",6,5,21,0.2381,1,"{2789, 3767, 3012, 3768, 3770}"
166,1,193,0,0.2094665,"\int \sin ^3(e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x])^3,x]","\frac{a \left(5 a^2+12 b^2\right) \cos ^3(e+f x)}{15 f}-\frac{a \left(5 a^2+12 b^2\right) \cos (e+f x)}{5 f}-\frac{b \left(18 a^2+5 b^2\right) \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{b \left(18 a^2+5 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} b x \left(18 a^2+5 b^2\right)-\frac{13 a b^2 \sin ^4(e+f x) \cos (e+f x)}{30 f}-\frac{b^2 \sin ^4(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{6 f}","\frac{a \left(a^2+6 b^2\right) \cos ^3(e+f x)}{3 f}-\frac{a \left(a^2+3 b^2\right) \cos (e+f x)}{f}-\frac{b \left(18 a^2+5 b^2\right) \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{b \left(18 a^2+5 b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} b x \left(18 a^2+5 b^2\right)-\frac{3 a b^2 \cos ^5(e+f x)}{5 f}-\frac{b^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}",1,"(b*(18*a^2 + 5*b^2)*x)/16 - (a*(5*a^2 + 12*b^2)*Cos[e + f*x])/(5*f) + (a*(5*a^2 + 12*b^2)*Cos[e + f*x]^3)/(15*f) - (b*(18*a^2 + 5*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (b*(18*a^2 + 5*b^2)*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (13*a*b^2*Cos[e + f*x]*Sin[e + f*x]^4)/(30*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^4*(a + b*Sin[e + f*x]))/(6*f)","A",8,6,21,0.2857,1,"{2793, 3023, 2748, 2633, 2635, 8}"
167,1,180,0,0.2157378,"\int \sin ^2(e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Sin[e + f*x]^2*(a + b*Sin[e + f*x])^3,x]","\frac{\left(-52 a^2 b^2+3 a^4-16 b^4\right) \cos (e+f x)}{30 b f}+\frac{\left(3 a^2-16 b^2\right) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}+\frac{a \left(6 a^2-71 b^2\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} a x \left(4 a^2+9 b^2\right)-\frac{\cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}+\frac{a \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}","\frac{b \left(15 a^2+4 b^2\right) \cos ^3(e+f x)}{15 f}-\frac{b \left(15 a^2+4 b^2\right) \cos (e+f x)}{5 f}-\frac{a \left(4 a^2+9 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a x \left(4 a^2+9 b^2\right)-\frac{11 a b^2 \sin ^3(e+f x) \cos (e+f x)}{20 f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x) (a+b \sin (e+f x))}{5 f}",1,"(a*(4*a^2 + 9*b^2)*x)/8 + ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Cos[e + f*x])/(30*b*f) + (a*(6*a^2 - 71*b^2)*Cos[e + f*x]*Sin[e + f*x])/(120*f) + ((3*a^2 - 16*b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(60*b*f) + (a*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(20*b*f) - (Cos[e + f*x]*(a + b*Sin[e + f*x])^4)/(5*b*f)","A",4,3,21,0.1429,1,"{2791, 2753, 2734}"
168,1,121,0,0.1140295,"\int \sin (e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Sin[e + f*x]*(a + b*Sin[e + f*x])^3,x]","-\frac{a \left(a^2+4 b^2\right) \cos (e+f x)}{2 f}-\frac{b \left(2 a^2+3 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} b x \left(4 a^2+b^2\right)-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f}","-\frac{a \left(a^2+4 b^2\right) \cos (e+f x)}{2 f}-\frac{b \left(2 a^2+3 b^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} b x \left(4 a^2+b^2\right)-\frac{\cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{a \cos (e+f x) (a+b \sin (e+f x))^2}{4 f}",1,"(3*b*(4*a^2 + b^2)*x)/8 - (a*(a^2 + 4*b^2)*Cos[e + f*x])/(2*f) - (b*(2*a^2 + 3*b^2)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(4*f) - (Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*f)","A",3,2,19,0.1053,1,"{2753, 2734}"
169,1,90,0,0.0656265,"\int (a+b \sin (e+f x))^3 \, dx","Int[(a + b*Sin[e + f*x])^3,x]","-\frac{2 b \left(4 a^2+b^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}","-\frac{2 b \left(4 a^2+b^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}",1,"(a*(2*a^2 + 3*b^2)*x)/2 - (2*b*(4*a^2 + b^2)*Cos[e + f*x])/(3*f) - (5*a*b^2*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (b*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)","A",2,2,12,0.1667,1,"{2656, 2734}"
170,1,74,0,0.1146373,"\int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Csc[e + f*x]*(a + b*Sin[e + f*x])^3,x]","\frac{1}{2} b x \left(6 a^2+b^2\right)-\frac{a^3 \tanh ^{-1}(\cos (e+f x))}{f}-\frac{5 a b^2 \cos (e+f x)}{2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f}","\frac{1}{2} b x \left(6 a^2+b^2\right)-\frac{a^3 \tanh ^{-1}(\cos (e+f x))}{f}-\frac{5 a b^2 \cos (e+f x)}{2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 f}",1,"(b*(6*a^2 + b^2)*x)/2 - (a^3*ArcTanh[Cos[e + f*x]])/f - (5*a*b^2*Cos[e + f*x])/(2*f) - (b^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(2*f)","A",4,4,19,0.2105,1,"{2793, 3023, 2735, 3770}"
171,1,68,0,0.1198848,"\int \csc ^2(e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Csc[e + f*x]^2*(a + b*Sin[e + f*x])^3,x]","\frac{b \left(a^2-b^2\right) \cos (e+f x)}{f}-\frac{3 a^2 b \tanh ^{-1}(\cos (e+f x))}{f}-\frac{a^2 \cot (e+f x) (a+b \sin (e+f x))}{f}+3 a b^2 x","\frac{b \left(a^2-b^2\right) \cos (e+f x)}{f}-\frac{3 a^2 b \tanh ^{-1}(\cos (e+f x))}{f}-\frac{a^2 \cot (e+f x) (a+b \sin (e+f x))}{f}+3 a b^2 x",1,"3*a*b^2*x - (3*a^2*b*ArcTanh[Cos[e + f*x]])/f + (b*(a^2 - b^2)*Cos[e + f*x])/f - (a^2*Cot[e + f*x]*(a + b*Sin[e + f*x]))/f","A",4,4,21,0.1905,1,"{2792, 3023, 2735, 3770}"
172,1,79,0,0.1331648,"\int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Csc[e + f*x]^3*(a + b*Sin[e + f*x])^3,x]","-\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{5 a^2 b \cot (e+f x)}{2 f}-\frac{a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+b^3 x","-\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{5 a^2 b \cot (e+f x)}{2 f}-\frac{a^2 \cot (e+f x) \csc (e+f x) (a+b \sin (e+f x))}{2 f}+b^3 x",1,"b^3*x - (a*(a^2 + 6*b^2)*ArcTanh[Cos[e + f*x]])/(2*f) - (5*a^2*b*Cot[e + f*x])/(2*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]*(a + b*Sin[e + f*x]))/(2*f)","A",4,4,21,0.1905,1,"{2792, 3021, 2735, 3770}"
173,1,109,0,0.1813922,"\int \csc ^4(e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Csc[e + f*x]^4*(a + b*Sin[e + f*x])^3,x]","-\frac{a \left(2 a^2+9 b^2\right) \cot (e+f x)}{3 f}-\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}","-\frac{a \left(2 a^2+9 b^2\right) \cot (e+f x)}{3 f}-\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{7 a^2 b \cot (e+f x) \csc (e+f x)}{6 f}-\frac{a^2 \cot (e+f x) \csc ^2(e+f x) (a+b \sin (e+f x))}{3 f}",1,"-(b*(3*a^2 + 2*b^2)*ArcTanh[Cos[e + f*x]])/(2*f) - (a*(2*a^2 + 9*b^2)*Cot[e + f*x])/(3*f) - (7*a^2*b*Cot[e + f*x]*Csc[e + f*x])/(6*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^2*(a + b*Sin[e + f*x]))/(3*f)","A",6,6,21,0.2857,1,"{2792, 3021, 2748, 3767, 8, 3770}"
174,1,134,0,0.2053917,"\int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx","Int[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3,x]","-\frac{b \left(2 a^2+b^2\right) \cot (e+f x)}{f}-\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{3 a \left(a^2+4 b^2\right) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac{a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}","-\frac{b \left(2 a^2+b^2\right) \cot (e+f x)}{f}-\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{3 a \left(a^2+4 b^2\right) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac{a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}",1,"(-3*a*(a^2 + 4*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - (b*(2*a^2 + b^2)*Cot[e + f*x])/f - (3*a*(a^2 + 4*b^2)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (3*a^2*b*Cot[e + f*x]*Csc[e + f*x]^2)/(4*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^3*(a + b*Sin[e + f*x]))/(4*f)","A",7,7,21,0.3333,1,"{2792, 3021, 2748, 3768, 3770, 3767, 8}"
175,1,137,0,0.145565,"\int (a+b \sin (e+f x))^4 \, dx","Int[(a + b*Sin[e + f*x])^4,x]","-\frac{a b \left(19 a^2+16 b^2\right) \cos (e+f x)}{6 f}-\frac{b^2 \left(26 a^2+9 b^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(24 a^2 b^2+8 a^4+3 b^4\right)-\frac{b \cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{7 a b \cos (e+f x) (a+b \sin (e+f x))^2}{12 f}","-\frac{a b \left(19 a^2+16 b^2\right) \cos (e+f x)}{6 f}-\frac{b^2 \left(26 a^2+9 b^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(24 a^2 b^2+8 a^4+3 b^4\right)-\frac{b \cos (e+f x) (a+b \sin (e+f x))^3}{4 f}-\frac{7 a b \cos (e+f x) (a+b \sin (e+f x))^2}{12 f}",1,"((8*a^4 + 24*a^2*b^2 + 3*b^4)*x)/8 - (a*b*(19*a^2 + 16*b^2)*Cos[e + f*x])/(6*f) - (b^2*(26*a^2 + 9*b^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (7*a*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*f) - (b*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*f)","A",3,3,12,0.2500,1,"{2656, 2753, 2734}"
176,1,110,0,0.2776656,"\int \frac{\sin ^4(x)}{a+b \sin (x)} \, dx","Int[Sin[x]^4/(a + b*Sin[x]),x]","-\frac{a x \left(2 a^2+b^2\right)}{2 b^4}-\frac{\left(3 a^2+2 b^2\right) \cos (x)}{3 b^3}+\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \sqrt{a^2-b^2}}+\frac{a \sin (x) \cos (x)}{2 b^2}-\frac{\sin ^2(x) \cos (x)}{3 b}","-\frac{a x \left(2 a^2+b^2\right)}{2 b^4}-\frac{\left(3 a^2+2 b^2\right) \cos (x)}{3 b^3}+\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \sqrt{a^2-b^2}}+\frac{a \sin (x) \cos (x)}{2 b^2}-\frac{\sin ^2(x) \cos (x)}{3 b}",1,"-(a*(2*a^2 + b^2)*x)/(2*b^4) + (2*a^4*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]) - ((3*a^2 + 2*b^2)*Cos[x])/(3*b^3) + (a*Cos[x]*Sin[x])/(2*b^2) - (Cos[x]*Sin[x]^2)/(3*b)","A",7,7,13,0.5385,1,"{2793, 3049, 3023, 2735, 2660, 618, 204}"
177,1,82,0,0.1635223,"\int \frac{\sin ^3(x)}{a+b \sin (x)} \, dx","Int[Sin[x]^3/(a + b*Sin[x]),x]","\frac{x \left(2 a^2+b^2\right)}{2 b^3}-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \sqrt{a^2-b^2}}+\frac{a \cos (x)}{b^2}-\frac{\sin (x) \cos (x)}{2 b}","\frac{x \left(2 a^2+b^2\right)}{2 b^3}-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \sqrt{a^2-b^2}}+\frac{a \cos (x)}{b^2}-\frac{\sin (x) \cos (x)}{2 b}",1,"((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]) + (a*Cos[x])/b^2 - (Cos[x]*Sin[x])/(2*b)","A",6,6,13,0.4615,1,"{2793, 3023, 2735, 2660, 618, 204}"
178,1,61,0,0.1035339,"\int \frac{\sin ^2(x)}{a+b \sin (x)} \, dx","Int[Sin[x]^2/(a + b*Sin[x]),x]","\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \sqrt{a^2-b^2}}-\frac{a x}{b^2}-\frac{\cos (x)}{b}","\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \sqrt{a^2-b^2}}-\frac{a x}{b^2}-\frac{\cos (x)}{b}",1,"-((a*x)/b^2) + (2*a^2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Cos[x]/b","A",6,6,13,0.4615,1,"{2746, 12, 2735, 2660, 618, 204}"
179,1,50,0,0.0552835,"\int \frac{\sin (x)}{a+b \sin (x)} \, dx","Int[Sin[x]/(a + b*Sin[x]),x]","\frac{x}{b}-\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2}}","\frac{x}{b}-\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2}}",1,"x/b - (2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])","A",4,4,11,0.3636,1,"{2735, 2660, 618, 204}"
180,1,40,0,0.0307481,"\int \frac{1}{a+b \sin (x)} \, dx","Int[(a + b*Sin[x])^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}",1,"(2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]","A",3,3,8,0.3750,1,"{2660, 618, 204}"
181,1,53,0,0.0662638,"\int \frac{\csc (x)}{a+b \sin (x)} \, dx","Int[Csc[x]/(a + b*Sin[x]),x]","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a}","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"(-2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]) - ArcTanh[Cos[x]]/a","A",5,5,11,0.4545,1,"{2747, 3770, 2660, 618, 204}"
182,1,62,0,0.1114668,"\int \frac{\csc ^2(x)}{a+b \sin (x)} \, dx","Int[Csc[x]^2/(a + b*Sin[x]),x]","\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \sqrt{a^2-b^2}}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\cot (x)}{a}","\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \sqrt{a^2-b^2}}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\cot (x)}{a}",1,"(2*b^2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) + (b*ArcTanh[Cos[x]])/a^2 - Cot[x]/a","A",7,7,13,0.5385,1,"{2802, 12, 2747, 3770, 2660, 618, 204}"
183,1,84,0,0.2713975,"\int \frac{\csc ^3(x)}{a+b \sin (x)} \, dx","Int[Csc[x]^3/(a + b*Sin[x]),x]","-\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \sqrt{a^2-b^2}}-\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{b \cot (x)}{a^2}-\frac{\cot (x) \csc (x)}{2 a}","-\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \sqrt{a^2-b^2}}-\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac{b \cot (x)}{a^2}-\frac{\cot (x) \csc (x)}{2 a}",1,"(-2*b^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) - ((a^2 + 2*b^2)*ArcTanh[Cos[x]])/(2*a^3) + (b*Cot[x])/a^2 - (Cot[x]*Csc[x])/(2*a)","A",7,7,13,0.5385,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
184,1,112,0,0.4333358,"\int \frac{\csc ^4(x)}{a+b \sin (x)} \, dx","Int[Csc[x]^4/(a + b*Sin[x]),x]","\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \sqrt{a^2-b^2}}-\frac{\left(2 a^2+3 b^2\right) \cot (x)}{3 a^3}+\frac{b \left(a^2+2 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^4}+\frac{b \cot (x) \csc (x)}{2 a^2}-\frac{\cot (x) \csc ^2(x)}{3 a}","\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \sqrt{a^2-b^2}}-\frac{\left(2 a^2+3 b^2\right) \cot (x)}{3 a^3}+\frac{b \left(a^2+2 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^4}+\frac{b \cot (x) \csc (x)}{2 a^2}-\frac{\cot (x) \csc ^2(x)}{3 a}",1,"(2*b^4*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) + (b*(a^2 + 2*b^2)*ArcTanh[Cos[x]])/(2*a^4) - ((2*a^2 + 3*b^2)*Cot[x])/(3*a^3) + (b*Cot[x]*Csc[x])/(2*a^2) - (Cot[x]*Csc[x]^2)/(3*a)","A",8,7,13,0.5385,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
185,1,169,0,0.3827917,"\int \frac{\sin ^4(x)}{(a+b \sin (x))^2} \, dx","Int[Sin[x]^4/(a + b*Sin[x])^2,x]","\frac{x \left(6 a^2+b^2\right)}{2 b^4}+\frac{a \left(3 a^2-2 b^2\right) \cos (x)}{b^3 \left(a^2-b^2\right)}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \sin ^2(x) \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\left(3 a^2-b^2\right) \sin (x) \cos (x)}{2 b^2 \left(a^2-b^2\right)}","\frac{x \left(6 a^2+b^2\right)}{2 b^4}+\frac{a \left(3 a^2-2 b^2\right) \cos (x)}{b^3 \left(a^2-b^2\right)}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \sin ^2(x) \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\left(3 a^2-b^2\right) \sin (x) \cos (x)}{2 b^2 \left(a^2-b^2\right)}",1,"((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)) + (a*(3*a^2 - 2*b^2)*Cos[x])/(b^3*(a^2 - b^2)) - ((3*a^2 - b^2)*Cos[x]*Sin[x])/(2*b^2*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x]^2)/(b*(a^2 - b^2)*(a + b*Sin[x]))","A",7,7,13,0.5385,1,"{2792, 3049, 3023, 2735, 2660, 618, 204}"
186,1,124,0,0.2182845,"\int \frac{\sin ^3(x)}{(a+b \sin (x))^2} \, dx","Int[Sin[x]^3/(a + b*Sin[x])^2,x]","-\frac{\left(2 a^2-b^2\right) \cos (x)}{b^2 \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \sin (x) \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}-\frac{2 a x}{b^3}","-\frac{\left(2 a^2-b^2\right) \cos (x)}{b^2 \left(a^2-b^2\right)}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \sin (x) \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}-\frac{2 a x}{b^3}",1,"(-2*a*x)/b^3 + (2*a^2*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)) - ((2*a^2 - b^2)*Cos[x])/(b^2*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x])/(b*(a^2 - b^2)*(a + b*Sin[x]))","A",6,6,13,0.4615,1,"{2792, 3023, 2735, 2660, 618, 204}"
187,1,87,0,0.1251012,"\int \frac{\sin ^2(x)}{(a+b \sin (x))^2} \, dx","Int[Sin[x]^2/(a + b*Sin[x])^2,x]","-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}+\frac{x}{b^2}","-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \left(a^2-b^2\right)^{3/2}}+\frac{a^2 \cos (x)}{b \left(a^2-b^2\right) (a+b \sin (x))}+\frac{x}{b^2}",1,"x/b^2 - (2*a*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)) + (a^2*Cos[x])/(b*(a^2 - b^2)*(a + b*Sin[x]))","A",5,5,13,0.3846,1,"{2790, 2735, 2660, 618, 204}"
188,1,66,0,0.0633352,"\int \frac{\sin (x)}{(a+b \sin (x))^2} \, dx","Int[Sin[x]/(a + b*Sin[x])^2,x]","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{a \cos (x)}{\left(a^2-b^2\right) (a+b \sin (x))}","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{a \cos (x)}{\left(a^2-b^2\right) (a+b \sin (x))}",1,"(-2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (a*Cos[x])/((a^2 - b^2)*(a + b*Sin[x]))","A",5,5,11,0.4545,1,"{2754, 12, 2660, 618, 204}"
189,1,65,0,0.0508134,"\int \frac{1}{(a+b \sin (x))^2} \, dx","Int[(a + b*Sin[x])^(-2),x]","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b \cos (x)}{\left(a^2-b^2\right) (a+b \sin (x))}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b \cos (x)}{\left(a^2-b^2\right) (a+b \sin (x))}",1,"(2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (b*Cos[x])/((a^2 - b^2)*(a + b*Sin[x]))","A",5,5,8,0.6250,1,"{2664, 12, 2660, 618, 204}"
190,1,93,0,0.1798235,"\int \frac{\csc (x)}{(a+b \sin (x))^2} \, dx","Int[Csc[x]/(a + b*Sin[x])^2,x]","-\frac{2 b \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right)^{3/2}}-\frac{b^2 \cos (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\tanh ^{-1}(\cos (x))}{a^2}","-\frac{2 b \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right)^{3/2}}-\frac{b^2 \cos (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\tanh ^{-1}(\cos (x))}{a^2}",1,"(-2*b*(2*a^2 - b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)) - ArcTanh[Cos[x]]/a^2 - (b^2*Cos[x])/(a*(a^2 - b^2)*(a + b*Sin[x]))","A",6,6,11,0.5455,1,"{2802, 3001, 3770, 2660, 618, 204}"
191,1,123,0,0.3348205,"\int \frac{\csc ^2(x)}{(a+b \sin (x))^2} \, dx","Int[Csc[x]^2/(a + b*Sin[x])^2,x]","\frac{2 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \left(a^2-b^2\right)^{3/2}}-\frac{\left(a^2-2 b^2\right) \cot (x)}{a^2 \left(a^2-b^2\right)}-\frac{b^2 \cot (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}+\frac{2 b \tanh ^{-1}(\cos (x))}{a^3}","\frac{2 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \left(a^2-b^2\right)^{3/2}}-\frac{\left(a^2-2 b^2\right) \cot (x)}{a^2 \left(a^2-b^2\right)}-\frac{b^2 \cot (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}+\frac{2 b \tanh ^{-1}(\cos (x))}{a^3}",1,"(2*b^2*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)) + (2*b*ArcTanh[Cos[x]])/a^3 - ((a^2 - 2*b^2)*Cot[x])/(a^2*(a^2 - b^2)) - (b^2*Cot[x])/(a*(a^2 - b^2)*(a + b*Sin[x]))","A",7,7,13,0.5385,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
192,1,168,0,0.5744289,"\int \frac{\csc ^3(x)}{(a+b \sin (x))^2} \, dx","Int[Csc[x]^3/(a + b*Sin[x])^2,x]","-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \left(a^2-b^2\right)^{3/2}}+\frac{b \left(2 a^2-3 b^2\right) \cot (x)}{a^3 \left(a^2-b^2\right)}-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^4}-\frac{\left(a^2-3 b^2\right) \cot (x) \csc (x)}{2 a^2 \left(a^2-b^2\right)}-\frac{b^2 \cot (x) \csc (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}","-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \left(a^2-b^2\right)^{3/2}}+\frac{b \left(2 a^2-3 b^2\right) \cot (x)}{a^3 \left(a^2-b^2\right)}-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^4}-\frac{\left(a^2-3 b^2\right) \cot (x) \csc (x)}{2 a^2 \left(a^2-b^2\right)}-\frac{b^2 \cot (x) \csc (x)}{a \left(a^2-b^2\right) (a+b \sin (x))}",1,"(-2*b^3*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2)) - ((a^2 + 6*b^2)*ArcTanh[Cos[x]])/(2*a^4) + (b*(2*a^2 - 3*b^2)*Cot[x])/(a^3*(a^2 - b^2)) - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/(2*a^2*(a^2 - b^2)) - (b^2*Cot[x]*Csc[x])/(a*(a^2 - b^2)*(a + b*Sin[x]))","A",8,7,13,0.5385,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
193,1,243,0,0.6610846,"\int \frac{\sin ^5(x)}{(a+b \sin (x))^3} \, dx","Int[Sin[x]^5/(a + b*Sin[x])^3,x]","\frac{x \left(12 a^2+b^2\right)}{2 b^5}+\frac{3 a \left(-7 a^2 b^2+4 a^4+2 b^4\right) \cos (x)}{2 b^4 \left(a^2-b^2\right)^2}-\frac{a^3 \left(-29 a^2 b^2+12 a^4+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 \left(a^2-b^2\right)^{5/2}}+\frac{a^2 \left(4 a^2-7 b^2\right) \sin ^2(x) \cos (x)}{2 b^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{a^2 \sin ^3(x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{\left(-10 a^2 b^2+6 a^4+b^4\right) \sin (x) \cos (x)}{2 b^3 \left(a^2-b^2\right)^2}","\frac{x \left(12 a^2+b^2\right)}{2 b^5}+\frac{3 a \left(-7 a^2 b^2+4 a^4+2 b^4\right) \cos (x)}{2 b^4 \left(a^2-b^2\right)^2}-\frac{a^3 \left(-29 a^2 b^2+12 a^4+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 \left(a^2-b^2\right)^{5/2}}+\frac{a^2 \left(4 a^2-7 b^2\right) \sin ^2(x) \cos (x)}{2 b^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{a^2 \sin ^3(x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{\left(-10 a^2 b^2+6 a^4+b^4\right) \sin (x) \cos (x)}{2 b^3 \left(a^2-b^2\right)^2}",1,"((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)) + (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[x])/(2*b^4*(a^2 - b^2)^2) - ((6*a^4 - 10*a^2*b^2 + b^4)*Cos[x]*Sin[x])/(2*b^3*(a^2 - b^2)^2) + (a^2*Cos[x]*Sin[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(4*a^2 - 7*b^2)*Cos[x]*Sin[x]^2)/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",8,8,13,0.6154,1,"{2792, 3047, 3049, 3023, 2735, 2660, 618, 204}"
194,1,179,0,0.4110728,"\int \frac{\sin ^4(x)}{(a+b \sin (x))^3} \, dx","Int[Sin[x]^4/(a + b*Sin[x])^3,x]","-\frac{\left(3 a^2-2 b^2\right) \cos (x)}{2 b^3 \left(a^2-b^2\right)}+\frac{3 a^2 \left(-5 a^2 b^2+2 a^4+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{5/2}}+\frac{a^2 \sin ^2(x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{3 a^3 \left(a^2-2 b^2\right) \cos (x)}{2 b^3 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{3 a x}{b^4}","-\frac{\left(3 a^2-2 b^2\right) \cos (x)}{2 b^3 \left(a^2-b^2\right)}+\frac{3 a^2 \left(-5 a^2 b^2+2 a^4+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{5/2}}+\frac{a^2 \sin ^2(x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{3 a^3 \left(a^2-2 b^2\right) \cos (x)}{2 b^3 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{3 a x}{b^4}",1,"(-3*a*x)/b^4 + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)) - ((3*a^2 - 2*b^2)*Cos[x])/(2*b^3*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x]^2)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) - (3*a^3*(a^2 - 2*b^2)*Cos[x])/(2*b^3*(a^2 - b^2)^2*(a + b*Sin[x]))","A",7,7,13,0.5385,1,"{2792, 3031, 3023, 2735, 2660, 618, 204}"
195,1,144,0,0.2445928,"\int \frac{\sin ^3(x)}{(a+b \sin (x))^3} \, dx","Int[Sin[x]^3/(a + b*Sin[x])^3,x]","-\frac{a \left(-5 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \left(a^2-b^2\right)^{5/2}}+\frac{a^2 \left(2 a^2-5 b^2\right) \cos (x)}{2 b^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{a^2 \sin (x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}+\frac{x}{b^3}","-\frac{a \left(-5 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 \left(a^2-b^2\right)^{5/2}}+\frac{a^2 \left(2 a^2-5 b^2\right) \cos (x)}{2 b^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{a^2 \sin (x) \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}+\frac{x}{b^3}",1,"x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)) + (a^2*Cos[x]*Sin[x])/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(2*a^2 - 5*b^2)*Cos[x])/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",6,6,13,0.4615,1,"{2792, 3021, 2735, 2660, 618, 204}"
196,1,118,0,0.1436044,"\int \frac{\sin ^2(x)}{(a+b \sin (x))^3} \, dx","Int[Sin[x]^2/(a + b*Sin[x])^3,x]","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a^2 \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{a \left(a^2-4 b^2\right) \cos (x)}{2 b \left(a^2-b^2\right)^2 (a+b \sin (x))}","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a^2 \cos (x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{a \left(a^2-4 b^2\right) \cos (x)}{2 b \left(a^2-b^2\right)^2 (a+b \sin (x))}",1,"((a^2 + 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a^2*Cos[x])/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) - (a*(a^2 - 4*b^2)*Cos[x])/(2*b*(a^2 - b^2)^2*(a + b*Sin[x]))","A",6,6,13,0.4615,1,"{2790, 2754, 12, 2660, 618, 204}"
197,1,103,0,0.1025214,"\int \frac{\sin (x)}{(a+b \sin (x))^3} \, dx","Int[Sin[x]/(a + b*Sin[x])^3,x]","-\frac{3 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\left(a^2+2 b^2\right) \cos (x)}{2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{a \cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))^2}","-\frac{3 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\left(a^2+2 b^2\right) \cos (x)}{2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{a \cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))^2}",1,"(-3*a*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (a*Cos[x])/(2*(a^2 - b^2)*(a + b*Sin[x])^2) - ((a^2 + 2*b^2)*Cos[x])/(2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",6,5,11,0.4545,1,"{2754, 12, 2660, 618, 204}"
198,1,102,0,0.0946794,"\int \frac{1}{(a+b \sin (x))^3} \, dx","Int[(a + b*Sin[x])^(-3),x]","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{3 a b \cos (x)}{2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{b \cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))^2}","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{3 a b \cos (x)}{2 \left(a^2-b^2\right)^2 (a+b \sin (x))}+\frac{b \cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))^2}",1,"((2*a^2 + b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*Cos[x])/(2*(a^2 - b^2)*(a + b*Sin[x])^2) + (3*a*b*Cos[x])/(2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",6,6,8,0.7500,1,"{2664, 2754, 12, 2660, 618, 204}"
199,1,145,0,0.3704854,"\int \frac{\csc (x)}{(a+b \sin (x))^3} \, dx","Int[Csc[x]/(a + b*Sin[x])^3,x]","-\frac{b \left(-5 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \left(a^2-b^2\right)^{5/2}}-\frac{b^2 \left(5 a^2-2 b^2\right) \cos (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cos (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{\tanh ^{-1}(\cos (x))}{a^3}","-\frac{b \left(-5 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 \left(a^2-b^2\right)^{5/2}}-\frac{b^2 \left(5 a^2-2 b^2\right) \cos (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cos (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{\tanh ^{-1}(\cos (x))}{a^3}",1,"-((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2))) - ArcTanh[Cos[x]]/a^3 - (b^2*Cos[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (b^2*(5*a^2 - 2*b^2)*Cos[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",7,7,11,0.6364,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
200,1,187,0,0.6393963,"\int \frac{\csc ^2(x)}{(a+b \sin (x))^3} \, dx","Int[Csc[x]^2/(a + b*Sin[x])^3,x]","\frac{3 b^2 \left(-5 a^2 b^2+4 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \left(a^2-b^2\right)^{5/2}}-\frac{\left(-11 a^2 b^2+2 a^4+6 b^4\right) \cot (x)}{2 a^3 \left(a^2-b^2\right)^2}-\frac{3 b^2 \left(2 a^2-b^2\right) \cot (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cot (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}+\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}","\frac{3 b^2 \left(-5 a^2 b^2+4 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 \left(a^2-b^2\right)^{5/2}}-\frac{\left(-11 a^2 b^2+2 a^4+6 b^4\right) \cot (x)}{2 a^3 \left(a^2-b^2\right)^2}-\frac{3 b^2 \left(2 a^2-b^2\right) \cot (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cot (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}+\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}",1,"(3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(5/2)) + (3*b*ArcTanh[Cos[x]])/a^4 - ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Cot[x])/(2*a^3*(a^2 - b^2)^2) - (b^2*Cot[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (3*b^2*(2*a^2 - b^2)*Cot[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",8,7,13,0.5385,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
201,1,241,0,0.8730674,"\int \frac{\csc ^3(x)}{(a+b \sin (x))^3} \, dx","Int[Csc[x]^3/(a + b*Sin[x])^3,x]","-\frac{b^3 \left(-29 a^2 b^2+20 a^4+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 \left(a^2-b^2\right)^{5/2}}+\frac{3 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \cot (x)}{2 a^4 \left(a^2-b^2\right)^2}-\frac{\left(a^2+12 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^5}-\frac{\left(-10 a^2 b^2+a^4+6 b^4\right) \cot (x) \csc (x)}{2 a^3 \left(a^2-b^2\right)^2}-\frac{b^2 \left(7 a^2-4 b^2\right) \cot (x) \csc (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cot (x) \csc (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}","-\frac{b^3 \left(-29 a^2 b^2+20 a^4+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 \left(a^2-b^2\right)^{5/2}}+\frac{3 b \left(-7 a^2 b^2+2 a^4+4 b^4\right) \cot (x)}{2 a^4 \left(a^2-b^2\right)^2}-\frac{\left(a^2+12 b^2\right) \tanh ^{-1}(\cos (x))}{2 a^5}-\frac{\left(-10 a^2 b^2+a^4+6 b^4\right) \cot (x) \csc (x)}{2 a^3 \left(a^2-b^2\right)^2}-\frac{b^2 \left(7 a^2-4 b^2\right) \cot (x) \csc (x)}{2 a^2 \left(a^2-b^2\right)^2 (a+b \sin (x))}-\frac{b^2 \cot (x) \csc (x)}{2 a \left(a^2-b^2\right) (a+b \sin (x))^2}",1,"-((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(5/2))) - ((a^2 + 12*b^2)*ArcTanh[Cos[x]])/(2*a^5) + (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Cot[x])/(2*a^4*(a^2 - b^2)^2) - ((a^4 - 10*a^2*b^2 + 6*b^4)*Cot[x]*Csc[x])/(2*a^3*(a^2 - b^2)^2) - (b^2*Cot[x]*Csc[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (b^2*(7*a^2 - 4*b^2)*Cot[x]*Csc[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))","A",9,7,13,0.5385,1,"{2802, 3055, 3001, 3770, 2660, 618, 204}"
202,1,182,0,0.2245641,"\int \frac{1}{(a+b \sin (c+d x))^4} \, dx","Int[(a + b*Sin[c + d*x])^(-4),x]","\frac{a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{b \left(11 a^2+4 b^2\right) \cos (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{5 a b \cos (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \cos (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^3}","\frac{a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{b \left(11 a^2+4 b^2\right) \cos (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{5 a b \cos (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \cos (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^3}",1,"(a*(2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (b*Cos[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^3) + (5*a*b*Cos[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (b*(11*a^2 + 4*b^2)*Cos[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",7,6,12,0.5000,1,"{2664, 2754, 12, 2660, 618, 204}"
203,1,172,0,0.1697823,"\int \sin (e+f x) \sqrt{a+b \sin (e+f x)} \, dx","Int[Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b f \sqrt{a+b \sin (e+f x)}}-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2 a \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b f \sqrt{a+b \sin (e+f x)}}-\frac{2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f}+\frac{2 a \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}",1,"(-2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*f) + (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(3*b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*(a^2 - b^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(3*b*f*Sqrt[a + b*Sin[e + f*x]])","A",6,6,21,0.2857,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
204,1,62,0,0.0365646,"\int \sqrt{a+b \sin (e+f x)} \, dx","Int[Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}","\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}",1,"(2*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)])","A",2,2,14,0.1429,1,"{2655, 2653}"
205,1,128,0,0.2352123,"\int \csc (e+f x) \sqrt{a+b \sin (e+f x)} \, dx","Int[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}","\frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(2*b*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (2*a*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",5,5,21,0.2381,1,"{2803, 2663, 2661, 2807, 2805}"
206,1,213,0,0.4809744,"\int \csc ^2(e+f x) \sqrt{a+b \sin (e+f x)} \, dx","Int[Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]],x]","-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\frac{a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}","-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\frac{a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/f) - (EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) + (a*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (b*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",9,9,23,0.3913,1,"{2796, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
207,1,132,0,0.1069001,"\int \frac{\sin (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[Sin[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b f \sqrt{a+b \sin (e+f x)}}",1,"(2*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(b*f*Sqrt[a + b*Sin[e + f*x]])","A",5,5,21,0.2381,1,"{2752, 2663, 2661, 2655, 2653}"
208,1,62,0,0.0367861,"\int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[1/Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(2*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",2,2,14,0.1429,1,"{2663, 2661}"
209,1,63,0,0.1299183,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[Csc[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}","\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}",1,"(2*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])","A",2,2,21,0.09524,1,"{2807, 2805}"
210,1,222,0,0.4953319,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]],x]","-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a f \sqrt{a+b \sin (e+f x)}}","-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a f \sqrt{a+b \sin (e+f x)}}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(a*f)) - (EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(a*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) + (EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) - (b*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(a*f*Sqrt[a + b*Sin[e + f*x]])","A",9,9,23,0.3913,1,"{2802, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
211,1,371,0,0.5630213,"\int \sqrt{\sin (c+d x)} \sqrt{a+b \sin (c+d x)} \, dx","Int[Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}-\frac{\sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{a \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}","-\frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}-\frac{\sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{a \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"-((Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])) + ((a - b)*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticE[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d) - (Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/d + (a*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(b*d)","A",7,7,25,0.2800,1,"{2821, 3054, 2809, 12, 2801, 2816, 2994}"
212,1,109,0,0.0677167,"\int \frac{1}{\sqrt{\sin (c+d x)} \sqrt{a+b \sin (c+d x)}} \, dx","Int[1/(Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]),x]","-\frac{2 \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}","-\frac{2 \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d)","A",1,1,25,0.04000,1,"{2816}"
213,1,270,0,0.3797733,"\int (d \sin (e+f x))^m (a+b \sin (e+f x))^3 \, dx","Int[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^3,x]","\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) (m+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left(a^2 (m+2)+3 b^2 (m+1)\right) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 m+7) \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2) (m+3)}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{m+1}}{d f (m+3)}","\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) (m+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left(a^2 (m+2)+3 b^2 (m+1)\right) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 m+7) \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2) (m+3)}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (d \sin (e+f x))^{m+1}}{d f (m+3)}",1,"-((a*b^2*(7 + 2*m)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m))/(d*f*(2 + m)*(3 + m))) + (a*(3*b^2*(1 + m) + a^2*(2 + m))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + m))/(d*f*(1 + m)*(2 + m)*Sqrt[Cos[e + f*x]^2]) + (b*(b^2*(2 + m) + 3*a^2*(3 + m))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + m))/(d^2*f*(2 + m)*(3 + m)*Sqrt[Cos[e + f*x]^2]) - (b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(a + b*Sin[e + f*x]))/(d*f*(3 + m))","A",5,4,23,0.1739,1,"{2793, 3023, 2748, 2643}"
214,1,194,0,0.1346542,"\int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx","Int[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^2,x]","\frac{\left(a^2 (m+2)+b^2 (m+1)\right) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2)}","\frac{\left(a^2 (m+2)+b^2 (m+1)\right) \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) (m+2) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2)}",1,"-((b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m))/(d*f*(2 + m))) + ((b^2*(1 + m) + a^2*(2 + m))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + m))/(d*f*(1 + m)*(2 + m)*Sqrt[Cos[e + f*x]^2]) + (2*a*b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + m))/(d^2*f*(2 + m)*Sqrt[Cos[e + f*x]^2])","A",4,3,23,0.1304,1,"{2789, 2643, 3014}"
215,1,139,0,0.0722351,"\int (d \sin (e+f x))^m (a+b \sin (e+f x)) \, dx","Int[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x]),x]","\frac{a \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) \sqrt{\cos ^2(e+f x)}}","\frac{a \cos (e+f x) (d \sin (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{d f (m+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \cos (e+f x) (d \sin (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right)}{d^2 f (m+2) \sqrt{\cos ^2(e+f x)}}",1,"(a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + m))/(d*f*(1 + m)*Sqrt[Cos[e + f*x]^2]) + (b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + m))/(d^2*f*(2 + m)*Sqrt[Cos[e + f*x]^2])","A",3,2,21,0.09524,1,"{2748, 2643}"
216,1,195,0,0.2484972,"\int \frac{(d \sin (e+f x))^m}{a+b \sin (e+f x)} \, dx","Int[(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x]),x]","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{a d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{a d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"-((a*d*AppellF1[1/2, (1 - m)/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((a^2 - b^2)*f)) + (b*AppellF1[1/2, -m/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((a^2 - b^2)*f*(Sin[e + f*x]^2)^(m/2))","A",5,3,23,0.1304,1,"{2823, 3189, 429}"
217,1,306,0,0.420374,"\int \frac{(d \sin (e+f x))^m}{(a+b \sin (e+f x))^2} \, dx","Int[(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^2,x]","-\frac{b^2 \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)^2}-\frac{a^2 d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}","-\frac{b^2 \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)^2}-\frac{a^2 d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"-((b^2*AppellF1[1/2, (-1 - m)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(Sin[e + f*x]^2)^((-1 - m)/2))/((a^2 - b^2)^2*d*f)) - (a^2*d*AppellF1[1/2, (1 - m)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((a^2 - b^2)^2*f) + (2*a*b*AppellF1[1/2, -m/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((a^2 - b^2)^2*f*(Sin[e + f*x]^2)^(m/2))","A",10,4,23,0.1739,1,"{2824, 3189, 429, 16}"
218,1,406,0,0.5485101,"\int \frac{(d \sin (e+f x))^m}{(a+b \sin (e+f x))^3} \, dx","Int[(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^3,x]","-\frac{3 a b^2 \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)^3}-\frac{a^3 d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{b^3 \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};\frac{1}{2} (-m-2),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}","-\frac{3 a b^2 \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-m-1)} (d \sin (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)^3}-\frac{a^3 d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{b^3 \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};\frac{1}{2} (-m-2),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-m/2} (d \sin (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}",1,"(-3*a*b^2*AppellF1[1/2, (-1 - m)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(Sin[e + f*x]^2)^((-1 - m)/2))/((a^2 - b^2)^3*d*f) - (a^3*d*AppellF1[1/2, (1 - m)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((a^2 - b^2)^3*f) + (b^3*AppellF1[1/2, (-2 - m)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((a^2 - b^2)^3*f*(Sin[e + f*x]^2)^(m/2)) + (3*a^2*b*AppellF1[1/2, -m/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((a^2 - b^2)^3*f*(Sin[e + f*x]^2)^(m/2))","A",13,4,23,0.1739,1,"{2824, 3189, 429, 16}"
219,1,142,0,0.12323,"\int \sin ^{-1-\frac{a^2}{a^2+b^2}}(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sin[c + d*x]^(-1 - a^2/(a^2 + b^2))*(a + b*Sin[c + d*x])^2,x]","\frac{2 a \left(a^2+b^2\right) \cos (c+d x) \sin ^{\frac{b^2}{a^2+b^2}}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{b^2}{2 \left(a^2+b^2\right)};\frac{1}{2} \left(3-\frac{a^2}{a^2+b^2}\right);\sin ^2(c+d x)\right)}{b d \sqrt{\cos ^2(c+d x)}}-\frac{\left(a^2+b^2\right) \cos (c+d x) \sin ^{-\frac{a^2}{a^2+b^2}}(c+d x)}{d}","\frac{2 a \left(a^2+b^2\right) \cos (c+d x) \sin ^{\frac{b^2}{a^2+b^2}}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{b^2}{2 \left(a^2+b^2\right)};\frac{1}{2} \left(3-\frac{a^2}{a^2+b^2}\right);\sin ^2(c+d x)\right)}{b d \sqrt{\cos ^2(c+d x)}}-\frac{\left(a^2+b^2\right) \cos (c+d x) \sin ^{-\frac{a^2}{a^2+b^2}}(c+d x)}{d}",1,"-(((a^2 + b^2)*Cos[c + d*x])/(d*Sin[c + d*x]^(a^2/(a^2 + b^2)))) + (2*a*(a^2 + b^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, b^2/(2*(a^2 + b^2)), (3 - a^2/(a^2 + b^2))/2, Sin[c + d*x]^2]*Sin[c + d*x]^(b^2/(a^2 + b^2)))/(b*d*Sqrt[Cos[c + d*x]^2])","A",3,3,36,0.08333,1,"{2789, 2643, 3011}"
220,1,73,0,0.0738084,"\int \frac{(1+2 \sin (c+d x))^2}{\sin ^{\frac{6}{5}}(c+d x)} \, dx","Int[(1 + 2*Sin[c + d*x])^2/Sin[c + d*x]^(6/5),x]","\frac{5 \sin ^{\frac{4}{5}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{2}{5},\frac{1}{2};\frac{7}{5};\sin ^2(c+d x)\right)}{d \sqrt{\cos ^2(c+d x)}}-\frac{5 \cos (c+d x)}{d \sqrt[5]{\sin (c+d x)}}","\frac{5 \sin ^{\frac{4}{5}}(c+d x) \cos (c+d x) \, _2F_1\left(\frac{2}{5},\frac{1}{2};\frac{7}{5};\sin ^2(c+d x)\right)}{d \sqrt{\cos ^2(c+d x)}}-\frac{5 \cos (c+d x)}{d \sqrt[5]{\sin (c+d x)}}",1,"(-5*Cos[c + d*x])/(d*Sin[c + d*x]^(1/5)) + (5*Cos[c + d*x]*Hypergeometric2F1[2/5, 1/2, 7/5, Sin[c + d*x]^2]*Sin[c + d*x]^(4/5))/(d*Sqrt[Cos[c + d*x]^2])","A",3,3,23,0.1304,1,"{2789, 2643, 3011}"
221,0,0,0,0.0397752,"\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx","Int[Sin[c + d*x]^m*(a + b*Sin[c + d*x])^n,x]","\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx","\text{Int}\left(\sin ^m(c+d x) (a+b \sin (c+d x))^n,x\right)",0,"Defer[Int][Sin[c + d*x]^m*(a + b*Sin[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
222,1,351,0,0.4855287,"\int \sin ^3(c+d x) (a+b \sin (c+d x))^n \, dx","Int[Sin[c + d*x]^3*(a + b*Sin[c + d*x])^n,x]","\frac{\sqrt{2} a \left(2 a^2+b^2 \left(n^2+5 n+4\right)\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^3 d (n+2) (n+3) \sqrt{\sin (c+d x)+1}}-\frac{\sqrt{2} (a+b) \left(2 a^2+b^2 (n+2)^2\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^3 d (n+2) (n+3) \sqrt{\sin (c+d x)+1}}+\frac{2 a \cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b^2 d (n+2) (n+3)}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+3)}","\frac{\sqrt{2} a \left(2 a^2+b^2 \left(n^2+5 n+4\right)\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^3 d (n+2) (n+3) \sqrt{\sin (c+d x)+1}}-\frac{\sqrt{2} (a+b) \left(2 a^2+b^2 (n+2)^2\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^3 d (n+2) (n+3) \sqrt{\sin (c+d x)+1}}+\frac{2 a \cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b^2 d (n+2) (n+3)}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+3)}",1,"(2*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1 + n))/(b^2*d*(2 + n)*(3 + n)) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(1 + n))/(b*d*(3 + n)) - (Sqrt[2]*(a + b)*(2*a^2 + b^2*(2 + n)^2)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(b^3*d*(2 + n)*(3 + n)*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n) + (Sqrt[2]*a*(2*a^2 + b^2*(4 + 5*n + n^2))*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(b^3*d*(2 + n)*(3 + n)*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n)","A",9,6,21,0.2857,1,"{2793, 3023, 2756, 2665, 139, 138}"
223,1,274,0,0.2982984,"\int \sin ^2(c+d x) (a+b \sin (c+d x))^n \, dx","Int[Sin[c + d*x]^2*(a + b*Sin[c + d*x])^n,x]","-\frac{\sqrt{2} \left(a^2+b^2 (n+1)\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^2 d (n+2) \sqrt{\sin (c+d x)+1}}+\frac{\sqrt{2} a (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^2 d (n+2) \sqrt{\sin (c+d x)+1}}-\frac{\cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+2)}","-\frac{\sqrt{2} \left(a^2+b^2 (n+1)\right) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^2 d (n+2) \sqrt{\sin (c+d x)+1}}+\frac{\sqrt{2} a (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b^2 d (n+2) \sqrt{\sin (c+d x)+1}}-\frac{\cos (c+d x) (a+b \sin (c+d x))^{n+1}}{b d (n+2)}",1,"-((Cos[c + d*x]*(a + b*Sin[c + d*x])^(1 + n))/(b*d*(2 + n))) + (Sqrt[2]*a*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(b^2*d*(2 + n)*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n) - (Sqrt[2]*(a^2 + b^2*(1 + n))*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(b^2*d*(2 + n)*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n)","A",8,5,21,0.2381,1,"{2791, 2756, 2665, 139, 138}"
224,1,220,0,0.1884081,"\int \sin (c+d x) (a+b \sin (c+d x))^n \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x])^n,x]","\frac{\sqrt{2} a \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b d \sqrt{\sin (c+d x)+1}}-\frac{\sqrt{2} (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b d \sqrt{\sin (c+d x)+1}}","\frac{\sqrt{2} a \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b d \sqrt{\sin (c+d x)+1}}-\frac{\sqrt{2} (a+b) \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{b d \sqrt{\sin (c+d x)+1}}",1,"-((Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(b*d*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n)) + (Sqrt[2]*a*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(b*d*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n)","A",7,4,19,0.2105,1,"{2756, 2665, 139, 138}"
225,1,104,0,0.0647978,"\int (a+b \sin (c+d x))^n \, dx","Int[(a + b*Sin[c + d*x])^n,x]","-\frac{\sqrt{2} \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1}}","-\frac{\sqrt{2} \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1}}",1,"-((Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[c + d*x])/2, (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(d*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^n))","A",3,3,12,0.2500,1,"{2665, 139, 138}"
226,0,0,0,0.0314893,"\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx","Int[Csc[c + d*x]*(a + b*Sin[c + d*x])^n,x]","\int \csc (c+d x) (a+b \sin (c+d x))^n \, dx","\text{Int}\left(\csc (c+d x) (a+b \sin (c+d x))^n,x\right)",0,"Defer[Int][Csc[c + d*x]*(a + b*Sin[c + d*x])^n, x]","A",0,0,0,0,-1,"{}"
227,1,116,0,0.1620335,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]","\frac{7 a c^4 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left(c^2-c^2 \sin (e+f x)\right)^2}{5 f}+\frac{7 a \cos ^3(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{20 f}+\frac{7 a c^4 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{7}{8} a c^4 x","\frac{7 a c^4 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left(c^2-c^2 \sin (e+f x)\right)^2}{5 f}+\frac{7 a \cos ^3(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{20 f}+\frac{7 a c^4 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{7}{8} a c^4 x",1,"(7*a*c^4*x)/8 + (7*a*c^4*Cos[e + f*x]^3)/(12*f) + (7*a*c^4*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*(c^2 - c^2*Sin[e + f*x])^2)/(5*f) + (7*a*Cos[e + f*x]^3*(c^4 - c^4*Sin[e + f*x]))/(20*f)","A",6,5,24,0.2083,1,"{2736, 2678, 2669, 2635, 8}"
228,1,83,0,0.1125325,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3,x]","\frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left(c^3-c^3 \sin (e+f x)\right)}{4 f}+\frac{5 a c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a c^3 x","\frac{5 a c^3 \cos ^3(e+f x)}{12 f}+\frac{a \cos ^3(e+f x) \left(c^3-c^3 \sin (e+f x)\right)}{4 f}+\frac{5 a c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a c^3 x",1,"(5*a*c^3*x)/8 + (5*a*c^3*Cos[e + f*x]^3)/(12*f) + (5*a*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*(c^3 - c^3*Sin[e + f*x]))/(4*f)","A",5,5,24,0.2083,1,"{2736, 2678, 2669, 2635, 8}"
229,1,52,0,0.0649879,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2,x]","\frac{a c^2 \cos ^3(e+f x)}{3 f}+\frac{a c^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a c^2 x","\frac{a c^2 \cos ^3(e+f x)}{3 f}+\frac{a c^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a c^2 x",1,"(a*c^2*x)/2 + (a*c^2*Cos[e + f*x]^3)/(3*f) + (a*c^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",4,4,24,0.1667,1,"{2736, 2669, 2635, 8}"
230,1,29,0,0.0183067,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]","\frac{a c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a c x}{2}","\frac{a c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a c x}{2}",1,"(a*c*x)/2 + (a*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,22,0.04545,1,"{2734}"
231,1,33,0,0.0484159,"\int \frac{a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x]),x]","\frac{2 a \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{a x}{c}","\frac{2 a \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{a x}{c}",1,"-((a*x)/c) + (2*a*Cos[e + f*x])/(f*(c - c*Sin[e + f*x]))","A",2,2,24,0.08333,1,"{2735, 2648}"
232,1,30,0,0.0729124,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^2,x]","\frac{a c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}","\frac{a c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"(a*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3)","A",2,2,24,0.08333,1,"{2736, 2671}"
233,1,60,0,0.1168963,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^3,x]","\frac{a \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^3}+\frac{a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}","\frac{a \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^3}+\frac{a c \cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}",1,"(a*c*Cos[e + f*x]^3)/(5*f*(c - c*Sin[e + f*x])^4) + (a*Cos[e + f*x]^3)/(15*f*(c - c*Sin[e + f*x])^3)","A",3,3,24,0.1250,1,"{2736, 2672, 2671}"
234,1,92,0,0.1701798,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^4,x]","\frac{2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3}+\frac{2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac{a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}","\frac{2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3}+\frac{2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac{a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}",1,"(a*c*Cos[e + f*x]^3)/(7*f*(c - c*Sin[e + f*x])^5) + (2*a*Cos[e + f*x]^3)/(35*f*(c - c*Sin[e + f*x])^4) + (2*a*Cos[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^3)","A",4,3,24,0.1250,1,"{2736, 2672, 2671}"
235,1,126,0,0.2196172,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]","\frac{2 a c \cos ^3(e+f x)}{315 f \left(c^2-c^2 \sin (e+f x)\right)^3}+\frac{2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac{a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac{a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}","\frac{2 a c \cos ^3(e+f x)}{315 f \left(c^2-c^2 \sin (e+f x)\right)^3}+\frac{2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac{a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac{a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}",1,"(a*c*Cos[e + f*x]^3)/(9*f*(c - c*Sin[e + f*x])^6) + (a*Cos[e + f*x]^3)/(21*f*(c - c*Sin[e + f*x])^5) + (2*a*Cos[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^4) + (2*a*c*Cos[e + f*x]^3)/(315*f*(c^2 - c^2*Sin[e + f*x])^3)","A",5,3,24,0.1250,1,"{2736, 2672, 2671}"
236,1,152,0,0.1966058,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5 \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5,x]","\frac{3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac{3 a^2 c^5 \sin (e+f x) \cos ^3(e+f x)}{8 f}+\frac{a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac{3 a^2 \cos ^5(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{14 f}+\frac{9 a^2 c^5 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{9}{16} a^2 c^5 x","\frac{3 a^2 c^5 \cos ^5(e+f x)}{10 f}+\frac{3 a^2 c^5 \sin (e+f x) \cos ^3(e+f x)}{8 f}+\frac{a^2 c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{7 f}+\frac{3 a^2 \cos ^5(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{14 f}+\frac{9 a^2 c^5 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{9}{16} a^2 c^5 x",1,"(9*a^2*c^5*x)/16 + (3*a^2*c^5*Cos[e + f*x]^5)/(10*f) + (9*a^2*c^5*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (3*a^2*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(8*f) + (a^2*c^3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(7*f) + (3*a^2*Cos[e + f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(14*f)","A",7,5,26,0.1923,1,"{2736, 2678, 2669, 2635, 8}"
237,1,118,0,0.1446998,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4 \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4,x]","\frac{7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac{a^2 \cos ^5(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{6 f}+\frac{7 a^2 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{7 a^2 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{7}{16} a^2 c^4 x","\frac{7 a^2 c^4 \cos ^5(e+f x)}{30 f}+\frac{a^2 \cos ^5(e+f x) \left(c^4-c^4 \sin (e+f x)\right)}{6 f}+\frac{7 a^2 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{7 a^2 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{7}{16} a^2 c^4 x",1,"(7*a^2*c^4*x)/16 + (7*a^2*c^4*Cos[e + f*x]^5)/(30*f) + (7*a^2*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (7*a^2*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^2*Cos[e + f*x]^5*(c^4 - c^4*Sin[e + f*x]))/(6*f)","A",6,5,26,0.1923,1,"{2736, 2678, 2669, 2635, 8}"
238,1,85,0,0.0974407,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3,x]","\frac{a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac{a^2 c^3 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^3 x","\frac{a^2 c^3 \cos ^5(e+f x)}{5 f}+\frac{a^2 c^3 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^3 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^3 x",1,"(3*a^2*c^3*x)/8 + (a^2*c^3*Cos[e + f*x]^5)/(5*f) + (3*a^2*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)","A",5,4,26,0.1538,1,"{2736, 2669, 2635, 8}"
239,1,64,0,0.0687816,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2,x]","\frac{a^2 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^2 x","\frac{a^2 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^2 x",1,"(3*a^2*c^2*x)/8 + (3*a^2*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)","A",4,3,26,0.1154,1,"{2736, 2635, 8}"
240,1,52,0,0.0767365,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c \cos ^3(e+f x)}{3 f}+\frac{a^2 c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a^2 c x","-\frac{a^2 c \cos ^3(e+f x)}{3 f}+\frac{a^2 c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a^2 c x",1,"(a^2*c*x)/2 - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",4,4,24,0.1667,1,"{2736, 2669, 2635, 8}"
241,1,57,0,0.1409056,"\int \frac{(a+a \sin (e+f x))^2}{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x]),x]","\frac{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac{3 a^2 x}{c}","\frac{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac{3 a^2 x}{c}",1,"(-3*a^2*x)/c + (3*a^2*Cos[e + f*x])/(c*f) + (2*a^2*c*Cos[e + f*x]^3)/(f*(c - c*Sin[e + f*x])^2)","A",4,4,26,0.1538,1,"{2736, 2680, 2682, 8}"
242,1,72,0,0.1367445,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^2,x]","-\frac{2 a^2 \cos (e+f x)}{f \left(c^2-c^2 \sin (e+f x)\right)}+\frac{a^2 x}{c^2}+\frac{2 a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}","-\frac{2 a^2 \cos (e+f x)}{f \left(c^2-c^2 \sin (e+f x)\right)}+\frac{a^2 x}{c^2}+\frac{2 a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"(a^2*x)/c^2 + (2*a^2*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3) - (2*a^2*Cos[e + f*x])/(f*(c^2 - c^2*Sin[e + f*x]))","A",4,3,26,0.1154,1,"{2736, 2680, 8}"
243,1,34,0,0.0938423,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^3,x]","\frac{a^2 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}","\frac{a^2 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}",1,"(a^2*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^5)","A",2,2,26,0.07692,1,"{2736, 2671}"
244,1,67,0,0.1359293,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^4} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^4,x]","\frac{a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5}","\frac{a^2 c^2 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^5}",1,"(a^2*c^2*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^6) + (a^2*c*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^5)","A",3,3,26,0.1154,1,"{2736, 2672, 2671}"
245,1,98,0,0.1822843,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]","\frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}","\frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}",1,"(a^2*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (2*a^2*c*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^6) + (2*a^2*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)","A",4,3,26,0.1154,1,"{2736, 2672, 2671}"
246,1,132,0,0.2327939,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^6,x]","\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}","\frac{a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac{2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5}+\frac{2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac{a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}",1,"(a^2*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*c*Cos[e + f*x]^5)/(33*f*(c - c*Sin[e + f*x])^7) + (2*a^2*Cos[e + f*x]^5)/(231*f*(c - c*Sin[e + f*x])^6) + (2*a^2*Cos[e + f*x]^5)/(1155*c*f*(c - c*Sin[e + f*x])^5)","A",5,3,26,0.1154,1,"{2736, 2672, 2671}"
247,1,180,0,0.2059283,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6,x]","\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left(c^3-c^3 \sin (e+f x)\right)^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left(c^6-c^6 \sin (e+f x)\right)}{72 f}+\frac{11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{55}{128} a^3 c^6 x","\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left(c^3-c^3 \sin (e+f x)\right)^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left(c^6-c^6 \sin (e+f x)\right)}{72 f}+\frac{11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{55}{128} a^3 c^6 x",1,"(55*a^3*c^6*x)/128 + (11*a^3*c^6*Cos[e + f*x]^7)/(56*f) + (55*a^3*c^6*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (55*a^3*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (11*a^3*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) + (a^3*Cos[e + f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(9*f) + (11*a^3*Cos[e + f*x]^7*(c^6 - c^6*Sin[e + f*x]))/(72*f)","A",8,5,26,0.1923,1,"{2736, 2678, 2669, 2635, 8}"
248,1,145,0,0.1570835,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5,x]","\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{8 f}+\frac{3 a^3 c^5 \sin (e+f x) \cos ^5(e+f x)}{16 f}+\frac{15 a^3 c^5 \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{45 a^3 c^5 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{45}{128} a^3 c^5 x","\frac{9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left(c^5-c^5 \sin (e+f x)\right)}{8 f}+\frac{3 a^3 c^5 \sin (e+f x) \cos ^5(e+f x)}{16 f}+\frac{15 a^3 c^5 \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{45 a^3 c^5 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{45}{128} a^3 c^5 x",1,"(45*a^3*c^5*x)/128 + (9*a^3*c^5*Cos[e + f*x]^7)/(56*f) + (45*a^3*c^5*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (15*a^3*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (3*a^3*c^5*Cos[e + f*x]^5*Sin[e + f*x])/(16*f) + (a^3*Cos[e + f*x]^7*(c^5 - c^5*Sin[e + f*x]))/(8*f)","A",7,5,26,0.1923,1,"{2736, 2678, 2669, 2635, 8}"
249,1,112,0,0.1077702,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4 \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4,x]","\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^4 x","\frac{a^3 c^4 \cos ^7(e+f x)}{7 f}+\frac{a^3 c^4 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^4 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^4 x",1,"(5*a^3*c^4*x)/16 + (a^3*c^4*Cos[e + f*x]^7)/(7*f) + (5*a^3*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^4*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)","A",6,4,26,0.1538,1,"{2736, 2669, 2635, 8}"
250,1,91,0,0.0793481,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3,x]","\frac{a^3 c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^3 x","\frac{a^3 c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{5 a^3 c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} a^3 c^3 x",1,"(5*a^3*c^3*x)/16 + (5*a^3*c^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^3*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)","A",5,3,26,0.1154,1,"{2736, 2635, 8}"
251,1,85,0,0.0924749,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2,x]","-\frac{a^3 c^2 \cos ^5(e+f x)}{5 f}+\frac{a^3 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^3 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^3 c^2 x","-\frac{a^3 c^2 \cos ^5(e+f x)}{5 f}+\frac{a^3 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^3 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^3 c^2 x",1,"(3*a^3*c^2*x)/8 - (a^3*c^2*Cos[e + f*x]^5)/(5*f) + (3*a^3*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^3*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)","A",5,4,26,0.1538,1,"{2736, 2669, 2635, 8}"
252,1,82,0,0.0984802,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x]),x]","-\frac{5 a^3 c \cos ^3(e+f x)}{12 f}-\frac{c \cos ^3(e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{4 f}+\frac{5 a^3 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a^3 c x","-\frac{5 a^3 c \cos ^3(e+f x)}{12 f}-\frac{c \cos ^3(e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{4 f}+\frac{5 a^3 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a^3 c x",1,"(5*a^3*c*x)/8 - (5*a^3*c*Cos[e + f*x]^3)/(12*f) + (5*a^3*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (c*Cos[e + f*x]^3*(a^3 + a^3*Sin[e + f*x]))/(4*f)","A",5,5,24,0.2083,1,"{2736, 2678, 2669, 2635, 8}"
253,1,94,0,0.1825528,"\int \frac{(a+a \sin (e+f x))^3}{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x]),x]","\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{15 a^3 x}{2 c}","\frac{2 a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^3}+\frac{15 a^3 \cos (e+f x)}{2 c f}+\frac{5 a^3 \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))}-\frac{15 a^3 x}{2 c}",1,"(-15*a^3*x)/(2*c) + (15*a^3*Cos[e + f*x])/(2*c*f) + (2*a^3*c^2*Cos[e + f*x]^5)/(f*(c - c*Sin[e + f*x])^3) + (5*a^3*Cos[e + f*x]^3)/(2*f*(c - c*Sin[e + f*x]))","A",5,5,26,0.1923,1,"{2736, 2680, 2679, 2682, 8}"
254,1,92,0,0.1860496,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^2,x]","-\frac{5 a^3 \cos (e+f x)}{c^2 f}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}+\frac{5 a^3 x}{c^2}-\frac{10 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2}","-\frac{5 a^3 \cos (e+f x)}{c^2 f}+\frac{2 a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^4}+\frac{5 a^3 x}{c^2}-\frac{10 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^2}",1,"(5*a^3*x)/c^2 - (5*a^3*Cos[e + f*x])/(c^2*f) + (2*a^3*c^2*Cos[e + f*x]^5)/(3*f*(c - c*Sin[e + f*x])^4) - (10*a^3*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^2)","A",5,4,26,0.1538,1,"{2736, 2680, 2682, 8}"
255,1,106,0,0.1889668,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^3,x]","\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^3 \cos (e+f x)}{f \left(c^3-c^3 \sin (e+f x)\right)}-\frac{a^3 x}{c^3}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}","\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac{2 a^3 \cos (e+f x)}{f \left(c^3-c^3 \sin (e+f x)\right)}-\frac{a^3 x}{c^3}-\frac{2 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^3}",1,"-((a^3*x)/c^3) + (2*a^3*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^5) - (2*a^3*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3) + (2*a^3*Cos[e + f*x])/(f*(c^3 - c^3*Sin[e + f*x]))","A",5,3,26,0.1154,1,"{2736, 2680, 8}"
256,1,34,0,0.0902245,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4,x]","\frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}","\frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}",1,"(a^3*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7)","A",2,2,26,0.07692,1,"{2736, 2671}"
257,1,69,0,0.1393549,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^5,x]","\frac{a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}+\frac{a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}","\frac{a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}+\frac{a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}",1,"(a^3*c^3*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^8) + (a^3*c^2*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^7)","A",3,3,26,0.1154,1,"{2736, 2672, 2671}"
258,1,101,0,0.182151,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]","\frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7}","\frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7}",1,"(a^3*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c^2*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])^8) + (2*a^3*c*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^7)","A",4,3,26,0.1154,1,"{2736, 2672, 2671}"
259,1,132,0,0.2299534,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7,x]","\frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}","\frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}",1,"(a^3*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (3*a^3*c^2*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^8) + (2*a^3*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e + f*x])^7)","A",5,3,26,0.1154,1,"{2736, 2672, 2671}"
260,1,166,0,0.2872971,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]","\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}","\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}",1,"(a^3*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (4*a^3*c^2*Cos[e + f*x]^7)/(195*f*(c - c*Sin[e + f*x])^10) + (4*a^3*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (8*a^3*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^8) + (8*a^3*Cos[e + f*x]^7)/(45045*c*f*(c - c*Sin[e + f*x])^7)","A",6,3,26,0.1154,1,"{2736, 2672, 2671}"
261,1,118,0,0.1956052,"\int \frac{(c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])^4/(a + a*Sin[e + f*x]),x]","-\frac{2 a^3 c^4 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}-\frac{35 c^4 \cos ^3(e+f x)}{3 a f}-\frac{14 a c^4 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2}-\frac{35 c^4 \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{35 c^4 x}{2 a}","-\frac{2 a^3 c^4 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}-\frac{35 c^4 \cos ^3(e+f x)}{3 a f}-\frac{14 a c^4 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2}-\frac{35 c^4 \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{35 c^4 x}{2 a}",1,"(-35*c^4*x)/(2*a) - (35*c^4*Cos[e + f*x]^3)/(3*a*f) - (35*c^4*Cos[e + f*x]*Sin[e + f*x])/(2*a*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) - (14*a*c^4*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)","A",6,5,26,0.1923,1,"{2736, 2680, 2682, 2635, 8}"
262,1,92,0,0.1762599,"\int \frac{(c-c \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])^3/(a + a*Sin[e + f*x]),x]","-\frac{2 a^2 c^3 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac{15 c^3 \cos (e+f x)}{2 a f}-\frac{5 c^3 \cos ^3(e+f x)}{2 f (a \sin (e+f x)+a)}-\frac{15 c^3 x}{2 a}","-\frac{2 a^2 c^3 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac{15 c^3 \cos (e+f x)}{2 a f}-\frac{5 c^3 \cos ^3(e+f x)}{2 f (a \sin (e+f x)+a)}-\frac{15 c^3 x}{2 a}",1,"(-15*c^3*x)/(2*a) - (15*c^3*Cos[e + f*x])/(2*a*f) - (2*a^2*c^3*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^3) - (5*c^3*Cos[e + f*x]^3)/(2*f*(a + a*Sin[e + f*x]))","A",5,5,26,0.1923,1,"{2736, 2680, 2679, 2682, 8}"
263,1,56,0,0.1356552,"\int \frac{(c-c \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])^2/(a + a*Sin[e + f*x]),x]","-\frac{3 c^2 \cos (e+f x)}{a f}-\frac{2 a c^2 \cos ^3(e+f x)}{f (a \sin (e+f x)+a)^2}-\frac{3 c^2 x}{a}","-\frac{3 c^2 \cos (e+f x)}{a f}-\frac{2 a c^2 \cos ^3(e+f x)}{f (a \sin (e+f x)+a)^2}-\frac{3 c^2 x}{a}",1,"(-3*c^2*x)/a - (3*c^2*Cos[e + f*x])/(a*f) - (2*a*c^2*Cos[e + f*x]^3)/(f*(a + a*Sin[e + f*x])^2)","A",4,4,26,0.1538,1,"{2736, 2680, 2682, 8}"
264,1,32,0,0.0439464,"\int \frac{c-c \sin (e+f x)}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])/(a + a*Sin[e + f*x]),x]","-\frac{2 c \cos (e+f x)}{f (a \sin (e+f x)+a)}-\frac{c x}{a}","-\frac{2 c \cos (e+f x)}{f (a \sin (e+f x)+a)}-\frac{c x}{a}",1,"-((c*x)/a) - (2*c*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))","A",2,2,24,0.08333,1,"{2735, 2648}"
265,1,16,0,0.0674709,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])),x]","\frac{\tan (e+f x)}{a c f}","\frac{\tan (e+f x)}{a c f}",1,"Tan[e + f*x]/(a*c*f)","A",3,3,26,0.1154,1,"{2736, 3767, 8}"
266,1,53,0,0.1104501,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2),x]","\frac{2 \tan (e+f x)}{3 a c^2 f}+\frac{\sec (e+f x)}{3 a f \left(c^2-c^2 \sin (e+f x)\right)}","\frac{2 \tan (e+f x)}{3 a c^2 f}+\frac{\sec (e+f x)}{3 a f \left(c^2-c^2 \sin (e+f x)\right)}",1,"Sec[e + f*x]/(3*a*f*(c^2 - c^2*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*a*c^2*f)","A",4,4,26,0.1538,1,"{2736, 2672, 3767, 8}"
267,1,85,0,0.1578699,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3),x]","\frac{2 \tan (e+f x)}{5 a c^3 f}+\frac{\sec (e+f x)}{5 a f \left(c^3-c^3 \sin (e+f x)\right)}+\frac{\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}","\frac{2 \tan (e+f x)}{5 a c^3 f}+\frac{\sec (e+f x)}{5 a f \left(c^3-c^3 \sin (e+f x)\right)}+\frac{\sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}",1,"Sec[e + f*x]/(5*a*c*f*(c - c*Sin[e + f*x])^2) + Sec[e + f*x]/(5*a*f*(c^3 - c^3*Sin[e + f*x])) + (2*Tan[e + f*x])/(5*a*c^3*f)","A",5,4,26,0.1538,1,"{2736, 2672, 3767, 8}"
268,1,118,0,0.2068912,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx","Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^4),x]","\frac{8 \tan (e+f x)}{35 a c^4 f}+\frac{4 \sec (e+f x)}{35 a f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{4 \sec (e+f x)}{35 a f \left(c^2-c^2 \sin (e+f x)\right)^2}+\frac{\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}","\frac{8 \tan (e+f x)}{35 a c^4 f}+\frac{4 \sec (e+f x)}{35 a f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{4 \sec (e+f x)}{35 a f \left(c^2-c^2 \sin (e+f x)\right)^2}+\frac{\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}",1,"Sec[e + f*x]/(7*a*c*f*(c - c*Sin[e + f*x])^3) + (4*Sec[e + f*x])/(35*a*f*(c^2 - c^2*Sin[e + f*x])^2) + (4*Sec[e + f*x])/(35*a*f*(c^4 - c^4*Sin[e + f*x])) + (8*Tan[e + f*x])/(35*a*c^4*f)","A",6,4,26,0.1538,1,"{2736, 2672, 3767, 8}"
269,1,148,0,0.2398552,"\int \frac{(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]","\frac{35 c^5 \cos ^3(e+f x)}{a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac{105 c^5 \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac{105 c^5 x}{2 a^2}+\frac{42 c^5 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2}","\frac{35 c^5 \cos ^3(e+f x)}{a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac{105 c^5 \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac{105 c^5 x}{2 a^2}+\frac{42 c^5 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2}",1,"(105*c^5*x)/(2*a^2) + (35*c^5*Cos[e + f*x]^3)/(a^2*f) + (105*c^5*Cos[e + f*x]*Sin[e + f*x])/(2*a^2*f) - (2*a^4*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^6) + (6*a^2*c^5*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) + (42*c^5*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)","A",7,5,26,0.1923,1,"{2736, 2680, 2682, 2635, 8}"
270,1,135,0,0.2241055,"\int \frac{(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^2,x]","\frac{35 c^4 \cos (e+f x)}{2 a^2 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{14 a^4 c^4 \cos ^5(e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{35 c^4 x}{2 a^2}","\frac{35 c^4 \cos (e+f x)}{2 a^2 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac{14 a^4 c^4 \cos ^5(e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)^3}+\frac{35 c^4 \cos ^3(e+f x)}{6 f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{35 c^4 x}{2 a^2}",1,"(35*c^4*x)/(2*a^2) + (35*c^4*Cos[e + f*x])/(2*a^2*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(3*f*(a + a*Sin[e + f*x])^5) + (14*a^4*c^4*Cos[e + f*x]^5)/(3*f*(a^2 + a^2*Sin[e + f*x])^3) + (35*c^4*Cos[e + f*x]^3)/(6*f*(a^2 + a^2*Sin[e + f*x]))","A",6,5,26,0.1923,1,"{2736, 2680, 2679, 2682, 8}"
271,1,90,0,0.1753769,"\int \frac{(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^2,x]","\frac{5 c^3 \cos (e+f x)}{a^2 f}-\frac{2 a^2 c^3 \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^4}+\frac{5 c^3 x}{a^2}+\frac{10 c^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^2}","\frac{5 c^3 \cos (e+f x)}{a^2 f}-\frac{2 a^2 c^3 \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^4}+\frac{5 c^3 x}{a^2}+\frac{10 c^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"(5*c^3*x)/a^2 + (5*c^3*Cos[e + f*x])/(a^2*f) - (2*a^2*c^3*Cos[e + f*x]^5)/(3*f*(a + a*Sin[e + f*x])^4) + (10*c^3*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^2)","A",5,4,26,0.1538,1,"{2736, 2680, 2682, 8}"
272,1,70,0,0.1317856,"\int \frac{(c-c \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^2,x]","\frac{2 c^2 \cos (e+f x)}{f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{c^2 x}{a^2}-\frac{2 a c^2 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}","\frac{2 c^2 \cos (e+f x)}{f \left(a^2 \sin (e+f x)+a^2\right)}+\frac{c^2 x}{a^2}-\frac{2 a c^2 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"(c^2*x)/a^2 - (2*a*c^2*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3) + (2*c^2*Cos[e + f*x])/(f*(a^2 + a^2*Sin[e + f*x]))","A",4,3,26,0.1154,1,"{2736, 2680, 8}"
273,1,29,0,0.0668245,"\int \frac{c-c \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]","-\frac{a c \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}","-\frac{a c \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"-(a*c*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3)","A",2,2,24,0.08333,1,"{2736, 2671}"
274,1,52,0,0.1066797,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])),x]","\frac{2 \tan (e+f x)}{3 a^2 c f}-\frac{\sec (e+f x)}{3 c f \left(a^2 \sin (e+f x)+a^2\right)}","\frac{2 \tan (e+f x)}{3 a^2 c f}-\frac{\sec (e+f x)}{3 c f \left(a^2 \sin (e+f x)+a^2\right)}",1,"-Sec[e + f*x]/(3*c*f*(a^2 + a^2*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*a^2*c*f)","A",4,4,26,0.1538,1,"{2736, 2672, 3767, 8}"
275,1,38,0,0.0632528,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2),x]","\frac{\tan ^3(e+f x)}{3 a^2 c^2 f}+\frac{\tan (e+f x)}{a^2 c^2 f}","\frac{\tan ^3(e+f x)}{3 a^2 c^2 f}+\frac{\tan (e+f x)}{a^2 c^2 f}",1,"Tan[e + f*x]/(a^2*c^2*f) + Tan[e + f*x]^3/(3*a^2*c^2*f)","A",3,2,26,0.07692,1,"{2736, 3767}"
276,1,76,0,0.1132856,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3),x]","\frac{4 \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac{4 \tan (e+f x)}{5 a^2 c^3 f}+\frac{\sec ^3(e+f x)}{5 a^2 f \left(c^3-c^3 \sin (e+f x)\right)}","\frac{4 \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac{4 \tan (e+f x)}{5 a^2 c^3 f}+\frac{\sec ^3(e+f x)}{5 a^2 f \left(c^3-c^3 \sin (e+f x)\right)}",1,"Sec[e + f*x]^3/(5*a^2*f*(c^3 - c^3*Sin[e + f*x])) + (4*Tan[e + f*x])/(5*a^2*c^3*f) + (4*Tan[e + f*x]^3)/(15*a^2*c^3*f)","A",4,3,26,0.1154,1,"{2736, 2672, 3767}"
277,1,111,0,0.1594831,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4),x]","\frac{4 \tan ^3(e+f x)}{21 a^2 c^4 f}+\frac{4 \tan (e+f x)}{7 a^2 c^4 f}+\frac{\sec ^3(e+f x)}{7 a^2 f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{\sec ^3(e+f x)}{7 a^2 f \left(c^2-c^2 \sin (e+f x)\right)^2}","\frac{4 \tan ^3(e+f x)}{21 a^2 c^4 f}+\frac{4 \tan (e+f x)}{7 a^2 c^4 f}+\frac{\sec ^3(e+f x)}{7 a^2 f \left(c^4-c^4 \sin (e+f x)\right)}+\frac{\sec ^3(e+f x)}{7 a^2 f \left(c^2-c^2 \sin (e+f x)\right)^2}",1,"Sec[e + f*x]^3/(7*a^2*f*(c^2 - c^2*Sin[e + f*x])^2) + Sec[e + f*x]^3/(7*a^2*f*(c^4 - c^4*Sin[e + f*x])) + (4*Tan[e + f*x])/(7*a^2*c^4*f) + (4*Tan[e + f*x]^3)/(21*a^2*c^4*f)","A",5,3,26,0.1154,1,"{2736, 2672, 3767}"
278,1,144,0,0.2151912,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5),x]","\frac{8 \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac{8 \tan (e+f x)}{21 a^2 c^5 f}+\frac{2 \sec ^3(e+f x)}{21 a^2 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac{\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}","\frac{8 \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac{8 \tan (e+f x)}{21 a^2 c^5 f}+\frac{2 \sec ^3(e+f x)}{21 a^2 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac{\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}",1,"Sec[e + f*x]^3/(9*a^2*c^2*f*(c - c*Sin[e + f*x])^3) + (2*Sec[e + f*x]^3)/(21*a^2*c^3*f*(c - c*Sin[e + f*x])^2) + (2*Sec[e + f*x]^3)/(21*a^2*f*(c^5 - c^5*Sin[e + f*x])) + (8*Tan[e + f*x])/(21*a^2*c^5*f) + (8*Tan[e + f*x]^3)/(63*a^2*c^5*f)","A",6,3,26,0.1154,1,"{2736, 2672, 3767}"
279,1,161,0,0.27595,"\int \frac{(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^3,x]","-\frac{63 c^5 \cos (e+f x)}{2 a^3 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac{21 c^5 \cos ^3(e+f x)}{2 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{63 c^5 x}{2 a^3}-\frac{42 c^5 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^3}","-\frac{63 c^5 \cos (e+f x)}{2 a^3 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac{21 c^5 \cos ^3(e+f x)}{2 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{63 c^5 x}{2 a^3}-\frac{42 c^5 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"(-63*c^5*x)/(2*a^3) - (63*c^5*Cos[e + f*x])/(2*a^3*f) - (2*a^4*c^5*Cos[e + f*x]^9)/(5*f*(a + a*Sin[e + f*x])^7) + (6*a^2*c^5*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^5) - (42*c^5*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^3) - (21*c^5*Cos[e + f*x]^3)/(2*f*(a^3 + a^3*Sin[e + f*x]))","A",7,5,26,0.1923,1,"{2736, 2680, 2679, 2682, 8}"
280,1,124,0,0.2211136,"\int \frac{(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^3,x]","-\frac{7 c^4 \cos (e+f x)}{a^3 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac{7 c^4 x}{a^3}+\frac{14 a c^4 \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}-\frac{14 c^4 \cos ^3(e+f x)}{3 a f (a \sin (e+f x)+a)^2}","-\frac{7 c^4 \cos (e+f x)}{a^3 f}-\frac{2 a^3 c^4 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac{7 c^4 x}{a^3}+\frac{14 a c^4 \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}-\frac{14 c^4 \cos ^3(e+f x)}{3 a f (a \sin (e+f x)+a)^2}",1,"(-7*c^4*x)/a^3 - (7*c^4*Cos[e + f*x])/(a^3*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^6) + (14*a*c^4*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^4) - (14*c^4*Cos[e + f*x]^3)/(3*a*f*(a + a*Sin[e + f*x])^2)","A",6,4,26,0.1538,1,"{2736, 2680, 2682, 8}"
281,1,103,0,0.179353,"\int \frac{(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^3,x]","-\frac{2 a^2 c^3 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac{2 c^3 \cos (e+f x)}{f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{c^3 x}{a^3}+\frac{2 c^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}","-\frac{2 a^2 c^3 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac{2 c^3 \cos (e+f x)}{f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{c^3 x}{a^3}+\frac{2 c^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^3}",1,"-((c^3*x)/a^3) - (2*a^2*c^3*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^5) + (2*c^3*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3) - (2*c^3*Cos[e + f*x])/(f*(a^3 + a^3*Sin[e + f*x]))","A",5,3,26,0.1154,1,"{2736, 2680, 8}"
282,1,33,0,0.0858378,"\int \frac{(c-c \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^3,x]","-\frac{a^2 c^2 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}","-\frac{a^2 c^2 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^5}",1,"-(a^2*c^2*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^5)","A",2,2,26,0.07692,1,"{2736, 2671}"
283,1,58,0,0.1066666,"\int \frac{c-c \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]","-\frac{c \cos ^3(e+f x)}{15 f (a \sin (e+f x)+a)^3}-\frac{a c \cos ^3(e+f x)}{5 f (a \sin (e+f x)+a)^4}","-\frac{c \cos ^3(e+f x)}{15 f (a \sin (e+f x)+a)^3}-\frac{a c \cos ^3(e+f x)}{5 f (a \sin (e+f x)+a)^4}",1,"-(a*c*Cos[e + f*x]^3)/(5*f*(a + a*Sin[e + f*x])^4) - (c*Cos[e + f*x]^3)/(15*f*(a + a*Sin[e + f*x])^3)","A",3,3,24,0.1250,1,"{2736, 2672, 2671}"
284,1,83,0,0.147634,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])),x]","\frac{2 \tan (e+f x)}{5 a^3 c f}-\frac{\sec (e+f x)}{5 c f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\sec (e+f x)}{5 a c f (a \sin (e+f x)+a)^2}","\frac{2 \tan (e+f x)}{5 a^3 c f}-\frac{\sec (e+f x)}{5 c f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\sec (e+f x)}{5 a c f (a \sin (e+f x)+a)^2}",1,"-Sec[e + f*x]/(5*a*c*f*(a + a*Sin[e + f*x])^2) - Sec[e + f*x]/(5*c*f*(a^3 + a^3*Sin[e + f*x])) + (2*Tan[e + f*x])/(5*a^3*c*f)","A",5,4,26,0.1538,1,"{2736, 2672, 3767, 8}"
285,1,75,0,0.1101161,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2),x]","\frac{4 \tan ^3(e+f x)}{15 a^3 c^2 f}+\frac{4 \tan (e+f x)}{5 a^3 c^2 f}-\frac{\sec ^3(e+f x)}{5 c^2 f \left(a^3 \sin (e+f x)+a^3\right)}","\frac{4 \tan ^3(e+f x)}{15 a^3 c^2 f}+\frac{4 \tan (e+f x)}{5 a^3 c^2 f}-\frac{\sec ^3(e+f x)}{5 c^2 f \left(a^3 \sin (e+f x)+a^3\right)}",1,"-Sec[e + f*x]^3/(5*c^2*f*(a^3 + a^3*Sin[e + f*x])) + (4*Tan[e + f*x])/(5*a^3*c^2*f) + (4*Tan[e + f*x]^3)/(15*a^3*c^2*f)","A",4,3,26,0.1154,1,"{2736, 2672, 3767}"
286,1,59,0,0.0725032,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3),x]","\frac{\tan ^5(e+f x)}{5 a^3 c^3 f}+\frac{2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac{\tan (e+f x)}{a^3 c^3 f}","\frac{\tan ^5(e+f x)}{5 a^3 c^3 f}+\frac{2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac{\tan (e+f x)}{a^3 c^3 f}",1,"Tan[e + f*x]/(a^3*c^3*f) + (2*Tan[e + f*x]^3)/(3*a^3*c^3*f) + Tan[e + f*x]^5/(5*a^3*c^3*f)","A",3,2,26,0.07692,1,"{2736, 3767}"
287,1,97,0,0.1196333,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4),x]","\frac{6 \tan ^5(e+f x)}{35 a^3 c^4 f}+\frac{4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac{6 \tan (e+f x)}{7 a^3 c^4 f}+\frac{\sec ^5(e+f x)}{7 a^3 f \left(c^4-c^4 \sin (e+f x)\right)}","\frac{6 \tan ^5(e+f x)}{35 a^3 c^4 f}+\frac{4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac{6 \tan (e+f x)}{7 a^3 c^4 f}+\frac{\sec ^5(e+f x)}{7 a^3 f \left(c^4-c^4 \sin (e+f x)\right)}",1,"Sec[e + f*x]^5/(7*a^3*f*(c^4 - c^4*Sin[e + f*x])) + (6*Tan[e + f*x])/(7*a^3*c^4*f) + (4*Tan[e + f*x]^3)/(7*a^3*c^4*f) + (6*Tan[e + f*x]^5)/(35*a^3*c^4*f)","A",4,3,26,0.1154,1,"{2736, 2672, 3767}"
288,1,131,0,0.1708233,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]","\frac{2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac{4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac{2 \tan (e+f x)}{3 a^3 c^5 f}+\frac{\sec ^5(e+f x)}{9 a^3 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}","\frac{2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac{4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac{2 \tan (e+f x)}{3 a^3 c^5 f}+\frac{\sec ^5(e+f x)}{9 a^3 f \left(c^5-c^5 \sin (e+f x)\right)}+\frac{\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}",1,"Sec[e + f*x]^5/(9*a^3*c^3*f*(c - c*Sin[e + f*x])^2) + Sec[e + f*x]^5/(9*a^3*f*(c^5 - c^5*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*a^3*c^5*f) + (4*Tan[e + f*x]^3)/(9*a^3*c^5*f) + (2*Tan[e + f*x]^5)/(15*a^3*c^5*f)","A",5,3,26,0.1154,1,"{2736, 2672, 3767}"
289,1,167,0,0.2172618,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]","\frac{16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{16 \tan (e+f x)}{33 a^3 c^6 f}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left(c^6-c^6 \sin (e+f x)\right)}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left(c^3-c^3 \sin (e+f x)\right)^2}+\frac{\sec ^5(e+f x)}{11 a^3 f \left(c^2-c^2 \sin (e+f x)\right)^3}","\frac{16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac{32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac{16 \tan (e+f x)}{33 a^3 c^6 f}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left(c^6-c^6 \sin (e+f x)\right)}+\frac{8 \sec ^5(e+f x)}{99 a^3 f \left(c^3-c^3 \sin (e+f x)\right)^2}+\frac{\sec ^5(e+f x)}{11 a^3 f \left(c^2-c^2 \sin (e+f x)\right)^3}",1,"Sec[e + f*x]^5/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (16*Tan[e + f*x])/(33*a^3*c^6*f) + (32*Tan[e + f*x]^3)/(99*a^3*c^6*f) + (16*Tan[e + f*x]^5)/(165*a^3*c^6*f)","A",6,3,26,0.1154,1,"{2736, 2672, 3767}"
290,1,137,0,0.2948725,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]","\frac{256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}","\frac{256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}",1,"(256*a*c^5*Cos[e + f*x]^3)/(315*f*(c - c*Sin[e + f*x])^(3/2)) + (64*a*c^4*Cos[e + f*x]^3)/(105*f*Sqrt[c - c*Sin[e + f*x]]) + (8*a*c^3*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(21*f) + (2*a*c^2*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(9*f)","A",5,3,26,0.1154,1,"{2736, 2674, 2673}"
291,1,103,0,0.222452,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2),x]","\frac{64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}","\frac{64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}",1,"(64*a*c^4*Cos[e + f*x]^3)/(105*f*(c - c*Sin[e + f*x])^(3/2)) + (16*a*c^3*Cos[e + f*x]^3)/(35*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a*c^2*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(7*f)","A",4,3,26,0.1154,1,"{2736, 2674, 2673}"
292,1,69,0,0.1572682,"\int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]","\frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}","\frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}",1,"(8*a*c^3*Cos[e + f*x]^3)/(15*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a*c^2*Cos[e + f*x]^3)/(5*f*Sqrt[c - c*Sin[e + f*x]])","A",3,3,26,0.1154,1,"{2736, 2674, 2673}"
293,1,34,0,0.0918206,"\int (a+a \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}","\frac{2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}",1,"(2*a*c^2*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2))","A",2,2,26,0.07692,1,"{2736, 2673}"
294,1,77,0,0.1384759,"\int \frac{a+a \sin (e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 \sqrt{2} a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}-\frac{2 a \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}","\frac{2 \sqrt{2} a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}-\frac{2 a \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}",1,"(2*Sqrt[2]*a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])","A",4,4,26,0.1538,1,"{2736, 2679, 2649, 206}"
295,1,76,0,0.1407068,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{2} c^{3/2} f}","\frac{a \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{2} c^{3/2} f}",1,"-((a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*c^(3/2)*f)) + (a*Cos[e + f*x])/(f*(c - c*Sin[e + f*x])^(3/2))","A",4,4,26,0.1538,1,"{2736, 2680, 2649, 206}"
296,1,113,0,0.1634809,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} c^{5/2} f}-\frac{a \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} c^{5/2} f}-\frac{a \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x)}{2 f (c-c \sin (e+f x))^{5/2}}",1,"-(a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*c^(5/2)*f) + (a*Cos[e + f*x])/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (a*Cos[e + f*x])/(8*c*f*(c - c*Sin[e + f*x])^(3/2))","A",5,5,26,0.1923,1,"{2736, 2680, 2650, 2649, 206}"
297,1,145,0,0.1934648,"\int \frac{a+a \sin (e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{32 \sqrt{2} c^{7/2} f}-\frac{a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}","-\frac{a \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{32 \sqrt{2} c^{7/2} f}-\frac{a \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{a \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}",1,"-(a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(32*Sqrt[2]*c^(7/2)*f) + (a*Cos[e + f*x])/(3*f*(c - c*Sin[e + f*x])^(7/2)) - (a*Cos[e + f*x])/(24*c*f*(c - c*Sin[e + f*x])^(5/2)) - (a*Cos[e + f*x])/(32*c^2*f*(c - c*Sin[e + f*x])^(3/2))","A",6,5,26,0.1923,1,"{2736, 2680, 2650, 2649, 206}"
298,1,145,0,0.3272258,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2),x]","\frac{256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{11 f}","\frac{256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{11 f}",1,"(256*a^2*c^6*Cos[e + f*x]^5)/(1155*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*c^5*Cos[e + f*x]^5)/(231*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^2*c^4*Cos[e + f*x]^5)/(33*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c^3*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(11*f)","A",5,3,28,0.1071,1,"{2736, 2674, 2673}"
299,1,109,0,0.2596772,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2),x]","\frac{2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt{c-c \sin (e+f x)}}+\frac{16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}","\frac{2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt{c-c \sin (e+f x)}}+\frac{16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}",1,"(64*a^2*c^5*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^(5/2)) + (16*a^2*c^4*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^2*c^3*Cos[e + f*x]^5)/(9*f*Sqrt[c - c*Sin[e + f*x]])","A",4,3,28,0.1071,1,"{2736, 2674, 2673}"
300,1,73,0,0.1950611,"\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2),x]","\frac{2 a^2 c^3 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}","\frac{2 a^2 c^3 \cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 \cos ^5(e+f x)}{35 f (c-c \sin (e+f x))^{5/2}}",1,"(8*a^2*c^4*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^2*c^3*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^(3/2))","A",3,3,28,0.1071,1,"{2736, 2674, 2673}"
301,1,36,0,0.1251237,"\int (a+a \sin (e+f x))^2 \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a^2 c^3 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}","\frac{2 a^2 c^3 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}",1,"(2*a^2*c^3*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2))","A",2,2,28,0.07143,1,"{2736, 2673}"
302,1,115,0,0.2441652,"\int \frac{(a+a \sin (e+f x))^2}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^2/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{4 a^2 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{4 \sqrt{2} a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}","-\frac{2 a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{4 a^2 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{4 \sqrt{2} a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}",1,"(4*Sqrt[2]*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^2*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (4*a^2*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])","A",5,4,28,0.1429,1,"{2736, 2679, 2649, 206}"
303,1,115,0,0.2440844,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{3 \sqrt{2} a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{c^{3/2} f}+\frac{a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^{5/2}}+\frac{3 a^2 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}","-\frac{3 \sqrt{2} a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{c^{3/2} f}+\frac{a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^{5/2}}+\frac{3 a^2 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}",1,"(-3*Sqrt[2]*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f) + (a^2*c*Cos[e + f*x]^3)/(f*(c - c*Sin[e + f*x])^(5/2)) + (3*a^2*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])","A",5,5,28,0.1786,1,"{2736, 2680, 2679, 2649, 206}"
304,1,122,0,0.2404523,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(5/2),x]","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} c^{5/2} f}+\frac{a^2 c \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}-\frac{3 a^2 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{3/2}}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} c^{5/2} f}+\frac{a^2 c \cos ^3(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}-\frac{3 a^2 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{3/2}}",1,"(3*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*c^(5/2)*f) + (a^2*c*Cos[e + f*x]^3)/(2*f*(c - c*Sin[e + f*x])^(7/2)) - (3*a^2*Cos[e + f*x])/(4*c*f*(c - c*Sin[e + f*x])^(3/2))","A",5,4,28,0.1429,1,"{2736, 2680, 2649, 206}"
305,1,156,0,0.2729215,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^2 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{16 \sqrt{2} c^{7/2} f}+\frac{a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{5/2}}","\frac{a^2 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{16 \sqrt{2} c^{7/2} f}+\frac{a^2 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x)}{4 c f (c-c \sin (e+f x))^{5/2}}",1,"(a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(16*Sqrt[2]*c^(7/2)*f) + (a^2*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*Cos[e + f*x])/(4*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*Cos[e + f*x])/(16*c^2*f*(c - c*Sin[e + f*x])^(3/2))","A",6,5,28,0.1786,1,"{2736, 2680, 2650, 2649, 206}"
306,1,190,0,0.3032949,"\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(9/2),x]","\frac{3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{256 \sqrt{2} c^{9/2} f}+\frac{a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}","\frac{3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{256 \sqrt{2} c^{9/2} f}+\frac{a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}",1,"(3*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(256*Sqrt[2]*c^(9/2)*f) + (a^2*c*Cos[e + f*x]^3)/(4*f*(c - c*Sin[e + f*x])^(11/2)) - (a^2*Cos[e + f*x])/(8*c*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*Cos[e + f*x])/(64*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (3*a^2*Cos[e + f*x])/(256*c^3*f*(c - c*Sin[e + f*x])^(3/2))","A",7,5,28,0.1786,1,"{2736, 2680, 2650, 2649, 206}"
307,1,145,0,0.3302161,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2),x]","\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}","\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}",1,"(256*a^3*c^7*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*c^6*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^(5/2)) + (24*a^3*c^5*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(13*f*Sqrt[c - c*Sin[e + f*x]])","A",5,3,28,0.1071,1,"{2736, 2674, 2673}"
308,1,109,0,0.2682879,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2),x]","\frac{2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a^3 c^5 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}","\frac{2 a^3 c^4 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a^3 c^5 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}",1,"(64*a^3*c^6*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^(7/2)) + (16*a^3*c^5*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2))","A",4,3,28,0.1071,1,"{2736, 2674, 2673}"
309,1,73,0,0.2028572,"\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2),x]","\frac{2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}+\frac{8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}","\frac{2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}+\frac{8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}",1,"(8*a^3*c^5*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^(5/2))","A",3,3,28,0.1071,1,"{2736, 2674, 2673}"
310,1,36,0,0.1271291,"\int (a+a \sin (e+f x))^3 \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a^3 c^4 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}","\frac{2 a^3 c^4 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}",1,"(2*a^3*c^4*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^(7/2))","A",2,2,28,0.07143,1,"{2736, 2673}"
311,1,151,0,0.3158708,"\int \frac{(a+a \sin (e+f x))^3}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^3/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{8 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}","-\frac{2 a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{8 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{c} f}",1,"(8*Sqrt[2]*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^3*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2)) - (4*a^3*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (8*a^3*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])","A",6,4,28,0.1429,1,"{2736, 2679, 2649, 206}"
312,1,150,0,0.3204583,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}-\frac{10 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{c^{3/2} f}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}","\frac{a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}-\frac{10 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{c^{3/2} f}+\frac{5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{10 a^3 \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}",1,"(-10*Sqrt[2]*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f) + (a^3*c^2*Cos[e + f*x]^5)/(f*(c - c*Sin[e + f*x])^(7/2)) + (5*a^3*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) + (10*a^3*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])","A",6,5,28,0.1786,1,"{2736, 2680, 2679, 2649, 206}"
313,1,157,0,0.3252915,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^3 c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac{15 a^3 \cos (e+f x)}{4 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{2 \sqrt{2} c^{5/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}","\frac{a^3 c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac{15 a^3 \cos (e+f x)}{4 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{2 \sqrt{2} c^{5/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{5/2}}",1,"(15*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(2*Sqrt[2]*c^(5/2)*f) + (a^3*c^2*Cos[e + f*x]^5)/(2*f*(c - c*Sin[e + f*x])^(9/2)) - (5*a^3*Cos[e + f*x]^3)/(4*f*(c - c*Sin[e + f*x])^(5/2)) - (15*a^3*Cos[e + f*x])/(4*c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",6,5,28,0.1786,1,"{2736, 2680, 2679, 2649, 206}"
314,1,157,0,0.323023,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}+\frac{5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} c^{7/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}","\frac{a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}+\frac{5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} c^{7/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}",1,"(-5*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*c^(7/2)*f) + (a^3*c^2*Cos[e + f*x]^5)/(3*f*(c - c*Sin[e + f*x])^(11/2)) - (5*a^3*Cos[e + f*x]^3)/(12*f*(c - c*Sin[e + f*x])^(7/2)) + (5*a^3*Cos[e + f*x])/(8*c^2*f*(c - c*Sin[e + f*x])^(3/2))","A",6,4,28,0.1429,1,"{2736, 2680, 2649, 206}"
315,1,191,0,0.3532278,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{128 \sqrt{2} c^{9/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}","\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{128 \sqrt{2} c^{9/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}",1,"(-5*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*c^(9/2)*f) + (a^3*c^2*Cos[e + f*x]^5)/(4*f*(c - c*Sin[e + f*x])^(13/2)) - (5*a^3*Cos[e + f*x]^3)/(24*f*(c - c*Sin[e + f*x])^(9/2)) + (5*a^3*Cos[e + f*x])/(32*c^2*f*(c - c*Sin[e + f*x])^(5/2)) - (5*a^3*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(3/2))","A",7,5,28,0.1786,1,"{2736, 2680, 2650, 2649, 206}"
316,1,225,0,0.3876115,"\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(11/2),x]","\frac{a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac{3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac{a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac{3 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{512 \sqrt{2} c^{11/2} f}-\frac{a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}","\frac{a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac{3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac{a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac{3 a^3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{512 \sqrt{2} c^{11/2} f}-\frac{a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}",1,"(-3*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(512*Sqrt[2]*c^(11/2)*f) + (a^3*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(15/2)) - (a^3*Cos[e + f*x]^3)/(8*f*(c - c*Sin[e + f*x])^(11/2)) + (a^3*Cos[e + f*x])/(16*c^2*f*(c - c*Sin[e + f*x])^(7/2)) - (a^3*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a^3*Cos[e + f*x])/(512*c^4*f*(c - c*Sin[e + f*x])^(3/2))","A",8,5,28,0.1786,1,"{2736, 2680, 2650, 2649, 206}"
317,1,132,0,0.344804,"\int \frac{(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x]),x]","\frac{64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac{256 c^3 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{5 a f}+\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac{8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}","\frac{64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}-\frac{256 c^3 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{5 a f}+\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f}+\frac{8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}",1,"(-256*c^3*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(5*a*f) + (64*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(5*a*f) + (8*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(5*a*f) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(5*a*f)","A",5,3,28,0.1071,1,"{2736, 2674, 2673}"
318,1,98,0,0.2694961,"\int \frac{(c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x]),x]","-\frac{64 c^2 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a f}+\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac{16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}","-\frac{64 c^2 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{3 a f}+\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac{16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}",1,"(-64*c^2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a*f) + (16*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*a*f) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f)","A",4,3,28,0.1071,1,"{2736, 2674, 2673}"
319,1,60,0,0.2038949,"\int \frac{(c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx","Int[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x]),x]","\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{a f}-\frac{8 c \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a f}","\frac{2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{a f}-\frac{8 c \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a f}",1,"(-8*c*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f)","A",3,3,28,0.1071,1,"{2736, 2674, 2673}"
320,1,29,0,0.1275099,"\int \frac{\sqrt{c-c \sin (e+f x)}}{a+a \sin (e+f x)} \, dx","Int[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x]),x]","-\frac{2 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a f}","-\frac{2 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a f}",1,"(-2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f)","A",2,2,28,0.07143,1,"{2736, 2673}"
321,1,83,0,0.1587306,"\int \frac{1}{(a+a \sin (e+f x)) \sqrt{c-c \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{2} a \sqrt{c} f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a c f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{\sqrt{2} a \sqrt{c} f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{a c f}",1,"ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(Sqrt[2]*a*Sqrt[c]*f) - (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*c*f)","A",4,4,28,0.1429,1,"{2736, 2675, 2649, 206}"
322,1,117,0,0.1924039,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} a c^{3/2} f}+\frac{3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{\sec (e+f x)}{a c f \sqrt{c-c \sin (e+f x)}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} a c^{3/2} f}+\frac{3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{\sec (e+f x)}{a c f \sqrt{c-c \sin (e+f x)}}",1,"(3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*a*c^(3/2)*f) + (3*Cos[e + f*x])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - Sec[e + f*x]/(a*c*f*Sqrt[c - c*Sin[e + f*x]])","A",5,5,28,0.1786,1,"{2736, 2687, 2650, 2649, 206}"
323,1,156,0,0.2668921,"\int \frac{1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{32 \sqrt{2} a c^{5/2} f}+\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}","-\frac{5 \sec (e+f x)}{8 a c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{15 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{32 \sqrt{2} a c^{5/2} f}+\frac{15 \cos (e+f x)}{32 a c f (c-c \sin (e+f x))^{3/2}}+\frac{\sec (e+f x)}{4 a c f (c-c \sin (e+f x))^{3/2}}",1,"(15*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(32*Sqrt[2]*a*c^(5/2)*f) + (15*Cos[e + f*x])/(32*a*c*f*(c - c*Sin[e + f*x])^(3/2)) + Sec[e + f*x]/(4*a*c*f*(c - c*Sin[e + f*x])^(3/2)) - (5*Sec[e + f*x])/(8*a*c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",6,6,28,0.2143,1,"{2736, 2681, 2687, 2650, 2649, 206}"
324,1,176,0,0.4199477,"\int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^2,x]","-\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}","-\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}",1,"(4096*c^3*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^2*f) - (1024*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(5*a^2*f) + (128*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(5*a^2*f) + (32*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(15*a^2*f) + (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(11/2))/(5*a^2*c*f)","A",6,3,28,0.1071,1,"{2736, 2674, 2673}"
325,1,136,0,0.3324573,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^2,x]","\frac{256 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 a^2 c f}+\frac{8 \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{a^2 f}-\frac{64 c \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 f}","\frac{256 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 a^2 c f}+\frac{8 \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{a^2 f}-\frac{64 c \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 f}",1,"(256*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - (64*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(a^2*f) + (8*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(a^2*f) + (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(3*a^2*c*f)","A",5,3,28,0.1071,1,"{2736, 2674, 2673}"
326,1,100,0,0.26357,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^2,x]","\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{a^2 c f}-\frac{16 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 f}+\frac{64 c \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}","\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{a^2 c f}-\frac{16 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 f}+\frac{64 c \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}",1,"(64*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - (16*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(a^2*f) + (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(a^2*c*f)","A",4,3,28,0.1071,1,"{2736, 2674, 2673}"
327,1,68,0,0.1956539,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx","Int[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^2,x]","\frac{8 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}-\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 c f}","\frac{8 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 f}-\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{a^2 c f}",1,"(8*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(a^2*c*f)","A",3,3,28,0.1071,1,"{2736, 2674, 2673}"
328,1,36,0,0.1362579,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx","Int[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^2,x]","-\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f}","-\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f}",1,"(-2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*c*f)","A",2,2,28,0.07143,1,"{2736, 2673}"
329,1,124,0,0.2241817,"\int \frac{1}{(a+a \sin (e+f x))^2 \sqrt{c-c \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c^2 f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{2 a^2 c f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{2 \sqrt{2} a^2 \sqrt{c} f}","-\frac{\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c^2 f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{2 a^2 c f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{2 \sqrt{2} a^2 \sqrt{c} f}",1,"ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(2*Sqrt[2]*a^2*Sqrt[c]*f) - (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(2*a^2*c*f) - (Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*c^2*f)","A",5,4,28,0.1429,1,"{2736, 2675, 2649, 206}"
330,1,155,0,0.249886,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{\sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 c^2 f}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} a^2 c^{3/2} f}+\frac{5 \cos (e+f x)}{8 a^2 f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{6 a^2 c f \sqrt{c-c \sin (e+f x)}}","-\frac{\sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{3 a^2 c^2 f}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{8 \sqrt{2} a^2 c^{3/2} f}+\frac{5 \cos (e+f x)}{8 a^2 f (c-c \sin (e+f x))^{3/2}}-\frac{5 \sec (e+f x)}{6 a^2 c f \sqrt{c-c \sin (e+f x)}}",1,"(5*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*a^2*c^(3/2)*f) + (5*Cos[e + f*x])/(8*a^2*f*(c - c*Sin[e + f*x])^(3/2)) - (5*Sec[e + f*x])/(6*a^2*c*f*Sqrt[c - c*Sin[e + f*x]]) - (Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*c^2*f)","A",6,6,28,0.2143,1,"{2736, 2675, 2687, 2650, 2649, 206}"
331,1,192,0,0.3307455,"\int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{35 \sec (e+f x)}{48 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{35 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{64 \sqrt{2} a^2 c^{5/2} f}+\frac{35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac{7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}","-\frac{\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{35 \sec (e+f x)}{48 a^2 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{35 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{64 \sqrt{2} a^2 c^{5/2} f}+\frac{35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac{7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}",1,"(35*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(64*Sqrt[2]*a^2*c^(5/2)*f) + (35*Cos[e + f*x])/(64*a^2*c*f*(c - c*Sin[e + f*x])^(3/2)) + (7*Sec[e + f*x])/(24*a^2*c*f*(c - c*Sin[e + f*x])^(3/2)) - (35*Sec[e + f*x])/(48*a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - Sec[e + f*x]^3/(3*a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",7,6,28,0.2143,1,"{2736, 2687, 2681, 2650, 2649, 206}"
332,1,174,0,0.4031166,"\int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^3,x]","\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}-\frac{4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}+\frac{32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}-\frac{128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac{1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 f}","\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}-\frac{4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}+\frac{32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}-\frac{128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac{1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 f}",1,"(-4096*c^2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(15*a^3*f) + (1024*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*a^3*f) - (128*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(a^3*f) + (32*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/(3*a^3*c*f) + (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(13/2))/(3*a^3*c^2*f)","A",6,3,28,0.1071,1,"{2736, 2674, 2673}"
333,1,134,0,0.3302416,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^3,x]","\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{a^3 c^2 f}-\frac{24 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c f}+\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{a^3 f}-\frac{256 c \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}","\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{a^3 c^2 f}-\frac{24 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c f}+\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{a^3 f}-\frac{256 c \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}",1,"(-256*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*f) + (64*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(a^3*f) - (24*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(a^3*c*f) + (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/(a^3*c^2*f)","A",5,3,28,0.1071,1,"{2736, 2674, 2673}"
334,1,104,0,0.2692067,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^3,x]","-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}+\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}","-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}+\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}",1,"(-64*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(15*a^3*f) + (16*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*a^3*c*f) - (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(a^3*c^2*f)","A",4,3,28,0.1071,1,"{2736, 2674, 2673}"
335,1,73,0,0.195981,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^3} \, dx","Int[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^3,x]","\frac{8 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 c f}-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c^2 f}","\frac{8 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 c f}-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c^2 f}",1,"(8*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(15*a^3*c*f) - (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*a^3*c^2*f)","A",3,3,28,0.1071,1,"{2736, 2674, 2673}"
336,1,36,0,0.1224187,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx","Int[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^3,x]","-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^2 f}","-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^2 f}",1,"(-2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*c^2*f)","A",2,2,28,0.07143,1,"{2736, 2673}"
337,1,160,0,0.294272,"\int \frac{1}{(a+a \sin (e+f x))^3 \sqrt{c-c \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^3 f}-\frac{\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{6 a^3 c^2 f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{4 a^3 c f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} a^3 \sqrt{c} f}","-\frac{\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 c^3 f}-\frac{\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{6 a^3 c^2 f}-\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)}}{4 a^3 c f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{4 \sqrt{2} a^3 \sqrt{c} f}",1,"ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(4*Sqrt[2]*a^3*Sqrt[c]*f) - (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(4*a^3*c*f) - (Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(6*a^3*c^2*f) - (Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*c^3*f)","A",6,4,28,0.1429,1,"{2736, 2675, 2649, 206}"
338,1,191,0,0.3281423,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac{7 \sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{16 \sqrt{2} a^3 c^{3/2} f}+\frac{7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac{7 \sec (e+f x)}{12 a^3 c f \sqrt{c-c \sin (e+f x)}}","-\frac{\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac{7 \sec ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{16 \sqrt{2} a^3 c^{3/2} f}+\frac{7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac{7 \sec (e+f x)}{12 a^3 c f \sqrt{c-c \sin (e+f x)}}",1,"(7*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(16*Sqrt[2]*a^3*c^(3/2)*f) + (7*Cos[e + f*x])/(16*a^3*f*(c - c*Sin[e + f*x])^(3/2)) - (7*Sec[e + f*x])/(12*a^3*c*f*Sqrt[c - c*Sin[e + f*x]]) - (7*Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(30*a^3*c^2*f) - (Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(5*a^3*c^3*f)","A",7,6,28,0.2143,1,"{2736, 2675, 2687, 2650, 2649, 206}"
339,1,228,0,0.4074212,"\int \frac{1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{\sec ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{5 a^3 c^3 f}-\frac{3 \sec ^3(e+f x)}{10 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{21 \sec (e+f x)}{32 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{63 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{128 \sqrt{2} a^3 c^{5/2} f}+\frac{63 \cos (e+f x)}{128 a^3 c f (c-c \sin (e+f x))^{3/2}}+\frac{21 \sec (e+f x)}{80 a^3 c f (c-c \sin (e+f x))^{3/2}}","-\frac{\sec ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{5 a^3 c^3 f}-\frac{3 \sec ^3(e+f x)}{10 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{21 \sec (e+f x)}{32 a^3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{63 \tanh ^{-1}\left(\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right)}{128 \sqrt{2} a^3 c^{5/2} f}+\frac{63 \cos (e+f x)}{128 a^3 c f (c-c \sin (e+f x))^{3/2}}+\frac{21 \sec (e+f x)}{80 a^3 c f (c-c \sin (e+f x))^{3/2}}",1,"(63*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*a^3*c^(5/2)*f) + (63*Cos[e + f*x])/(128*a^3*c*f*(c - c*Sin[e + f*x])^(3/2)) + (21*Sec[e + f*x])/(80*a^3*c*f*(c - c*Sin[e + f*x])^(3/2)) - (21*Sec[e + f*x])/(32*a^3*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - (3*Sec[e + f*x]^3)/(10*a^3*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - (Sec[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(5*a^3*c^3*f)","A",8,7,28,0.2500,1,"{2736, 2675, 2687, 2681, 2650, 2649, 206}"
340,1,43,0,0.0829605,"\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}}","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*Sqrt[a + a*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
341,1,43,0,0.0820688,"\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
342,1,43,0,0.0805129,"\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}","-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
343,1,41,0,0.0730127,"\int \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}","-\frac{a \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}",1,"-((a*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]))","A",1,1,30,0.03333,1,"{2738}"
344,1,52,0,0.1009856,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","-\frac{a \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-((a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))","A",3,3,30,0.1000,1,"{2737, 2667, 31}"
345,1,40,0,0.0823106,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}","\frac{a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}",1,"(a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))","A",1,1,30,0.03333,1,"{2738}"
346,1,43,0,0.0849277,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a \cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}","\frac{a \cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"(a*Cos[e + f*x])/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))","A",1,1,30,0.03333,1,"{2738}"
347,1,43,0,0.08369,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}","\frac{a \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}",1,"(a*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))","A",1,1,30,0.03333,1,"{2738}"
348,1,89,0,0.1879075,"\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 f}","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 f}",1,"-(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(10*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(5*f)","A",2,2,30,0.06667,1,"{2740, 2738}"
349,1,89,0,0.1793898,"\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 f}","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 f}",1,"-(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(6*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(4*f)","A",2,2,30,0.06667,1,"{2740, 2738}"
350,1,89,0,0.1784793,"\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}",1,"-(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(3*f)","A",2,2,30,0.06667,1,"{2740, 2738}"
351,1,43,0,0.0818158,"\int (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
352,1,96,0,0.1916905,"\int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}","-\frac{2 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}",1,"(-2*a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]])","A",4,4,30,0.1333,1,"{2740, 2737, 2667, 31}"
353,1,97,0,0.1952006,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}","\frac{a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}",1,"(a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,30,0.1333,1,"{2739, 2737, 2667, 31}"
354,1,42,0,0.0909264,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(5/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 f (c-c \sin (e+f x))^{5/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 f (c-c \sin (e+f x))^{5/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*f*(c - c*Sin[e + f*x])^(5/2))","A",1,1,30,0.03333,1,"{2742}"
355,1,88,0,0.1844312,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(7/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{7/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{7/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(6*f*(c - c*Sin[e + f*x])^(7/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(24*c*f*(c - c*Sin[e + f*x])^(5/2))","A",2,2,30,0.06667,1,"{2743, 2742}"
356,1,92,0,0.177274,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x)}{12 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}","\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x)}{12 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}",1,"(a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*Cos[e + f*x])/(12*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))","A",2,2,30,0.06667,1,"{2739, 2738}"
357,1,92,0,0.1762021,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(11/2),x]","\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 \cos (e+f x)}{20 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}","\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac{a^2 \cos (e+f x)}{20 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}",1,"(a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*f*(c - c*Sin[e + f*x])^(11/2)) - (a^2*Cos[e + f*x])/(20*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))","A",2,2,30,0.06667,1,"{2739, 2738}"
358,1,134,0,0.2701315,"\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}","-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 f}-\frac{a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 f}",1,"-(a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(15*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(6*f)","A",3,2,30,0.06667,1,"{2740, 2738}"
359,1,134,0,0.2712181,"\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}","-\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 f}",1,"(-2*a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(5*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(5*f)","A",3,2,30,0.06667,1,"{2740, 2738}"
360,1,89,0,0.1715898,"\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 f}","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 f}",1,"(c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*f*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(4*f)","A",2,2,30,0.06667,1,"{2740, 2738}"
361,1,43,0,0.0802235,"\int (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
362,1,141,0,0.2791053,"\int \frac{(a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{4 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}","-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{4 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}",1,"(-4*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]])","A",5,4,30,0.1333,1,"{2740, 2737, 2667, 31}"
363,1,144,0,0.2903304,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(3/2),x]","\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}","\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]])","A",5,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
364,1,147,0,0.2929866,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}","-\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*(c - c*Sin[e + f*x])^(3/2)) - (a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,4,30,0.1333,1,"{2739, 2737, 2667, 31}"
365,1,42,0,0.0920967,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(7/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f (c-c \sin (e+f x))^{7/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*f*(c - c*Sin[e + f*x])^(7/2))","A",1,1,30,0.03333,1,"{2742}"
366,1,88,0,0.1925584,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(9/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 c f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{9/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 c f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{9/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(48*c*f*(c - c*Sin[e + f*x])^(7/2))","A",2,2,30,0.06667,1,"{2743, 2742}"
367,1,133,0,0.2826068,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(11/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{240 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{240 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(40*c*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(240*c^2*f*(c - c*Sin[e + f*x])^(7/2))","A",3,2,30,0.06667,1,"{2743, 2742}"
368,1,140,0,0.2726277,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(13/2),x]","\frac{a^3 \cos (e+f x)}{60 c^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}","\frac{a^3 \cos (e+f x)}{60 c^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(6*f*(c - c*Sin[e + f*x])^(13/2)) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*c*f*(c - c*Sin[e + f*x])^(11/2)) + (a^3*Cos[e + f*x])/(60*c^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))","A",3,2,30,0.06667,1,"{2739, 2738}"
369,1,179,0,0.3655603,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2),x]","-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{28 f}-\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{14 f}-\frac{a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{8 f}","-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{28 f}-\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{14 f}-\frac{a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{8 f}",1,"-(a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(35*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(14*f) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(28*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(8*f)","A",4,2,30,0.06667,1,"{2740, 2738}"
370,1,179,0,0.3663978,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}","-\frac{2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}",1,"(-2*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(35*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(35*f) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f)","A",4,2,30,0.06667,1,"{2740, 2738}"
371,1,134,0,0.263819,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{2 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}","\frac{2 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}",1,"(c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(15*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(6*f)","A",3,2,30,0.06667,1,"{2740, 2738}"
372,1,89,0,0.1699824,"\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f}","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f}",1,"(c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f)","A",2,2,30,0.06667,1,"{2740, 2738}"
373,1,43,0,0.0815774,"\int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}","\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
374,1,184,0,0.376197,"\int \frac{(a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f \sqrt{c-c \sin (e+f x)}}-\frac{8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}","-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f \sqrt{c-c \sin (e+f x)}}-\frac{8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt{c-c \sin (e+f x)}}",1,"(-8*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]])","A",6,4,30,0.1333,1,"{2740, 2737, 2667, 31}"
375,1,192,0,0.3955155,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(3/2),x]","\frac{6 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{12 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}","\frac{6 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{12 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(f*(c - c*Sin[e + f*x])^(3/2)) + (12*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) + (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]])","A",6,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
376,1,195,0,0.4016645,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{3 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{6 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}","-\frac{3 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{6 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{3/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(3/2)) - (6*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",6,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
377,1,193,0,0.4075504,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{5/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}","\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{5/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*(c - c*Sin[e + f*x])^(7/2)) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) + (a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,4,30,0.1333,1,"{2739, 2737, 2667, 31}"
378,1,42,0,0.093497,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(9/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 f (c-c \sin (e+f x))^{9/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8*f*(c - c*Sin[e + f*x])^(9/2))","A",1,1,30,0.03333,1,"{2742}"
379,1,88,0,0.1862928,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(11/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*c*f*(c - c*Sin[e + f*x])^(9/2))","A",2,2,30,0.06667,1,"{2743, 2742}"
380,1,133,0,0.2941355,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(13/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{480 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{60 c f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(12*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(60*c*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(480*c^2*f*(c - c*Sin[e + f*x])^(9/2))","A",3,2,30,0.06667,1,"{2743, 2742}"
381,1,178,0,0.3804353,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(280*c^2*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2240*c^3*f*(c - c*Sin[e + f*x])^(9/2))","A",4,2,30,0.06667,1,"{2743, 2742}"
382,1,188,0,0.3841049,"\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx","Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(17/2),x]","-\frac{a^4 \cos (e+f x)}{280 c^3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}+\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}","-\frac{a^4 \cos (e+f x)}{280 c^3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}+\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}",1,"(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(17/2)) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(56*c^2*f*(c - c*Sin[e + f*x])^(13/2)) - (a^4*Cos[e + f*x])/(280*c^3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(11/2))","A",4,2,30,0.06667,1,"{2739, 2738}"
383,1,139,0,0.2808401,"\int \frac{(c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c - c*Sin[e + f*x])^(5/2)/Sqrt[a + a*Sin[e + f*x]],x]","\frac{2 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}+\frac{4 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}","\frac{2 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}+\frac{4 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}",1,"(4*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]) + (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])","A",5,4,30,0.1333,1,"{2740, 2737, 2667, 31}"
384,1,93,0,0.1871892,"\int \frac{(c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c - c*Sin[e + f*x])^(3/2)/Sqrt[a + a*Sin[e + f*x]],x]","\frac{2 c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}","\frac{2 c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}",1,"(2*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,30,0.1333,1,"{2740, 2737, 2667, 31}"
385,1,49,0,0.1004713,"\int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Sqrt[c - c*Sin[e + f*x]]/Sqrt[a + a*Sin[e + f*x]],x]","\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",3,3,30,0.1000,1,"{2737, 2667, 31}"
386,1,46,0,0.086145,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",2,2,30,0.06667,1,"{2741, 3770}"
387,1,95,0,0.1753755,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x)}{2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"Cos[e + f*x]/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",3,3,30,0.1000,1,"{2743, 2741, 3770}"
388,1,140,0,0.2687988,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x)}{4 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x)}{4 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x)}{4 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x)}{4 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"Cos[e + f*x]/(4*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + Cos[e + f*x]/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(4*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,3,30,0.1000,1,"{2743, 2741, 3770}"
389,1,191,0,0.376912,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{6 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}-\frac{12 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{f (a \sin (e+f x)+a)^{3/2}}","-\frac{6 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}-\frac{12 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{f (a \sin (e+f x)+a)^{3/2}}",1,"(-12*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (6*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - (3*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(f*(a + a*Sin[e + f*x])^(3/2))","A",6,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
390,1,143,0,0.2877499,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c - c*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{2 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{4 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)^{3/2}}","-\frac{2 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}-\frac{4 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)^{3/2}}",1,"(-4*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f*x])^(3/2))","A",5,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
391,1,97,0,0.1972585,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}","-\frac{c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}",1,"-((c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x])^(3/2))","A",4,4,30,0.1333,1,"{2739, 2737, 2667, 31}"
392,1,41,0,0.0854564,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{c \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}","-\frac{c \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-((c*Cos[e + f*x])/(f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]))","A",1,1,30,0.03333,1,"{2738}"
393,1,95,0,0.1762377,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",3,3,30,0.1000,1,"{2743, 2741, 3770}"
394,1,143,0,0.2781763,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + Cos[e + f*x]/(2*a*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,3,30,0.1000,1,"{2743, 2741, 3770}"
395,1,191,0,0.3744535,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 \cos (e+f x)}{8 a c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 \cos (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}","\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 \cos (e+f x)}{8 a c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 \cos (e+f x)}{8 a f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}",1,"-Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,3,30,0.1000,1,"{2743, 2741, 3770}"
396,1,237,0,0.4927471,"\int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{24 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a \sin (e+f x)+a)^{5/2}}","\frac{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{24 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a \sin (e+f x)+a)^{5/2}}",1,"(24*c^5*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (12*c^4*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*c^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(a*f*(a + a*Sin[e + f*x])^(3/2)) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(2*f*(a + a*Sin[e + f*x])^(5/2))","A",7,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
397,1,193,0,0.3868202,"\int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{3 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{6 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a \sin (e+f x)+a)^{5/2}}","\frac{3 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{6 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a \sin (e+f x)+a)^{5/2}}",1,"(6*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (3*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*(a + a*Sin[e + f*x])^(3/2)) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(2*f*(a + a*Sin[e + f*x])^(5/2))","A",6,5,30,0.1667,1,"{2739, 2740, 2737, 2667, 31}"
398,1,143,0,0.3028239,"\int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c - c*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(5/2),x]","\frac{c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{5/2}}","\frac{c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{5/2}}",1,"(c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*(a + a*Sin[e + f*x])^(3/2)) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*(a + a*Sin[e + f*x])^(5/2))","A",5,4,30,0.1333,1,"{2739, 2737, 2667, 31}"
399,1,42,0,0.0924086,"\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c - c*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"-(Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))","A",1,1,30,0.03333,1,"{2742}"
400,1,43,0,0.0850884,"\int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[Sqrt[c - c*Sin[e + f*x]]/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{c \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}","-\frac{c \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}",1,"-(c*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1,1,30,0.03333,1,"{2738}"
401,1,140,0,0.2707805,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}",1,"-Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - Cos[e + f*x]/(4*a*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,3,30,0.1000,1,"{2743, 2741, 3770}"
402,1,188,0,0.3832011,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{3 \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 \cos (e+f x)}{8 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}","\frac{3 \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 \cos (e+f x)}{8 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}",1,"-Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)) - (3*Cos[e + f*x])/(8*a*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + (3*Cos[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,3,30,0.1000,1,"{2743, 2741, 3770}"
403,1,236,0,0.478639,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 \cos (e+f x)}{8 a^2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{2 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}","\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 \cos (e+f x)}{8 a^2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{3 \cos (e+f x)}{8 a^2 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{2 a f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}",1,"-Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)) - Cos[e + f*x]/(2*a*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,3,30,0.1000,1,"{2743, 2741, 3770}"
404,1,110,0,0.1547171,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c 2^{n+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{\frac{1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (1-2 n);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}","\frac{c 2^{n+\frac{1}{2}} \cos (e+f x) (1-\sin (e+f x))^{\frac{1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (1-2 n);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}",1,"(2^(1/2 + n)*c*Cos[e + f*x]*Hypergeometric2F1[(1 + 2*m)/2, (1 - 2*n)/2, (3 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*(1 + 2*m))","A",4,4,26,0.1538,1,"{2745, 2689, 70, 69}"
405,1,86,0,0.1427806,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3,x]","-\frac{a^4 c^3 2^{m+\frac{1}{2}} \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{7}{2},\frac{1}{2}-m;\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f}","-\frac{a^4 c^3 2^{m+\frac{1}{2}} \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{7}{2},\frac{1}{2}-m;\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f}",1,"-(2^(1/2 + m)*a^4*c^3*Cos[e + f*x]^7*Hypergeometric2F1[7/2, 1/2 - m, 9/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-4 + m))/(7*f)","A",4,4,26,0.1538,1,"{2736, 2689, 70, 69}"
406,1,86,0,0.139956,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2,x]","-\frac{a^3 c^2 2^{m+\frac{1}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f}","-\frac{a^3 c^2 2^{m+\frac{1}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{5}{2},\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f}",1,"-(2^(1/2 + m)*a^3*c^2*Cos[e + f*x]^5*Hypergeometric2F1[5/2, 1/2 - m, 7/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(5*f)","A",4,4,26,0.1538,1,"{2736, 2689, 70, 69}"
407,1,84,0,0.1129764,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c 2^{m+\frac{1}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{3}{2},\frac{1}{2}-m;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f}","-\frac{a^2 c 2^{m+\frac{1}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{3}{2},\frac{1}{2}-m;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f}",1,"-(2^(1/2 + m)*a^2*c*Cos[e + f*x]^3*Hypergeometric2F1[3/2, 1/2 - m, 5/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-2 + m))/(3*f)","A",4,4,24,0.1667,1,"{2736, 2689, 70, 69}"
408,1,76,0,0.1327064,"\int \frac{(a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x]),x]","\frac{2^{m+\frac{1}{2}} \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{c f}","\frac{2^{m+\frac{1}{2}} \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{c f}",1,"(2^(1/2 + m)*Hypergeometric2F1[-1/2, 1/2 - m, 1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(c*f)","A",4,4,26,0.1538,1,"{2736, 2689, 70, 69}"
409,1,86,0,0.1316821,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^2,x]","\frac{2^{m+\frac{1}{2}} \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a c^2 f}","\frac{2^{m+\frac{1}{2}} \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{1}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a c^2 f}",1,"(2^(1/2 + m)*Hypergeometric2F1[-3/2, 1/2 - m, -1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*c^2*f)","A",4,4,26,0.1538,1,"{2736, 2689, 70, 69}"
410,1,86,0,0.1333669,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^3,x]","\frac{2^{m+\frac{1}{2}} \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{1}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a^2 c^3 f}","\frac{2^{m+\frac{1}{2}} \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{1}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a^2 c^3 f}",1,"(2^(1/2 + m)*Hypergeometric2F1[-5/2, 1/2 - m, -3/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]^5*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(2 + m))/(5*a^2*c^3*f)","A",4,4,26,0.1538,1,"{2736, 2689, 70, 69}"
411,1,160,0,0.2518379,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]","\frac{16 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f \left(4 m^2+16 m+15\right)}+\frac{64 c^3 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}","\frac{16 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f \left(4 m^2+16 m+15\right)}+\frac{64 c^3 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}",1,"(64*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(15 + 16*m + 4*m^2)) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m))","A",3,2,28,0.07143,1,"{2740, 2738}"
412,1,100,0,0.149104,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2),x]","\frac{8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}","\frac{8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}",1,"(8*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m))","A",2,2,28,0.07143,1,"{2740, 2738}"
413,1,46,0,0.0677496,"\int (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}","\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",1,1,28,0.03571,1,"{2738}"
414,1,68,0,0.1338014,"\int \frac{(a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^m/Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",3,3,28,0.1071,1,"{2745, 2667, 68}"
415,1,74,0,0.1572113,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(3/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(2,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{2 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(2,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{2 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[2, 1/2 + m, 3/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m)/(2*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",3,3,28,0.1071,1,"{2745, 2667, 68}"
416,1,74,0,0.1599108,"\int \frac{(a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(5/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(3,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{4 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(3,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{4 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[3, 1/2 + m, 3/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m)/(4*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",3,3,28,0.1071,1,"{2745, 2667, 68}"
417,1,68,0,0.1285413,"\int \frac{(a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^m/Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",3,3,28,0.1071,1,"{2745, 2667, 68}"
418,1,68,0,0.1379284,"\int \frac{(c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Int[(c + c*Sin[e + f*x])^m/Sqrt[a - a*Sin[e + f*x]],x]","\frac{\cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{a-a \sin (e+f x)}}","\frac{\cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left(1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1) \sqrt{a-a \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1 + Sin[e + f*x])/2]*(c + c*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[a - a*Sin[e + f*x]])","A",3,3,28,0.1071,1,"{2745, 2667, 68}"
419,1,164,0,0.2234015,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m),x]","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^2 f (2 m+5) \left(4 m^2+8 m+3\right)}+\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)}","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c^2 f (2 m+5) \left(4 m^2+8 m+3\right)}+\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{c f \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3}}{f (2 m+5)}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(f*(5 + 2*m)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c*f*(15 + 16*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*(5 + 2*m)*(3 + 8*m + 4*m^2))","A",3,2,30,0.06667,1,"{2743, 2742}"
420,1,101,0,0.1347023,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c f \left(4 m^2+8 m+3\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{f (2 m+3)}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{c f \left(4 m^2+8 m+3\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2}}{f (2 m+3)}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c*f*(3 + 8*m + 4*m^2))","A",2,2,30,0.06667,1,"{2743, 2742}"
421,1,46,0,0.0669966,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))","A",1,1,30,0.03333,1,"{2742}"
422,1,112,0,0.1576363,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","Int[(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^m,x]","\frac{c 2^{\frac{1}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}","\frac{c 2^{\frac{1}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}",1,"(2^(1/2 - m)*c*Cos[e + f*x]*Hypergeometric2F1[(1 + 2*m)/2, (1 + 2*m)/2, (3 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))","A",4,4,28,0.1429,1,"{2745, 2689, 70, 69}"
423,1,114,0,0.1860053,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m),x]","\frac{c^2 2^{\frac{3}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m-1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}","\frac{c^2 2^{\frac{3}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m-1),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}",1,"(2^(3/2 - m)*c^2*Cos[e + f*x]*Hypergeometric2F1[(-1 + 2*m)/2, (1 + 2*m)/2, (3 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))","A",4,4,30,0.1333,1,"{2745, 2689, 70, 69}"
424,1,114,0,0.1822758,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m} \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m),x]","\frac{c^3 2^{\frac{5}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}","\frac{c^3 2^{\frac{5}{2}-m} \cos (e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+1)}",1,"(2^(5/2 - m)*c^3*Cos[e + f*x]*Hypergeometric2F1[(-3 + 2*m)/2, (1 + 2*m)/2, (3 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))","A",4,4,30,0.1333,1,"{2745, 2689, 70, 69}"
425,1,227,0,0.281751,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^4 \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^4,x]","-\frac{a \left(112 c^2 d^2+95 c^3 d+12 c^4+80 c d^3+16 d^4\right) \cos (e+f x)}{30 f}-\frac{a \left(12 c^2+35 c d+16 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 f}-\frac{a d \left(130 c^2 d+24 c^3+116 c d^2+45 d^3\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} a x \left(24 c^2 d^2+16 c^3 d+8 c^4+12 c d^3+3 d^4\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^4}{5 f}-\frac{a (4 c+5 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 f}","-\frac{a \left(112 c^2 d^2+95 c^3 d+12 c^4+80 c d^3+16 d^4\right) \cos (e+f x)}{30 f}-\frac{a \left(12 c^2+35 c d+16 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 f}-\frac{a d \left(130 c^2 d+24 c^3+116 c d^2+45 d^3\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} a x \left(24 c^2 d^2+16 c^3 d+8 c^4+12 c d^3+3 d^4\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^4}{5 f}-\frac{a (4 c+5 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 f}",1,"(a*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*x)/8 - (a*(12*c^4 + 95*c^3*d + 112*c^2*d^2 + 80*c*d^3 + 16*d^4)*Cos[e + f*x])/(30*f) - (a*d*(24*c^3 + 130*c^2*d + 116*c*d^2 + 45*d^3)*Cos[e + f*x]*Sin[e + f*x])/(120*f) - (a*(12*c^2 + 35*c*d + 16*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*f) - (a*(4*c + 5*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*f) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*f)","A",4,2,23,0.08696,1,"{2753, 2734}"
426,1,162,0,0.1894305,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","-\frac{a \left(16 c^2 d+3 c^3+12 c d^2+4 d^3\right) \cos (e+f x)}{6 f}-\frac{a d \left(6 c^2+20 c d+9 d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a x \left(12 c^2 d+8 c^3+12 c d^2+3 d^3\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}-\frac{a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}","-\frac{a \left(16 c^2 d+3 c^3+12 c d^2+4 d^3\right) \cos (e+f x)}{6 f}-\frac{a d \left(6 c^2+20 c d+9 d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a x \left(12 c^2 d+8 c^3+12 c d^2+3 d^3\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}-\frac{a (3 c+4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}",1,"(a*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*x)/8 - (a*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^3)*Cos[e + f*x])/(6*f) - (a*d*(6*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (a*(3*c + 4*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(12*f) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f)","A",3,2,23,0.08696,1,"{2753, 2734}"
427,1,99,0,0.0926097,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{2 a \left(c^2+3 c d+d^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 c^2+2 c d+d^2\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}-\frac{a d (2 c+3 d) \sin (e+f x) \cos (e+f x)}{6 f}","-\frac{2 a \left(c^2+3 c d+d^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 c^2+2 c d+d^2\right)-\frac{a \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}-\frac{a d (2 c+3 d) \sin (e+f x) \cos (e+f x)}{6 f}",1,"(a*(2*c^2 + 2*c*d + d^2)*x)/2 - (2*a*(c^2 + 3*c*d + d^2)*Cos[e + f*x])/(3*f) - (a*d*(2*c + 3*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f)","A",2,2,23,0.08696,1,"{2753, 2734}"
428,1,48,0,0.023605,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{a (c+d) \cos (e+f x)}{f}+\frac{1}{2} a x (2 c+d)-\frac{a d \sin (e+f x) \cos (e+f x)}{2 f}","-\frac{a (c+d) \cos (e+f x)}{f}+\frac{1}{2} a x (2 c+d)-\frac{a d \sin (e+f x) \cos (e+f x)}{2 f}",1,"(a*(2*c + d)*x)/2 - (a*(c + d)*Cos[e + f*x])/f - (a*d*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,21,0.04762,1,"{2734}"
429,1,16,0,0.0082288,"\int (a+a \sin (e+f x)) \, dx","Int[a + a*Sin[e + f*x],x]","a x-\frac{a \cos (e+f x)}{f}","a x-\frac{a \cos (e+f x)}{f}",1,"a*x - (a*Cos[e + f*x])/f","A",2,1,10,0.1000,1,"{2638}"
430,1,63,0,0.0911328,"\int \frac{a+a \sin (e+f x)}{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x]),x]","\frac{a x}{d}-\frac{2 a (c-d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d f \sqrt{c^2-d^2}}","\frac{a x}{d}-\frac{2 a (c-d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d f \sqrt{c^2-d^2}}",1,"(a*x)/d - (2*a*(c - d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d*Sqrt[c^2 - d^2]*f)","A",4,4,23,0.1739,1,"{2735, 2660, 618, 204}"
431,1,83,0,0.091673,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^2,x]","\frac{2 a \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \sqrt{c^2-d^2}}-\frac{a \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}","\frac{2 a \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \sqrt{c^2-d^2}}-\frac{a \cos (e+f x)}{f (c+d) (c+d \sin (e+f x))}",1,"(2*a*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]*f) - (a*Cos[e + f*x])/((c + d)*f*(c + d*Sin[e + f*x]))","A",5,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
432,1,134,0,0.1842967,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^3,x]","\frac{a (2 c-d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \left(c^2-d^2\right)^{3/2}}-\frac{a (c-2 d) \cos (e+f x)}{2 f (c-d) (c+d)^2 (c+d \sin (e+f x))}-\frac{a \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}","\frac{a (2 c-d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \left(c^2-d^2\right)^{3/2}}-\frac{a (c-2 d) \cos (e+f x)}{2 f (c-d) (c+d)^2 (c+d \sin (e+f x))}-\frac{a \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2}",1,"(a*(2*c - d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(3/2)*f) - (a*Cos[e + f*x])/(2*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a*(c - 2*d)*Cos[e + f*x])/(2*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x]))","A",6,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
433,1,192,0,0.3349074,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^4,x]","\frac{a \left(2 c^2-2 c d+d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \left(c^2-d^2\right)^{5/2}}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac{a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac{a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^3}","\frac{a \left(2 c^2-2 c d+d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d) \left(c^2-d^2\right)^{5/2}}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac{a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac{a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^3}",1,"(a*(2*c^2 - 2*c*d + d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(5/2)*f) - (a*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a*(2*c - 3*d)*Cos[e + f*x])/(6*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (a*(c - 4*d)*(2*c - d)*Cos[e + f*x])/(6*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))","A",7,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
434,1,318,0,0.4625891,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4,x]","\frac{a^2 \left(-311 c^3 d^2-448 c^2 d^3-48 c^4 d+4 c^5-288 c d^4-64 d^5\right) \cos (e+f x)}{60 d f}+\frac{a^2 \left(4 c^2-48 c d-55 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 \left(-48 c^2 d+4 c^3-123 c d^2-64 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left(-438 c^2 d^2-96 c^3 d+8 c^4-464 c d^3-165 d^4\right) \sin (e+f x) \cos (e+f x)}{240 f}+\frac{1}{16} a^2 x \left(84 c^2 d^2+64 c^3 d+24 c^4+48 c d^3+11 d^4\right)-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}","\frac{a^2 \left(-311 c^3 d^2-448 c^2 d^3-48 c^4 d+4 c^5-288 c d^4-64 d^5\right) \cos (e+f x)}{60 d f}+\frac{a^2 \left(4 c^2-48 c d-55 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac{a^2 \left(-48 c^2 d+4 c^3-123 c d^2-64 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac{a^2 \left(-438 c^2 d^2-96 c^3 d+8 c^4-464 c d^3-165 d^4\right) \sin (e+f x) \cos (e+f x)}{240 f}+\frac{1}{16} a^2 x \left(84 c^2 d^2+64 c^3 d+24 c^4+48 c d^3+11 d^4\right)-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac{a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}",1,"(a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*x)/16 + (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c^2*d^3 - 288*c*d^4 - 64*d^5)*Cos[e + f*x])/(60*d*f) + (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d^4)*Cos[e + f*x]*Sin[e + f*x])/(240*f) + (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d*f) + (a^2*(4*c^2 - 48*c*d - 55*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d*f) + (a^2*(c - 12*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(6*d*f)","A",5,3,25,0.1200,1,"{2763, 2753, 2734}"
435,1,233,0,0.3085468,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3,x]","\frac{a^2 \left(-44 c^2 d^2-10 c^3 d+c^4-40 c d^3-12 d^4\right) \cos (e+f x)}{10 d f}+\frac{a^2 \left(c^2-10 c d-12 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{a^2 \left(-20 c^2 d+2 c^3-57 c d^2-30 d^3\right) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{3}{8} a^2 x (2 c+d) \left(2 c^2+3 c d+2 d^2\right)-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}","\frac{a^2 \left(-44 c^2 d^2-10 c^3 d+c^4-40 c d^3-12 d^4\right) \cos (e+f x)}{10 d f}+\frac{a^2 \left(c^2-10 c d-12 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{20 d f}+\frac{a^2 \left(-20 c^2 d+2 c^3-57 c d^2-30 d^3\right) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{3}{8} a^2 x (2 c+d) \left(2 c^2+3 c d+2 d^2\right)-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}+\frac{a^2 (c-10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}",1,"(3*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*x)/8 + (a^2*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4)*Cos[e + f*x])/(10*d*f) + (a^2*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3)*Cos[e + f*x]*Sin[e + f*x])/(40*f) + (a^2*(c^2 - 10*c*d - 12*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(20*d*f) + (a^2*(c - 10*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f)","A",4,3,25,0.1200,1,"{2763, 2753, 2734}"
436,1,156,0,0.2018967,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2,x]","-\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \cos (e+f x)}{6 f}-\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a^2 x \left(12 c^2+16 c d+7 d^2\right)-\frac{d (8 c-d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f}","-\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \cos (e+f x)}{6 f}-\frac{a^2 \left(12 c^2+16 c d+7 d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} a^2 x \left(12 c^2+16 c d+7 d^2\right)-\frac{d (8 c-d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f}",1,"(a^2*(12*c^2 + 16*c*d + 7*d^2)*x)/8 - (a^2*(12*c^2 + 16*c*d + 7*d^2)*Cos[e + f*x])/(6*f) - (a^2*(12*c^2 + 16*c*d + 7*d^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((8*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(12*f) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*a*f)","A",3,3,25,0.1200,1,"{2761, 2751, 2644}"
437,1,94,0,0.0623795,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x]),x]","-\frac{2 a^2 (3 c+2 d) \cos (e+f x)}{3 f}-\frac{a^2 (3 c+2 d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a^2 x (3 c+2 d)-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f}","-\frac{2 a^2 (3 c+2 d) \cos (e+f x)}{3 f}-\frac{a^2 (3 c+2 d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a^2 x (3 c+2 d)-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f}",1,"(a^2*(3*c + 2*d)*x)/2 - (2*a^2*(3*c + 2*d)*Cos[e + f*x])/(3*f) - (a^2*(3*c + 2*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*f)","A",2,2,23,0.08696,1,"{2751, 2644}"
438,1,45,0,0.0148743,"\int (a+a \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^2,x]","-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{3 a^2 x}{2}","-\frac{2 a^2 \cos (e+f x)}{f}-\frac{a^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{3 a^2 x}{2}",1,"(3*a^2*x)/2 - (2*a^2*Cos[e + f*x])/f - (a^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,12,0.08333,1,"{2644}"
439,1,92,0,0.2020154,"\int \frac{(a+a \sin (e+f x))^2}{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x]),x]","\frac{2 a^2 (c-d)^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \sqrt{c^2-d^2}}-\frac{a^2 x (c-2 d)}{d^2}-\frac{a^2 \cos (e+f x)}{d f}","\frac{2 a^2 (c-d)^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \sqrt{c^2-d^2}}-\frac{a^2 x (c-2 d)}{d^2}-\frac{a^2 \cos (e+f x)}{d f}",1,"-((a^2*(c - 2*d)*x)/d^2) + (2*a^2*(c - d)^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^2*Sqrt[c^2 - d^2]*f) - (a^2*Cos[e + f*x])/(d*f)","A",5,5,25,0.2000,1,"{2746, 2735, 2660, 618, 204}"
440,1,115,0,0.1823447,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2,x]","-\frac{2 a^2 (c-d) (c+2 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f (c+d) \sqrt{c^2-d^2}}+\frac{a^2 (c-d) \cos (e+f x)}{d f (c+d) (c+d \sin (e+f x))}+\frac{a^2 x}{d^2}","-\frac{2 a^2 (c-d)^2 (c+2 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \left(c^2-d^2\right)^{3/2}}+\frac{a^2 (c-d) \cos (e+f x)}{d f (c+d) (c+d \sin (e+f x))}+\frac{a^2 x}{d^2}",1,"(a^2*x)/d^2 - (2*a^2*(c - d)*(c + 2*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^2*(c + d)*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*(c + d*Sin[e + f*x]))","A",5,5,25,0.2000,1,"{2762, 2735, 2660, 618, 204}"
441,1,138,0,0.1819489,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3,x]","\frac{3 a^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d)^2 \sqrt{c^2-d^2}}-\frac{a^2 (c+4 d) \cos (e+f x)}{2 d f (c+d)^2 (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x)}{2 d f (c+d) (c+d \sin (e+f x))^2}","\frac{3 a^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d)^2 \sqrt{c^2-d^2}}-\frac{a^2 (c+4 d) \cos (e+f x)}{2 d f (c+d)^2 (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x)}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"(3*a^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)^2*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a^2*(c + 4*d)*Cos[e + f*x])/(2*d*(c + d)^2*f*(c + d*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2762, 2754, 12, 2660, 618, 204}"
442,1,207,0,0.3147237,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]","\frac{a^2 (3 c-2 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c-d) (c+d)^3 \sqrt{c^2-d^2}}-\frac{a^2 \left(c^2+6 c d-10 d^2\right) \cos (e+f x)}{6 d f (c-d) (c+d)^3 (c+d \sin (e+f x))}-\frac{a^2 (c+6 d) \cos (e+f x)}{6 d f (c+d)^2 (c+d \sin (e+f x))^2}+\frac{a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}","\frac{a^2 (3 c-2 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c-d) (c+d)^3 \sqrt{c^2-d^2}}-\frac{a^2 \left(c^2+6 c d-10 d^2\right) \cos (e+f x)}{6 d f (c-d) (c+d)^3 (c+d \sin (e+f x))}-\frac{a^2 (c+6 d) \cos (e+f x)}{6 d f (c+d)^2 (c+d \sin (e+f x))^2}+\frac{a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3}",1,"(a^2*(3*c - 2*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c - d)*(c + d)^3*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a^2*(c + 6*d)*Cos[e + f*x])/(6*d*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (a^2*(c^2 + 6*c*d - 10*d^2)*Cos[e + f*x])/(6*(c - d)*d*(c + d)^3*f*(c + d*Sin[e + f*x]))","A",7,6,25,0.2400,1,"{2762, 2754, 12, 2660, 618, 204}"
443,1,286,0,0.506972,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^5} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^5,x]","\frac{a^2 \left(12 c^2-16 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{4 f (c-d)^2 (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^2 \left(16 c^2 d+2 c^3-59 c d^2+32 d^3\right) \cos (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sin (e+f x))}-\frac{a^2 \left(2 c^2+16 c d-21 d^2\right) \cos (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sin (e+f x))^2}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{a^2 (c-d) \cos (e+f x)}{4 d f (c+d) (c+d \sin (e+f x))^4}","\frac{a^2 \left(12 c^2-16 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{4 f (c-d)^2 (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^2 \left(16 c^2 d+2 c^3-59 c d^2+32 d^3\right) \cos (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sin (e+f x))}-\frac{a^2 \left(2 c^2+16 c d-21 d^2\right) \cos (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sin (e+f x))^2}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{a^2 (c-d) \cos (e+f x)}{4 d f (c+d) (c+d \sin (e+f x))^4}",1,"(a^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(4*(c - d)^2*(c + d)^4*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) - (a^2*(c + 8*d)*Cos[e + f*x])/(12*d*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x])/(24*(c - d)*d*(c + d)^3*f*(c + d*Sin[e + f*x])^2) - (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*Cos[e + f*x])/(24*(c - d)^2*d*(c + d)^4*f*(c + d*Sin[e + f*x]))","A",8,6,25,0.2400,1,"{2762, 2754, 12, 2660, 618, 204}"
444,1,326,0,0.5446489,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]","-\frac{a^3 \left(107 c^3 d^2+472 c^2 d^3-18 c^4 d+2 c^5+456 c d^4+136 d^5\right) \cos (e+f x)}{60 d^2 f}-\frac{a^3 \left(2 c^2-18 c d+115 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}-\frac{a^3 \left(-18 c^2 d+2 c^3+111 c d^2+136 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac{a^3 \left(216 c^2 d^2-36 c^3 d+4 c^4+626 c d^3+345 d^4\right) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac{1}{16} a^3 x \left(90 c^2 d+40 c^3+78 c d^2+23 d^3\right)+\frac{a^3 (2 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac{\cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^4}{6 d f}","-\frac{a^3 d \left(18 c^2+54 c d+23 d^2\right) \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac{a^3 \left(90 c^2 d+24 c^3+78 c d^2+23 d^3\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} a^3 x \left(90 c^2 d+40 c^3+78 c d^2+23 d^3\right)-\frac{3 a^3 d^2 (c+d) \cos ^5(e+f x)}{5 f}+\frac{a^3 (c+d)^2 (c+7 d) \cos ^3(e+f x)}{3 f}-\frac{4 a^3 (c+d)^3 \cos (e+f x)}{f}-\frac{a^3 d^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}",1,"(a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*x)/16 - (a^3*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 136*d^5)*Cos[e + f*x])/(60*d^2*f) - (a^3*(4*c^4 - 36*c^3*d + 216*c^2*d^2 + 626*c*d^3 + 345*d^4)*Cos[e + f*x]*Sin[e + f*x])/(240*d*f) - (a^3*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^2*f) - (a^3*(2*c^2 - 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d^2*f) + (a^3*(2*c - 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d^2*f) - (Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(6*d*f)","A",6,5,25,0.2000,1,"{2763, 2968, 3023, 2753, 2734}"
445,1,189,0,0.2597661,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2,x]","\frac{a^3 \left(20 c^2+30 c d+13 d^2\right) \cos ^3(e+f x)}{60 f}-\frac{a^3 \left(20 c^2+30 c d+13 d^2\right) \cos (e+f x)}{5 f}-\frac{3 a^3 \left(20 c^2+30 c d+13 d^2\right) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{1}{8} a^3 x \left(20 c^2+30 c d+13 d^2\right)-\frac{d (10 c-d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f}","\frac{a^3 \left(c^2+6 c d+5 d^2\right) \cos ^3(e+f x)}{3 f}-\frac{a^3 \left(12 c^2+30 c d+13 d^2\right) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^3 x \left(20 c^2+30 c d+13 d^2\right)-\frac{4 a^3 (c+d)^2 \cos (e+f x)}{f}-\frac{a^3 d (2 c+3 d) \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{a^3 d^2 \cos ^5(e+f x)}{5 f}",1,"(a^3*(20*c^2 + 30*c*d + 13*d^2)*x)/8 - (a^3*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x])/(5*f) + (a^3*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]^3)/(60*f) - (3*a^3*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*Sin[e + f*x])/(40*f) - ((10*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(20*f) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^4)/(5*a*f)","A",9,7,25,0.2800,1,"{2761, 2751, 2645, 2638, 2635, 8, 2633}"
446,1,117,0,0.0967658,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x]),x]","\frac{a^3 (4 c+3 d) \cos ^3(e+f x)}{12 f}-\frac{a^3 (4 c+3 d) \cos (e+f x)}{f}-\frac{3 a^3 (4 c+3 d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a^3 x (4 c+3 d)-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^3}{4 f}","\frac{a^3 (c+3 d) \cos ^3(e+f x)}{3 f}-\frac{4 a^3 (c+d) \cos (e+f x)}{f}-\frac{3 a^3 (4 c+5 d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{5}{8} a^3 x (4 c+3 d)-\frac{a^3 d \sin ^3(e+f x) \cos (e+f x)}{4 f}",1,"(5*a^3*(4*c + 3*d)*x)/8 - (a^3*(4*c + 3*d)*Cos[e + f*x])/f + (a^3*(4*c + 3*d)*Cos[e + f*x]^3)/(12*f) - (3*a^3*(4*c + 3*d)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*f)","A",8,6,23,0.2609,1,"{2751, 2645, 2638, 2635, 8, 2633}"
447,1,63,0,0.0503981,"\int (a+a \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^3,x]","\frac{a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{3 a^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{5 a^3 x}{2}","\frac{a^3 \cos ^3(e+f x)}{3 f}-\frac{4 a^3 \cos (e+f x)}{f}-\frac{3 a^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{5 a^3 x}{2}",1,"(5*a^3*x)/2 - (4*a^3*Cos[e + f*x])/f + (a^3*Cos[e + f*x]^3)/(3*f) - (3*a^3*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",7,5,12,0.4167,1,"{2645, 2638, 2635, 8, 2633}"
448,1,143,0,0.3886797,"\int \frac{(a+a \sin (e+f x))^3}{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x]),x]","-\frac{2 a^3 (c-d)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \sqrt{c^2-d^2}}+\frac{a^3 x \left(2 c^2-6 c d+7 d^2\right)}{2 d^3}+\frac{a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac{\cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d f}","-\frac{2 a^3 (c-d)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \sqrt{c^2-d^2}}+\frac{a^3 x \left(2 c^2-6 c d+7 d^2\right)}{2 d^3}+\frac{a^3 (2 c-5 d) \cos (e+f x)}{2 d^2 f}-\frac{\cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d f}",1,"(a^3*(2*c^2 - 6*c*d + 7*d^2)*x)/(2*d^3) - (2*a^3*(c - d)^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*Sqrt[c^2 - d^2]*f) + (a^3*(2*c - 5*d)*Cos[e + f*x])/(2*d^2*f) - (Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d*f)","A",7,7,25,0.2800,1,"{2763, 2968, 3023, 2735, 2660, 618, 204}"
449,1,161,0,0.3805204,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2,x]","\frac{2 a^3 (c-d)^2 (2 c+3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f (c+d) \sqrt{c^2-d^2}}-\frac{2 a^3 c \cos (e+f x)}{d^2 f (c+d)}-\frac{a^3 x (2 c-3 d)}{d^3}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{d f (c+d) (c+d \sin (e+f x))}","\frac{2 a^3 (c-d)^2 (2 c+3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f (c+d) \sqrt{c^2-d^2}}-\frac{2 a^3 c \cos (e+f x)}{d^2 f (c+d)}-\frac{a^3 x (2 c-3 d)}{d^3}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{d f (c+d) (c+d \sin (e+f x))}",1,"-((a^3*(2*c - 3*d)*x)/d^3) + (2*a^3*(c - d)^2*(2*c + 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c + d)*Sqrt[c^2 - d^2]*f) - (2*a^3*c*Cos[e + f*x])/(d^2*(c + d)*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))","A",7,7,25,0.2800,1,"{2762, 2968, 3023, 2735, 2660, 618, 204}"
450,1,187,0,0.4774778,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3,x]","-\frac{a^3 (c-d) \left(2 c^2+6 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f (c+d)^2 \sqrt{c^2-d^2}}+\frac{a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac{a^3 x}{d^3}","-\frac{a^3 (c-d) \left(2 c^2+6 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f (c+d)^2 \sqrt{c^2-d^2}}+\frac{a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac{a^3 x}{d^3}",1,"(a^3*x)/d^3 - (a^3*(c - d)*(2*c^2 + 6*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c + d)^2*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) + (a^3*(c - d)*(2*c + 5*d)*Cos[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))","A",7,7,25,0.2800,1,"{2762, 2968, 3021, 2735, 2660, 618, 204}"
451,1,207,0,0.4750848,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4,x]","\frac{5 a^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d)^3 \sqrt{c^2-d^2}}-\frac{a^3 \left(2 c^2+9 c d+22 d^2\right) \cos (e+f x)}{6 d^2 f (c+d)^3 (c+d \sin (e+f x))}+\frac{a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 f (c+d)^2 (c+d \sin (e+f x))^2}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{3 d f (c+d) (c+d \sin (e+f x))^3}","\frac{5 a^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f (c+d)^3 \sqrt{c^2-d^2}}-\frac{a^3 \left(2 c^2+9 c d+22 d^2\right) \cos (e+f x)}{6 d^2 f (c+d)^3 (c+d \sin (e+f x))}+\frac{a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 f (c+d)^2 (c+d \sin (e+f x))^2}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{3 d f (c+d) (c+d \sin (e+f x))^3}",1,"(5*a^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)^3*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^3) + (a^3*(c - d)*(2*c + 7*d)*Cos[e + f*x])/(6*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (a^3*(2*c^2 + 9*c*d + 22*d^2)*Cos[e + f*x])/(6*d^2*(c + d)^3*f*(c + d*Sin[e + f*x]))","A",8,8,25,0.3200,1,"{2762, 2968, 3021, 2754, 12, 2660, 618, 204}"
452,1,289,0,0.6961388,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^5} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5,x]","\frac{5 a^3 (4 c-3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{4 f (c-d) (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^3 \left(12 c^2 d+2 c^3+43 c d^2-72 d^3\right) \cos (e+f x)}{24 d^2 f (c-d) (c+d)^4 (c+d \sin (e+f x))}-\frac{a^3 \left(2 c^2+12 c d+45 d^2\right) \cos (e+f x)}{24 d^2 f (c+d)^3 (c+d \sin (e+f x))^2}+\frac{a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{4 d f (c+d) (c+d \sin (e+f x))^4}","\frac{5 a^3 (4 c-3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{4 f (c-d) (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^3 \left(12 c^2 d+2 c^3+43 c d^2-72 d^3\right) \cos (e+f x)}{24 d^2 f (c-d) (c+d)^4 (c+d \sin (e+f x))}-\frac{a^3 \left(2 c^2+12 c d+45 d^2\right) \cos (e+f x)}{24 d^2 f (c+d)^3 (c+d \sin (e+f x))^2}+\frac{a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{(c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{4 d f (c+d) (c+d \sin (e+f x))^4}",1,"(5*a^3*(4*c - 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(4*(c - d)*(c + d)^4*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) + (a^3*(c - d)*(2*c + 9*d)*Cos[e + f*x])/(12*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^3*(2*c^2 + 12*c*d + 45*d^2)*Cos[e + f*x])/(24*d^2*(c + d)^3*f*(c + d*Sin[e + f*x])^2) - (a^3*(2*c^3 + 12*c^2*d + 43*c*d^2 - 72*d^3)*Cos[e + f*x])/(24*(c - d)*d^2*(c + d)^4*f*(c + d*Sin[e + f*x]))","A",9,8,25,0.3200,1,"{2762, 2968, 3021, 2754, 12, 2660, 618, 204}"
453,1,189,0,0.2273598,"\int \frac{(c+d \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x]),x]","\frac{2 d \left(-16 c^2 d+3 c^3+12 c d^2-4 d^3\right) \cos (e+f x)}{3 a f}+\frac{d^2 \left(6 c^2-20 c d+9 d^2\right) \sin (e+f x) \cos (e+f x)}{6 a f}+\frac{d x \left(-12 c^2 d+8 c^3+12 c d^2-3 d^3\right)}{2 a}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 c-4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}","\frac{2 d \left(-16 c^2 d+3 c^3+12 c d^2-4 d^3\right) \cos (e+f x)}{3 a f}+\frac{d^2 \left(6 c^2-20 c d+9 d^2\right) \sin (e+f x) \cos (e+f x)}{6 a f}+\frac{d x \left(-12 c^2 d+8 c^3+12 c d^2-3 d^3\right)}{2 a}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 c-4 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}",1,"(d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*x)/(2*a) + (2*d*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3)*Cos[e + f*x])/(3*a*f) + (d^2*(6*c^2 - 20*c*d + 9*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a*f) + ((3*c - 4*d)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))","A",3,3,25,0.1200,1,"{2767, 2753, 2734}"
454,1,121,0,0.1253983,"\int \frac{(c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x]),x]","\frac{2 d \left(c^2-3 c d+d^2\right) \cos (e+f x)}{a f}+\frac{3 d x \left(2 c^2-2 c d+d^2\right)}{2 a}+\frac{d^2 (2 c-3 d) \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}","\frac{2 d \left(c^2-3 c d+d^2\right) \cos (e+f x)}{a f}+\frac{3 d x \left(2 c^2-2 c d+d^2\right)}{2 a}+\frac{d^2 (2 c-3 d) \sin (e+f x) \cos (e+f x)}{2 a f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)}",1,"(3*d*(2*c^2 - 2*c*d + d^2)*x)/(2*a) + (2*d*(c^2 - 3*c*d + d^2)*Cos[e + f*x])/(a*f) + ((2*c - 3*d)*d^2*Cos[e + f*x]*Sin[e + f*x])/(2*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(f*(a + a*Sin[e + f*x]))","A",2,2,25,0.08000,1,"{2767, 2734}"
455,1,62,0,0.1365633,"\int \frac{(c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + a*Sin[e + f*x]),x]","-\frac{(c-d)^2 \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac{d x (2 c-d)}{a}-\frac{d^2 \cos (e+f x)}{a f}","-\frac{(c-d)^2 \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac{d x (2 c-d)}{a}-\frac{d^2 \cos (e+f x)}{a f}",1,"((2*c - d)*d*x)/a - (d^2*Cos[e + f*x])/(a*f) - ((c - d)^2*Cos[e + f*x])/(a*f*(1 + Sin[e + f*x]))","A",3,3,25,0.1200,1,"{2746, 2735, 2648}"
456,1,35,0,0.047232,"\int \frac{c+d \sin (e+f x)}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x]),x]","\frac{d x}{a}-\frac{(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)}","\frac{d x}{a}-\frac{(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)}",1,"(d*x)/a - ((c - d)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))","A",2,2,23,0.08696,1,"{2735, 2648}"
457,1,23,0,0.0124415,"\int \frac{1}{a+a \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(-1),x]","-\frac{\cos (e+f x)}{f (a \sin (e+f x)+a)}","-\frac{\cos (e+f x)}{f (a \sin (e+f x)+a)}",1,"-(Cos[e + f*x]/(f*(a + a*Sin[e + f*x])))","A",1,1,12,0.08333,1,"{2648}"
458,1,89,0,0.1299896,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","-\frac{2 d \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \sqrt{c^2-d^2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a)}","-\frac{2 d \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \sqrt{c^2-d^2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a)}",1,"(-2*d*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a*(c - d)*Sqrt[c^2 - d^2]*f) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x]))","A",5,5,25,0.2000,1,"{2747, 2648, 2660, 618, 204}"
459,1,150,0,0.1782823,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","-\frac{2 d (2 c+d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \left(c^2-d^2\right)^{3/2}}-\frac{d (c+2 d) \cos (e+f x)}{a f (c-d)^2 (c+d) (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))}","-\frac{2 d (2 c+d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \left(c^2-d^2\right)^{3/2}}-\frac{d (c+2 d) \cos (e+f x)}{a f (c-d)^2 (c+d) (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))}",1,"(-2*d*(2*c + d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(3/2)*f) - (d*(c + 2*d)*Cos[e + f*x])/(a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2768, 2754, 12, 2660, 618, 204}"
460,1,213,0,0.3227879,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]","-\frac{3 d \left(2 c^2+2 c d+d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \left(c^2-d^2\right)^{5/2}}-\frac{d (2 c+d) (c+4 d) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac{d (2 c+3 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}","-\frac{3 d \left(2 c^2+2 c d+d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a f (c-d) \left(c^2-d^2\right)^{5/2}}-\frac{d (2 c+d) (c+4 d) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac{d (2 c+3 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2}",1,"(-3*d*(2*c^2 + 2*c*d + d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(5/2)*f) - (d*(2*c + 3*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^2) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(2*c + d)*(c + 4*d)*Cos[e + f*x])/(2*a*(c - d)^3*(c + d)^2*f*(c + d*Sin[e + f*x]))","A",7,6,25,0.2400,1,"{2768, 2754, 12, 2660, 618, 204}"
461,1,260,0,0.4950915,"\int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]","\frac{2 d \left(-44 c^2 d^2+10 c^3 d+c^4+40 c d^3-12 d^4\right) \cos (e+f x)}{3 a^2 f}+\frac{d \left(c^2+10 c d-12 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}+\frac{d^2 \left(20 c^2 d+2 c^3-57 c d^2+30 d^3\right) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{5 d^2 x (2 c-d) \left(2 c^2-3 c d+2 d^2\right)}{2 a^2}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}","\frac{2 d \left(-44 c^2 d^2+10 c^3 d+c^4+40 c d^3-12 d^4\right) \cos (e+f x)}{3 a^2 f}+\frac{d \left(c^2+10 c d-12 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}+\frac{d^2 \left(20 c^2 d+2 c^3-57 c d^2+30 d^3\right) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{5 d^2 x (2 c-d) \left(2 c^2-3 c d+2 d^2\right)}{2 a^2}-\frac{(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2}",1,"(5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*x)/(2*a^2) + (2*d*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4)*Cos[e + f*x])/(3*a^2*f) + (d^2*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) + (d*(c^2 + 10*c*d - 12*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f) - ((c - d)*(c + 10*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(3*f*(a + a*Sin[e + f*x])^2)","A",4,4,25,0.1600,1,"{2765, 2977, 2753, 2734}"
462,1,195,0,0.3622953,"\int \frac{(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^2,x]","\frac{2 d \left(8 c^2 d+c^3-20 c d^2+8 d^3\right) \cos (e+f x)}{3 a^2 f}+\frac{d^2 \left(2 c^2+16 c d-21 d^2\right) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{d^2 x \left(12 c^2-16 c d+7 d^2\right)}{2 a^2}-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}","\frac{2 d \left(8 c^2 d+c^3-20 c d^2+8 d^3\right) \cos (e+f x)}{3 a^2 f}+\frac{d^2 \left(2 c^2+16 c d-21 d^2\right) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{d^2 x \left(12 c^2-16 c d+7 d^2\right)}{2 a^2}-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2}",1,"(d^2*(12*c^2 - 16*c*d + 7*d^2)*x)/(2*a^2) + (2*d*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3)*Cos[e + f*x])/(3*a^2*f) + (d^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) - ((c - d)*(c + 8*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*f*(a + a*Sin[e + f*x])^2)","A",3,3,25,0.1200,1,"{2765, 2977, 2734}"
463,1,120,0,0.3671955,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^2,x]","\frac{d^2 (c-4 d) \cos (e+f x)}{3 a^2 f}+\frac{d^2 x (3 c-2 d)}{a^2}-\frac{(c+6 d) (c-d)^2 \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}","\frac{d^2 (c-4 d) \cos (e+f x)}{3 a^2 f}+\frac{d^2 x (3 c-2 d)}{a^2}-\frac{(c+6 d) (c-d)^2 \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2}",1,"((3*c - 2*d)*d^2*x)/a^2 + ((c - 4*d)*d^2*Cos[e + f*x])/(3*a^2*f) - ((c - d)^2*(c + 6*d)*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f*(a + a*Sin[e + f*x])^2)","A",5,5,25,0.2000,1,"{2765, 2968, 3023, 2735, 2648}"
464,1,85,0,0.1415707,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^2,x]","-\frac{(c-d) (c+4 d) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac{d^2 x}{a^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{3 f (a \sin (e+f x)+a)^2}","-\frac{(c-d) (c+4 d) \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}+\frac{d^2 x}{a^2}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{3 f (a \sin (e+f x)+a)^2}",1,"(d^2*x)/a^2 - ((c - d)*(c + 4*d)*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(3*f*(a + a*Sin[e + f*x])^2)","A",3,3,25,0.1200,1,"{2760, 2735, 2648}"
465,1,65,0,0.0519345,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^2,x]","-\frac{(c+2 d) \cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{(c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}","-\frac{(c+2 d) \cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{(c-d) \cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"-((c - d)*Cos[e + f*x])/(3*f*(a + a*Sin[e + f*x])^2) - ((c + 2*d)*Cos[e + f*x])/(3*f*(a^2 + a^2*Sin[e + f*x]))","A",2,2,23,0.08696,1,"{2750, 2648}"
466,1,55,0,0.0278865,"\int \frac{1}{(a+a \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^(-2),x]","-\frac{\cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}","-\frac{\cos (e+f x)}{3 f \left(a^2 \sin (e+f x)+a^2\right)}-\frac{\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}",1,"-Cos[e + f*x]/(3*f*(a + a*Sin[e + f*x])^2) - Cos[e + f*x]/(3*f*(a^2 + a^2*Sin[e + f*x]))","A",2,2,12,0.1667,1,"{2650, 2648}"
467,1,131,0,0.2822646,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]","\frac{2 d^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^2 \sqrt{c^2-d^2}}-\frac{(c-4 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2}","\frac{2 d^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^2 \sqrt{c^2-d^2}}-\frac{(c-4 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2}",1,"(2*d^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^2*Sqrt[c^2 - d^2]*f) - ((c - 4*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2)","A",6,6,25,0.2400,1,"{2766, 2978, 12, 2660, 618, 204}"
468,1,221,0,0.4170814,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2),x]","\frac{2 d^2 (3 c+2 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^3 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(c^2-6 c d-10 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))}-\frac{(c-6 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}","\frac{2 d^2 (3 c+2 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^3 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(c^2-6 c d-10 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))}-\frac{(c-6 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}",1,"(2*d^2*(3*c + 2*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^3*(c + d)*Sqrt[c^2 - d^2]*f) - (d*(c^2 - 6*c*d - 10*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])) - ((c - 6*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x]))","A",7,7,25,0.2800,1,"{2766, 2978, 2754, 12, 2660, 618, 204}"
469,1,294,0,0.6154878,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]","\frac{d^2 \left(12 c^2+16 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^4 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left(-16 c^2 d+2 c^3-59 c d^2-32 d^3\right) \cos (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left(2 c^2-16 c d-21 d^2\right) \cos (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^2}-\frac{(c-8 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}","\frac{d^2 \left(12 c^2+16 c d+7 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^2 f (c-d)^4 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left(-16 c^2 d+2 c^3-59 c d^2-32 d^3\right) \cos (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left(2 c^2-16 c d-21 d^2\right) \cos (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^2}-\frac{(c-8 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}",1,"(d^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^4*(c + d)^2*Sqrt[c^2 - d^2]*f) - (d*(2*c^2 - 16*c*d - 21*d^2)*Cos[e + f*x])/(6*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((c - 8*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - (d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Cos[e + f*x])/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*Sin[e + f*x]))","A",8,7,25,0.2800,1,"{2766, 2978, 2754, 12, 2660, 618, 204}"
470,1,354,0,0.7897441,"\int \frac{(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^6/(a + a*Sin[e + f*x])^3,x]","\frac{2 d \left(107 c^3 d^2-472 c^2 d^3+18 c^4 d+2 c^5+456 c d^4-136 d^5\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left(2 c^2+18 c d+115 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d \left(18 c^2 d+2 c^3+111 c d^2-136 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}+\frac{d^2 \left(216 c^2 d^2+36 c^3 d+4 c^4-626 c d^3+345 d^4\right) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left(-90 c^2 d+40 c^3+78 c d^2-23 d^3\right)}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a \sin (e+f x)+a)^2}","\frac{2 d \left(107 c^3 d^2-472 c^2 d^3+18 c^4 d+2 c^5+456 c d^4-136 d^5\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left(2 c^2+18 c d+115 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d \left(18 c^2 d+2 c^3+111 c d^2-136 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a^3 f}+\frac{d^2 \left(216 c^2 d^2+36 c^3 d+4 c^4-626 c d^3+345 d^4\right) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left(-90 c^2 d+40 c^3+78 c d^2-23 d^3\right)}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^5}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+13 d) \cos (e+f x) (c+d \sin (e+f x))^4}{15 a f (a \sin (e+f x)+a)^2}",1,"(d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/(2*a^3) + (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*Cos[e + f*x]*Sin[e + f*x])/(30*a^3*f) + (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*a^3*f) - ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(5*f*(a + a*Sin[e + f*x])^3)","A",5,4,25,0.1600,1,"{2765, 2977, 2753, 2734}"
471,1,278,0,0.6140088,"\int \frac{(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^3,x]","\frac{2 d \left(72 c^2 d^2+15 c^3 d+2 c^4-180 c d^3+76 d^4\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left(2 c^2+15 c d+76 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^2 \left(30 c^2 d+4 c^3+146 c d^2-195 d^3\right) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left(20 c^2-30 c d+13 d^2\right)}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a \sin (e+f x)+a)^2}","\frac{2 d \left(72 c^2 d^2+15 c^3 d+2 c^4-180 c d^3+76 d^4\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d) \left(2 c^2+15 c d+76 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^2}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^2 \left(30 c^2 d+4 c^3+146 c d^2-195 d^3\right) \sin (e+f x) \cos (e+f x)}{30 a^3 f}+\frac{d^3 x \left(20 c^2-30 c d+13 d^2\right)}{2 a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+11 d) \cos (e+f x) (c+d \sin (e+f x))^3}{15 a f (a \sin (e+f x)+a)^2}",1,"(d^3*(20*c^2 - 30*c*d + 13*d^2)*x)/(2*a^3) + (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Cos[e + f*x]*Sin[e + f*x])/(30*a^3*f) - ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*f*(a + a*Sin[e + f*x])^3)","A",4,3,25,0.1200,1,"{2765, 2977, 2734}"
472,1,195,0,0.613012,"\int \frac{(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^3,x]","\frac{d^2 \left(2 c^2+10 c d-27 d^2\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d)^2 \left(2 c^2+12 c d+45 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x (4 c-3 d)}{a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a \sin (e+f x)+a)^2}","\frac{d^2 \left(2 c^2+10 c d-27 d^2\right) \cos (e+f x)}{15 a^3 f}-\frac{(c-d)^2 \left(2 c^2+12 c d+45 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x (4 c-3 d)}{a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+9 d) \cos (e+f x) (c+d \sin (e+f x))^2}{15 a f (a \sin (e+f x)+a)^2}",1,"((4*c - 3*d)*d^3*x)/a^3 + (d^2*(2*c^2 + 10*c*d - 27*d^2)*Cos[e + f*x])/(15*a^3*f) - ((c - d)^2*(2*c^2 + 12*c*d + 45*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(5*f*(a + a*Sin[e + f*x])^3)","A",6,6,25,0.2400,1,"{2765, 2977, 2968, 3023, 2735, 2648}"
473,1,142,0,0.332717,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^3,x]","-\frac{(c-d) \left(2 c^2+11 c d+29 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x}{a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}","-\frac{(c-d) \left(2 c^2+11 c d+29 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{d^3 x}{a^3}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}",1,"(d^3*x)/a^3 - ((c - d)^2*(2*c + 7*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*(2*c^2 + 11*c*d + 29*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*(a + a*Sin[e + f*x])^3)","A",5,5,25,0.2000,1,"{2765, 2968, 3019, 2735, 2648}"
474,1,125,0,0.1816933,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^3,x]","-\frac{\left(2 c^2+6 c d+7 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+5 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}","-\frac{\left(2 c^2+6 c d+7 d^2\right) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{5 f (a \sin (e+f x)+a)^3}-\frac{(c-d) (2 c+5 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}",1,"-((c - d)*(2*c + 5*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((2*c^2 + 6*c*d + 7*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(5*f*(a + a*Sin[e + f*x])^3)","A",3,3,25,0.1200,1,"{2760, 2750, 2648}"
475,1,102,0,0.0753259,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]","-\frac{(2 c+3 d) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 c+3 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}","-\frac{(2 c+3 d) \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 c+3 d) \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"-((c - d)*Cos[e + f*x])/(5*f*(a + a*Sin[e + f*x])^3) - ((2*c + 3*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((2*c + 3*d)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))","A",3,3,23,0.1304,1,"{2750, 2650, 2648}"
476,1,83,0,0.0469928,"\int \frac{1}{(a+a \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^(-3),x]","-\frac{2 \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}","-\frac{2 \cos (e+f x)}{15 f \left(a^3 \sin (e+f x)+a^3\right)}-\frac{2 \cos (e+f x)}{15 a f (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}",1,"-Cos[e + f*x]/(5*f*(a + a*Sin[e + f*x])^3) - (2*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - (2*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))","A",3,2,12,0.1667,1,"{2650, 2648}"
477,1,186,0,0.5215144,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]","-\frac{2 d^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^3 \sqrt{c^2-d^2}}-\frac{\left(2 c^2-9 c d+22 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 c-7 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3}","-\frac{2 d^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^3 \sqrt{c^2-d^2}}-\frac{\left(2 c^2-9 c d+22 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{(2 c-7 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3}",1,"(-2*d^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^3*Sqrt[c^2 - d^2]*f) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3) - ((2*c - 7*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2) - ((2*c^2 - 9*c*d + 22*d^2)*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x]))","A",7,6,25,0.2400,1,"{2766, 2978, 12, 2660, 618, 204}"
478,1,298,0,0.7300075,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2),x]","-\frac{2 d^3 (4 c+3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^4 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(-12 c^2 d+2 c^3+43 c d^2+72 d^3\right) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac{\left(2 c^2-12 c d+45 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))}-\frac{(2 c-9 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))}","-\frac{2 d^3 (4 c+3 d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^4 (c+d) \sqrt{c^2-d^2}}-\frac{d \left(-12 c^2 d+2 c^3+43 c d^2+72 d^3\right) \cos (e+f x)}{15 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))}-\frac{\left(2 c^2-12 c d+45 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))}-\frac{(2 c-9 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))}",1,"(-2*d^3*(4*c + 3*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^4*(c + d)*Sqrt[c^2 - d^2]*f) - (d*(2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*Cos[e + f*x])/(15*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])) - ((2*c - 9*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])) - ((2*c^2 - 12*c*d + 45*d^2)*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x]))","A",8,7,25,0.2800,1,"{2766, 2978, 2754, 12, 2660, 618, 204}"
479,1,378,0,0.9621097,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]","-\frac{d^3 \left(20 c^2+30 c d+13 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left(142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left(-30 c^2 d+4 c^3+146 c d^2+195 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{\left(2 c^2-15 c d+76 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}","-\frac{d^3 \left(20 c^2+30 c d+13 d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{a^3 f (c-d)^5 (c+d)^2 \sqrt{c^2-d^2}}-\frac{d \left(142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{d \left(-30 c^2 d+4 c^3+146 c d^2+195 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{\left(2 c^2-15 c d+76 d^2\right) \cos (e+f x)}{15 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^2}-\frac{(2 c-11 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^2}",1,"-((d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*Sqrt[c^2 - d^2]*f)) - (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^2) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*c - 11*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((2*c^2 - 15*c*d + 76*d^2)*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))","A",9,7,25,0.2800,1,"{2766, 2978, 2754, 12, 2660, 618, 204}"
480,1,75,0,0.0567041,"\int \frac{A+B \sin (x)}{(1+\sin (x))^4} \, dx","Int[(A + B*Sin[x])/(1 + Sin[x])^4,x]","-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac{(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac{(A-B) \cos (x)}{7 (\sin (x)+1)^4}","-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac{(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac{(A-B) \cos (x)}{7 (\sin (x)+1)^4}",1,"-((A - B)*Cos[x])/(7*(1 + Sin[x])^4) - ((3*A + 4*B)*Cos[x])/(35*(1 + Sin[x])^3) - (2*(3*A + 4*B)*Cos[x])/(105*(1 + Sin[x])^2) - (2*(3*A + 4*B)*Cos[x])/(105*(1 + Sin[x]))","A",4,3,13,0.2308,1,"{2750, 2650, 2648}"
481,1,81,0,0.0663277,"\int \frac{A+B \sin (x)}{(1-\sin (x))^4} \, dx","Int[(A + B*Sin[x])/(1 - Sin[x])^4,x]","\frac{2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))}+\frac{2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac{(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac{(A+B) \cos (x)}{7 (1-\sin (x))^4}","\frac{2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))}+\frac{2 (3 A-4 B) \cos (x)}{105 (1-\sin (x))^2}+\frac{(3 A-4 B) \cos (x)}{35 (1-\sin (x))^3}+\frac{(A+B) \cos (x)}{7 (1-\sin (x))^4}",1,"((A + B)*Cos[x])/(7*(1 - Sin[x])^4) + ((3*A - 4*B)*Cos[x])/(35*(1 - Sin[x])^3) + (2*(3*A - 4*B)*Cos[x])/(105*(1 - Sin[x])^2) + (2*(3*A - 4*B)*Cos[x])/(105*(1 - Sin[x]))","A",4,3,15,0.2000,1,"{2750, 2650, 2648}"
482,1,290,0,0.442811,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 a \left(15 c^2+56 c d+25 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 f}-\frac{2 a \left(c^2-d^2\right) \left(15 c^2+56 c d+25 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 a \left(161 c^2 d+15 c^3+145 c d^2+63 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}-\frac{2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}","-\frac{2 a \left(15 c^2+56 c d+25 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 f}-\frac{2 a \left(c^2-d^2\right) \left(15 c^2+56 c d+25 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 a \left(161 c^2 d+15 c^3+145 c d^2+63 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}-\frac{2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}",1,"(-2*a*(15*c^2 + 56*c*d + 25*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*f) - (2*a*(5*c + 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*f) - (2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + (2*a*(15*c^3 + 161*c^2*d + 145*c*d^2 + 63*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c^2 - d^2)*(15*c^2 + 56*c*d + 25*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d*f*Sqrt[c + d*Sin[e + f*x]])","A",8,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
483,1,231,0,0.3170235,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 a (3 c+5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 a \left(3 c^2+20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac{2 a (3 c+5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 f}","-\frac{2 a (3 c+5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 a \left(3 c^2+20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac{2 a (3 c+5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 f}",1,"(-2*a*(3*c + 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*f) - (2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (2*a*(3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(3*c + 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d*f*Sqrt[c + d*Sin[e + f*x]])","A",7,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
484,1,179,0,0.1977743,"\int (a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 a \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f}+\frac{2 a (c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{2 a \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f}+\frac{2 a (c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*a*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + (2*a*(c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
485,1,138,0,0.1222684,"\int \frac{a+a \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])/Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 a \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}","\frac{2 a \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}",1,"(2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])","A",5,5,25,0.2000,1,"{2752, 2663, 2661, 2655, 2653}"
486,1,169,0,0.2058368,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
487,1,237,0,0.3360743,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 a (c-3 d) \cos (e+f x)}{3 f (c-d) (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{2 a (c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f (c-d) (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{2 a (c-3 d) \cos (e+f x)}{3 f (c-d) (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{2 a (c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f (c-d) (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*a*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - (2*a*(c - 3*d)*Cos[e + f*x])/(3*(c - d)*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*(c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*(c - d)*d*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
488,1,318,0,0.5068824,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{2 a \left(3 c^2-20 c d+9 d^2\right) \cos (e+f x)}{15 f (c-d)^2 (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 a \left(3 c^2-20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f (c-d)^2 (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (3 c-5 d) \cos (e+f x)}{15 f (c-d) (c+d)^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}+\frac{2 a (3 c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f (c-d) (c+d)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{2 a \left(3 c^2-20 c d+9 d^2\right) \cos (e+f x)}{15 f (c-d)^2 (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 a \left(3 c^2-20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f (c-d)^2 (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (3 c-5 d) \cos (e+f x)}{15 f (c-d) (c+d)^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}+\frac{2 a (3 c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f (c-d) (c+d)^2 \sqrt{c+d \sin (e+f x)}}",1,"(-2*a*Cos[e + f*x])/(5*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) - (2*a*(3*c - 5*d)*Cos[e + f*x])/(15*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*a*(3*c^2 - 20*c*d + 9*d^2)*Cos[e + f*x])/(15*(c - d)^2*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*(3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*(c - d)^2*d*(c + d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*(3*c - 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*(c - d)*d*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
489,1,378,0,0.6696232,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]","\frac{4 a^2 \left(-45 c^2 d+5 c^3-141 c d^2-75 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 a^2 \left(c^2-d^2\right) \left(-45 c^2 d+5 c^3-141 c d^2-75 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(-381 c^2 d^2-45 c^3 d+5 c^4-435 c d^3-168 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 \left(5 c (c-9 d)-56 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac{4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}","\frac{4 a^2 \left(-45 c^2 d+5 c^3-141 c d^2-75 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 a^2 \left(c^2-d^2\right) \left(-45 c^2 d+5 c^3-141 c d^2-75 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(-381 c^2 d^2-45 c^3 d+5 c^4-435 c d^3-168 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 \left(5 c (c-9 d)-56 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac{4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}",1,"(4*a^2*(5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d*f) + (4*a^2*(5*c*(c - 9*d) - 56*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d*f) + (4*a^2*(c - 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d*f) - (4*a^2*(5*c^4 - 45*c^3*d - 381*c^2*d^2 - 435*c*d^3 - 168*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c^2 - d^2)*(5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",9,7,27,0.2593,1,"{2763, 2753, 2752, 2663, 2661, 2655, 2653}"
490,1,298,0,0.478873,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2),x]","\frac{4 a^2 \left(c^2-7 c d-10 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{35 d f}+\frac{4 a^2 \left(c^2-7 c d-10 d^2\right) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{35 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+3 d) \left(c^2-10 c d-7 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{35 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}","\frac{4 a^2 \left(c^2-7 c d-10 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{35 d f}+\frac{4 a^2 \left(c^2-7 c d-10 d^2\right) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{35 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+3 d) \left(c^2-10 c d-7 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{35 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac{4 a^2 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}",1,"(4*a^2*(c^2 - 7*c*d - 10*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(35*d*f) + (4*a^2*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) - (4*a^2*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(35*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c^2 - 7*c*d - 10*d^2)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(35*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2763, 2753, 2752, 2663, 2661, 2655, 2653}"
491,1,239,0,0.3360933,"\int (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]],x]","\frac{4 a^2 (c-5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(c^2-5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}+\frac{4 a^2 (c-5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d f}","\frac{4 a^2 (c-5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(c^2-5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}+\frac{4 a^2 (c-5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d f}",1,"(4*a^2*(c - 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d*f) - (4*a^2*(c^2 - 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2763, 2753, 2752, 2663, 2661, 2655, 2653}"
492,1,189,0,0.2464938,"\int \frac{(a+a \sin (e+f x))^2}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^2/Sqrt[c + d*Sin[e + f*x]],x]","\frac{4 a^2 (c-2 d) (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}","\frac{4 a^2 (c-2 d) (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}",1,"(-2*a^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) - (4*a^2*(c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c - 2*d)*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2763, 2752, 2663, 2661, 2655, 2653}"
493,1,189,0,0.2390026,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{4 a^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^2 c \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^2 c \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{c+d \sin (e+f x)}}",1,"(2*a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) + (4*a^2*c*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^2*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2762, 2752, 2663, 2661, 2655, 2653}"
494,1,247,0,0.3691419,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(5/2),x]","\frac{4 a^2 (c+2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^2 (c+3 d) \cos (e+f x)}{3 d f (c+d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}","\frac{4 a^2 (c+2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^2 (c+3 d) \cos (e+f x)}{3 d f (c+d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c + 3*d)*Cos[e + f*x])/(3*d*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^2*(c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c + 2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2762, 2754, 2752, 2663, 2661, 2655, 2653}"
495,1,320,0,0.577853,"\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{4 a^2 \left(c^2+5 c d-12 d^2\right) \cos (e+f x)}{15 d f (c-d) (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(c^2+5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f (c-d) (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 (c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}","-\frac{4 a^2 \left(c^2+5 c d-12 d^2\right) \cos (e+f x)}{15 d f (c-d) (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left(c^2+5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f (c-d) (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 (c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x])/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) - (4*a^2*(c + 5*d)*Cos[e + f*x])/(15*d*(c + d)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c^2 + 5*c*d - 12*d^2)*Cos[e + f*x])/(15*(c - d)*d*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^2*(c^2 + 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*(c - d)*d^2*(c + d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2762, 2754, 2752, 2663, 2661, 2655, 2653}"
496,1,467,0,1.0316691,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{4 a^3 \left(4 c^2-33 c d+189 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{4 a^3 \left(-33 c^2 d+4 c^3+182 c d^2+231 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left(177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{693 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 (c+3 d) \left(309 c^2 d^2-45 c^3 d+4 c^4+525 c d^3+231 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{693 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{7/2}}{11 d f}","-\frac{4 a^3 \left(4 c^2-33 c d+189 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{4 a^3 \left(-33 c^2 d+4 c^3+182 c d^2+231 d^3\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac{4 a^3 \left(177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{693 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(177 c^2 d^2-33 c^3 d+4 c^4+561 c d^3+315 d^4\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{693 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 (c+3 d) \left(309 c^2 d^2-45 c^3 d+4 c^4+525 c d^3+231 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{693 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{7/2}}{11 d f}",1,"(-4*a^3*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(693*d^2*f) - (4*a^3*(4*c^3 - 33*c^2*d + 182*c*d^2 + 231*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(693*d^2*f) - (4*a^3*(4*c^2 - 33*c*d + 189*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8*a^3*(c - 6*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(7/2))/(11*d*f) + (4*a^3*(c + 3*d)*(4*c^4 - 45*c^3*d + 309*c^2*d^2 + 525*c*d^3 + 231*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(693*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(693*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",11,9,27,0.3333,1,"{2763, 2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
497,1,390,0,0.7751343,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{4 a^3 \left(4 c^2-27 c d+119 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{4 a^3 \left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{5/2}}{9 d f}","-\frac{4 a^3 \left(4 c^2-27 c d+119 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{4 a^3 \left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{5/2}}{9 d f}",1,"(-4*a^3*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f) - (4*a^3*(4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d^2*f) + (8*a^3*(c - 5*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d*f) + (4*a^3*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",10,9,27,0.3333,1,"{2763, 2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
498,1,318,0,0.5797934,"\int (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{4 a^3 \left(4 c^2-21 c d+65 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(4 c^2-21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(-21 c^2 d+4 c^3+62 c d^2+147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{3/2}}{7 d f}","-\frac{4 a^3 \left(4 c^2-21 c d+65 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}-\frac{4 a^3 \left(c^2-d^2\right) \left(4 c^2-21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(-21 c^2 d+4 c^3+62 c d^2+147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{3/2}}{7 d f}",1,"(-4*a^3*(4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f) + (8*a^3*(c - 4*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d*f) + (4*a^3*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^2 - 21*c*d + 65*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2763, 2968, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
499,1,258,0,0.476269,"\int \frac{(a+a \sin (e+f x))^3}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{4 a^3 (c-d) \left(4 c^2-11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^2-15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) \sqrt{c+d \sin (e+f x)}}{5 d f}","-\frac{4 a^3 (c-d) \left(4 c^2-11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^2-15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) \sqrt{c+d \sin (e+f x)}}{5 d f}",1,"(8*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(5*d*f) + (4*a^3*(4*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c - d)*(4*c^2 - 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2763, 2968, 3023, 2752, 2663, 2661, 2655, 2653}"
500,1,270,0,0.4805512,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{4 a^3 \left(4 c^2-5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^3 (2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f (c+d)}+\frac{4 a^3 (4 c-5 d) (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{d f (c+d) \sqrt{c+d \sin (e+f x)}}","-\frac{4 a^3 \left(4 c^2-5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^3 (2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f (c+d)}+\frac{4 a^3 (4 c-5 d) (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{d f (c+d) \sqrt{c+d \sin (e+f x)}}",1,"(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^3*(2*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c + d)*f) - (4*a^3*(4*c^2 - 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^3*(4*c - 5*d)*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2762, 2968, 3023, 2752, 2663, 2661, 2655, 2653}"
501,1,280,0,0.5772852,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]","\frac{4 a^3 \left(4 c^2+5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 (c-d) (4 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}","\frac{4 a^3 \left(4 c^2+5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+2 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 (c-d) (4 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}",1,"(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + (8*a^3*(c - d)*(c + 2*d)*Cos[e + f*x])/(3*d^2*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (4*a^3*(4*c^2 + 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c - d)*(4*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2762, 2968, 3021, 2752, 2663, 2661, 2655, 2653}"
502,1,336,0,0.7204151,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{4 a^3 \left(4 c^2+15 c d+27 d^2\right) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^2+11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^2+15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+3 d) \cos (e+f x)}{15 d^2 f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}","-\frac{4 a^3 \left(4 c^2+15 c d+27 d^2\right) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left(4 c^2+11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^2+15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+3 d) \cos (e+f x)}{15 d^2 f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}",1,"(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*a^3*(c - d)*(c + 3*d)*Cos[e + f*x])/(15*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^3*(4*c^2 + 15*c*d + 27*d^2)*Cos[e + f*x])/(15*d^2*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^3*(4*c^2 + 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*(c + d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^3*(4*c^2 + 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2762, 2968, 3021, 2754, 2752, 2663, 2661, 2655, 2653}"
503,1,419,0,0.9282863,"\int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(9/2),x]","-\frac{4 a^3 \left(21 c^2 d+4 c^3+62 c d^2-147 d^3\right) \cos (e+f x)}{105 d^2 f (c-d) (c+d)^4 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^2+21 c d+65 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 (c+d \sin (e+f x))^{3/2}}+\frac{4 a^3 \left(4 c^2+21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(21 c^2 d+4 c^3+62 c d^2-147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f (c-d) (c+d)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 f (c+d)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}","-\frac{4 a^3 \left(21 c^2 d+4 c^3+62 c d^2-147 d^3\right) \cos (e+f x)}{105 d^2 f (c-d) (c+d)^4 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(4 c^2+21 c d+65 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 (c+d \sin (e+f x))^{3/2}}+\frac{4 a^3 \left(4 c^2+21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left(21 c^2 d+4 c^3+62 c d^2-147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f (c-d) (c+d)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 f (c+d)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (c-d) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right)}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}",1,"(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(7*d*(c + d)*f*(c + d*Sin[e + f*x])^(7/2)) + (8*a^3*(c - d)*(c + 4*d)*Cos[e + f*x])/(35*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^(5/2)) - (4*a^3*(4*c^2 + 21*c*d + 65*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^3*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*Cos[e + f*x])/(105*(c - d)*d^2*(c + d)^4*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^3*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*(c - d)*d^3*(c + d)^4*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^3*(4*c^2 + 21*c*d + 65*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",10,9,27,0.3333,1,"{2762, 2968, 3021, 2754, 2752, 2663, 2661, 2655, 2653}"
504,1,246,0,0.3813044,"\int \frac{(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x]),x]","\frac{(3 c-5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f \sqrt{c+d \sin (e+f x)}}-\frac{\left(3 c^2-20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}+\frac{d (3 c-5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a f}","\frac{(3 c-5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f \sqrt{c+d \sin (e+f x)}}-\frac{\left(3 c^2-20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}+\frac{d (3 c-5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a f}",1,"((3*c - 5*d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f*x])) - ((3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2767, 2753, 2752, 2663, 2661, 2655, 2653}"
505,1,186,0,0.2570378,"\int \frac{(c+d \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x]),x]","\frac{\left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{\left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"-(((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) - ((c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2767, 2752, 2663, 2661, 2655, 2653}"
506,1,170,0,0.207021,"\int \frac{\sqrt{c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x]),x]","-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"-((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) - (EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2769, 2752, 2663, 2661, 2655, 2653}"
507,1,181,0,0.2144328,"\int \frac{1}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"-((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x]))) - (EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2768, 2752, 2663, 2661, 2655, 2653}"
508,1,244,0,0.3269148,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{d (c+3 d) \cos (e+f x)}{a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d)^2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{d (c+3 d) \cos (e+f x)}{a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{a f (c-d)^2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"-((d*(c + 3*d)*Cos[e + f*x])/(a*(c - d)^2*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*(c - d)^2*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(c - d)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2768, 2754, 2752, 2663, 2661, 2655, 2653}"
509,1,333,0,0.4880608,"\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{d \left(3 c^2+20 c d+9 d^2\right) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(3 c^2+20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}","-\frac{d \left(3 c^2+20 c d+9 d^2\right) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(3 c^2+20 c d+9 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}",1,"-(d*(3*c + 5*d)*Cos[e + f*x])/(3*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*(c - d)^2*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2768, 2754, 2752, 2663, 2661, 2655, 2653}"
510,1,256,0,0.5534513,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^2,x]","\frac{(c+5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left(c^2+5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) (c+5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}","\frac{(c+5 d) \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left(c^2+5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) (c+5 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3 f (a \sin (e+f x)+a)^2}",1,"-((c - d)*(c + 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(3*f*(a + a*Sin[e + f*x])^2) - ((c^2 + 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2765, 2977, 2752, 2663, 2661, 2655, 2653}"
511,1,237,0,0.5377779,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^2,x]","-\frac{(c+3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}+\frac{(c+d) (c+2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}","-\frac{(c+3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (\sin (e+f x)+1)}+\frac{(c+d) (c+2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}",1,"-((c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f*(a + a*Sin[e + f*x])^2) - ((c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(c + 2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2765, 2978, 2752, 2663, 2661, 2655, 2653}"
512,1,233,0,0.4104891,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x])^2,x]","-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d) (\sin (e+f x)+1)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{c \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}","-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d) (\sin (e+f x)+1)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{c \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}",1,"-(c*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)*f*(1 + Sin[e + f*x])) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f*(a + a*Sin[e + f*x])^2) - (c*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2764, 2978, 2752, 2663, 2661, 2655, 2653}"
513,1,257,0,0.4413462,"\int \frac{1}{(a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}+\frac{(c-2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}","-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}+\frac{(c-2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}",1,"-((c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2) - ((c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c - 2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2766, 2978, 2752, 2663, 2661, 2655, 2653}"
514,1,326,0,0.6416993,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{d \left(c^2-5 c d-12 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(c^2-5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^3 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-5 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}+\frac{(c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{d \left(c^2-5 c d-12 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(c^2-5 c d-12 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^3 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-5 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}}+\frac{(c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}",1,"-(d*(c^2 - 5*c*d - 12*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - ((c - 5*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) - ((c^2 - 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^3*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c - 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653}"
515,1,405,0,0.8336478,"\int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{d (c+3 d) \left(c^2-10 c d-7 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(c^2-7 c d-10 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{\left(c^2-7 c d-10 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \left(c^2-10 c d-7 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}","-\frac{d (c+3 d) \left(c^2-10 c d-7 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(c^2-7 c d-10 d^2\right) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{\left(c^2-7 c d-10 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \left(c^2-10 c d-7 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}",1,"-(d*(c^2 - 7*c*d - 10*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - ((c - 7*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) - (d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^4*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((c + 3*d)*(c^2 - 10*c*d - 7*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^4*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c^2 - 7*c*d - 10*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",9,8,27,0.2963,1,"{2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653}"
516,1,322,0,0.833098,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^3,x]","-\frac{\left(4 c^2+15 c d+27 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{(c+d) \left(4 c^2+11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2+15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}-\frac{2 (c-d) (c+3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2}","-\frac{\left(4 c^2+15 c d+27 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f \left(a^3 \sin (e+f x)+a^3\right)}+\frac{(c+d) \left(4 c^2+11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2+15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}-\frac{2 (c-d) (c+3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2}",1,"(-2*(c - d)*(c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 + 15*c*d + 27*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f*(a + a*Sin[e + f*x])^3) - ((4*c^2 + 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(4*c^2 + 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2765, 2977, 2978, 2752, 2663, 2661, 2655, 2653}"
517,1,323,0,0.8768793,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^3,x]","-\frac{\left(4 c^2+5 c d-3 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d) \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\left(4 c^2+5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(c+d) (4 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 (c+2 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}","-\frac{\left(4 c^2+5 c d-3 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d) \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\left(4 c^2+5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(c+d) (4 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 (c+2 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}",1,"-((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(5*f*(a + a*Sin[e + f*x])^3) - (2*(c + 2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 + 5*c*d - 3*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*(c - d)*f*(a^3 + a^3*Sin[e + f*x])) - ((4*c^2 + 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(4*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2765, 2978, 2752, 2663, 2661, 2655, 2653}"
518,1,334,0,0.7677211,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x])^3,x]","-\frac{\left(4 c^2-5 c d-3 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^2 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\left(4 c^2-5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(4 c-5 d) (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d) (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}","-\frac{\left(4 c^2-5 c d-3 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^2 \left(a^3 \sin (e+f x)+a^3\right)}-\frac{\left(4 c^2-5 c d-3 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(4 c-5 d) (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d) (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}",1,"-(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(5*f*(a + a*Sin[e + f*x])^3) - ((2*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*(c - d)*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 - 5*c*d - 3*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*(c - d)^2*f*(a^3 + a^3*Sin[e + f*x])) - ((4*c^2 - 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c - 5*d)*(c + d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2764, 2978, 2752, 2663, 2661, 2655, 2653}"
519,1,344,0,0.7690779,"\int \frac{1}{(a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{\left(4 c^2-15 c d+27 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}+\frac{\left(4 c^2-11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2-15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}","-\frac{\left(4 c^2-15 c d+27 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right)}+\frac{\left(4 c^2-11 c d+15 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2-15 c d+27 d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}",1,"-(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3) - (2*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 - 15*c*d + 27*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])) - ((4*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^2 - 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2766, 2978, 2752, 2663, 2661, 2655, 2653}"
520,1,423,0,1.0253143,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{d \left(-21 c^2 d+4 c^3+62 c d^2+147 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2-21 c d+65 d^2\right) \cos (e+f x)}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) \sqrt{c+d \sin (e+f x)}}+\frac{\left(4 c^2-21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left(-21 c^2 d+4 c^3+62 c d^2+147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^4 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt{c+d \sin (e+f x)}}","-\frac{d \left(-21 c^2 d+4 c^3+62 c d^2+147 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(4 c^2-21 c d+65 d^2\right) \cos (e+f x)}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) \sqrt{c+d \sin (e+f x)}}+\frac{\left(4 c^2-21 c d+65 d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left(-21 c^2 d+4 c^3+62 c d^2+147 d^3\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^4 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt{c+d \sin (e+f x)}}",1,"-(d*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]) - (2*(c - 4*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^2 - 21*c*d + 65*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",9,8,27,0.2963,1,"{2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653}"
521,1,518,0,1.2148641,"\int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{d \left(111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\left(4 c^2-27 c d+119 d^2\right) \cos (e+f x)}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{3/2}}+\frac{\left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}","-\frac{d \left(111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\left(4 c^2-27 c d+119 d^2\right) \cos (e+f x)}{30 f (c-d)^3 \left(a^3 \sin (e+f x)+a^3\right) (c+d \sin (e+f x))^{3/2}}+\frac{\left(-27 c^2 d+4 c^3+114 c d^2+165 d^3\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left(111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}",1,"-(d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)) - (2*(c - 5*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) - ((4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])","A",10,8,27,0.2963,1,"{2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653}"
522,1,161,0,0.2787616,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3,x]","-\frac{4 a (c+d) \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{12 d^2 (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt{a \sin (e+f x)+a}}-\frac{8 d (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{35 f}","-\frac{4 a (c+d) \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{35 f \sqrt{a \sin (e+f x)+a}}-\frac{12 d^2 (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt{a \sin (e+f x)+a}}-\frac{8 d (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{35 f}",1,"(-4*a*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(35*f*Sqrt[a + a*Sin[e + f*x]]) - (8*(5*c - d)*d*(c + d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(35*f) - (12*d^2*(c + d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*a*f) - (2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(7*f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,27,0.1481,1,"{2770, 2761, 2751, 2646}"
523,1,112,0,0.1685839,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2,x]","-\frac{2 a \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{4 d (5 c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}","-\frac{2 a \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{4 d (5 c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}",1,"(-2*a*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (4*(5*c - d)*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*f)","A",3,3,27,0.1111,1,"{2761, 2751, 2646}"
524,1,62,0,0.0554464,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x)) \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]),x]","-\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}","-\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"(-2*a*(3*c + d)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)","A",2,2,25,0.08000,1,"{2751, 2646}"
525,1,26,0,0.0138016,"\int \sqrt{a+a \sin (e+f x)} \, dx","Int[Sqrt[a + a*Sin[e + f*x]],x]","-\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}","-\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}",1,"(-2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]])","A",1,1,14,0.07143,1,"{2646}"
526,1,61,0,0.1150065,"\int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x]),x]","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{d} f \sqrt{c+d}}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{d} f \sqrt{c+d}}",1,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[d]*Sqrt[c + d]*f)","A",2,2,27,0.07407,1,"{2773, 208}"
527,1,105,0,0.1856319,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^2} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^2,x]","-\frac{a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{d} f (c+d)^{3/2}}","-\frac{a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{d} f (c+d)^{3/2}}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[d]*(c + d)^(3/2)*f)) - (a*Cos[e + f*x])/((c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",3,3,27,0.1111,1,"{2772, 2773, 208}"
528,1,154,0,0.2691557,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^3} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^3,x]","-\frac{3 a \cos (e+f x)}{4 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{a \cos (e+f x)}{2 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{d} f (c+d)^{5/2}}","-\frac{3 a \cos (e+f x)}{4 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{a \cos (e+f x)}{2 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{d} f (c+d)^{5/2}}",1,"(-3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[d]*(c + d)^(5/2)*f) - (a*Cos[e + f*x])/(2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*a*Cos[e + f*x])/(4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",4,3,27,0.1111,1,"{2772, 2773, 208}"
529,1,231,0,0.3844278,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3,x]","\frac{4 a^2 (c-17 d) (c+d) \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{315 d f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt{a \sin (e+f x)+a}}+\frac{4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}","\frac{4 a^2 (c-17 d) (c+d) \left(15 c^2+10 c d+7 d^2\right) \cos (e+f x)}{315 d f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt{a \sin (e+f x)+a}}+\frac{4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}",1,"(4*a^2*(c - 17*d)*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(315*d*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(c - 17*d)*(5*c - d)*(c + d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) + (4*(c - 17*d)*d*(c + d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) + (2*a^2*(c - 17*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*d*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(9*d*f*Sqrt[a + a*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2763, 21, 2770, 2761, 2751, 2646}"
530,1,157,0,0.2294631,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2,x]","-\frac{8 a^2 \left(35 c^2+42 c d+19 d^2\right) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left(35 c^2+42 c d+19 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{4 d (7 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f}","-\frac{8 a^2 \left(35 c^2+42 c d+19 d^2\right) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left(35 c^2+42 c d+19 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{4 d (7 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f}",1,"(-8*a^2*(35*c^2 + 42*c*d + 19*d^2)*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(35*c^2 + 42*c*d + 19*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) - (4*(7*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*a*f)","A",4,4,27,0.1481,1,"{2761, 2751, 2647, 2646}"
531,1,101,0,0.0842337,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x]),x]","-\frac{8 a^2 (5 c+3 d) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a (5 c+3 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}","-\frac{8 a^2 (5 c+3 d) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a (5 c+3 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}",1,"(-8*a^2*(5*c + 3*d)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(5*c + 3*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f)","A",3,3,25,0.1200,1,"{2751, 2647, 2646}"
532,1,59,0,0.0294619,"\int (a+a \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2),x]","-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}","-\frac{8 a^2 \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"(-8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)","A",2,2,14,0.1429,1,"{2647, 2646}"
533,1,98,0,0.1998002,"\int \frac{(a+a \sin (e+f x))^{3/2}}{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x]),x]","\frac{2 a^{3/2} (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{3/2} f \sqrt{c+d}}-\frac{2 a^2 \cos (e+f x)}{d f \sqrt{a \sin (e+f x)+a}}","\frac{2 a^{3/2} (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{3/2} f \sqrt{c+d}}-\frac{2 a^2 \cos (e+f x)}{d f \sqrt{a \sin (e+f x)+a}}",1,"(2*a^(3/2)*(c - d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(3/2)*Sqrt[c + d]*f) - (2*a^2*Cos[e + f*x])/(d*f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,27,0.1481,1,"{2763, 21, 2773, 208}"
534,1,119,0,0.1929159,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^2,x]","\frac{a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{a^{3/2} (c+3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{3/2} f (c+d)^{3/2}}","\frac{a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{a^{3/2} (c+3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{3/2} f (c+d)^{3/2}}",1,"-((a^(3/2)*(c + 3*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(3/2)*(c + d)^(3/2)*f)) + (a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",4,4,27,0.1481,1,"{2762, 21, 2773, 208}"
535,1,179,0,0.2904027,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^3,x]","-\frac{a^{3/2} (c+7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{3/2} f (c+d)^{5/2}}-\frac{a^2 (c+7 d) \cos (e+f x)}{4 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x)}{2 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}","-\frac{a^{3/2} (c+7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{3/2} f (c+d)^{5/2}}-\frac{a^2 (c+7 d) \cos (e+f x)}{4 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x)}{2 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}",1,"-(a^(3/2)*(c + 7*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(3/2)*(c + d)^(5/2)*f) + (a^2*(c - d)*Cos[e + f*x])/(2*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (a^2*(c + 7*d)*Cos[e + f*x])/(4*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",5,5,27,0.1852,1,"{2762, 21, 2772, 2773, 208}"
536,1,328,0,0.6563147,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3,x]","-\frac{2 a^3 \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{4 a^3 (c+d) \left(15 c^2+10 c d+7 d^2\right) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}-\frac{4 a (c+d) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}","-\frac{2 a^3 \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{8 a^2 (5 c-d) (c+d) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{4 a^3 (c+d) \left(15 c^2+10 c d+7 d^2\right) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-23 d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f}-\frac{4 a (c+d) \left(3 c^2-38 c d+355 d^2\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}",1,"(-4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x])/(3465*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (8*a^2*(5*c - d)*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d*f) - (4*a*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) - (2*a^3*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^3*(3*c - 23*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)","A",6,6,27,0.2222,1,"{2763, 2981, 2770, 2761, 2751, 2646}"
537,1,202,0,0.2738985,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2,x]","-\frac{16 a^2 \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{64 a^3 \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x)}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac{4 d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}","-\frac{16 a^2 \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{64 a^3 \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x)}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left(21 c^2+30 c d+13 d^2\right) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac{4 d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f}",1,"(-64*a^3*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x])/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) - (2*a*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) - (4*(9*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(63*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f)","A",5,4,27,0.1481,1,"{2761, 2751, 2647, 2646}"
538,1,138,0,0.1079618,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x]),x]","-\frac{64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}","-\frac{64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 c+5 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f}",1,"(-64*a^3*(7*c + 5*d)*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(7*c + 5*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) - (2*a*(7*c + 5*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*f) - (2*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*f)","A",4,3,25,0.1200,1,"{2751, 2647, 2646}"
539,1,89,0,0.0487699,"\int (a+a \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2),x]","-\frac{64 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}","-\frac{64 a^3 \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}",1,"(-64*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f)","A",3,2,14,0.1429,1,"{2647, 2646}"
540,1,142,0,0.4113991,"\int \frac{(a+a \sin (e+f x))^{5/2}}{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x]),x]","\frac{2 a^3 (3 c-7 d) \cos (e+f x)}{3 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^{5/2} (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{5/2} f \sqrt{c+d}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f}","\frac{2 a^3 (3 c-7 d) \cos (e+f x)}{3 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^{5/2} (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{5/2} f \sqrt{c+d}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f}",1,"(-2*a^(5/2)*(c - d)^2*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(5/2)*Sqrt[c + d]*f) + (2*a^3*(3*c - 7*d)*Cos[e + f*x])/(3*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*d*f)","A",4,4,27,0.1481,1,"{2763, 2981, 2773, 208}"
541,1,166,0,0.3888631,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^2,x]","-\frac{a^3 (3 c+d) \cos (e+f x)}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a^{5/2} (c-d) (3 c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{5/2} f (c+d)^{3/2}}+\frac{a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) (c+d \sin (e+f x))}","-\frac{a^3 (3 c+d) \cos (e+f x)}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a^{5/2} (c-d) (3 c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{d^{5/2} f (c+d)^{3/2}}+\frac{a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) (c+d \sin (e+f x))}",1,"(a^(5/2)*(c - d)*(3*c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(5/2)*(c + d)^(3/2)*f) - (a^3*(3*c + d)*Cos[e + f*x])/(d^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(d*(c + d)*f*(c + d*Sin[e + f*x]))","A",4,4,27,0.1481,1,"{2762, 2981, 2773, 208}"
542,1,194,0,0.4425315,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^3,x]","-\frac{a^{5/2} \left(3 c^2+10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{5/2} f (c+d)^{5/2}}+\frac{3 a^3 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{2 d f (c+d) (c+d \sin (e+f x))^2}","-\frac{a^{5/2} \left(3 c^2+10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 d^{5/2} f (c+d)^{5/2}}+\frac{3 a^3 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{2 d f (c+d) (c+d \sin (e+f x))^2}",1,"-(a^(5/2)*(3*c^2 + 10*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(5/2)*(c + d)^(5/2)*f) + (a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) + (3*a^3*(c - d)*(c + 3*d)*Cos[e + f*x])/(4*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",4,4,27,0.1481,1,"{2762, 2980, 2773, 208}"
543,1,178,0,0.4395246,"\int \frac{(c+d \sin (e+f x))^3}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^3/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{4 d \left(21 c^2-12 c d+7 d^2\right) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d^2 (9 c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 a f}-\frac{2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}","-\frac{4 d \left(21 c^2-12 c d+7 d^2\right) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d^2 (9 c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 a f}-\frac{2 d \cos (e+f x) (c+d \sin (e+f x))^2}{5 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}",1,"-((Sqrt[2]*(c - d)^3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (4*d*(21*c^2 - 12*c*d + 7*d^2)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*(9*c - d)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*a*f) - (2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*Sqrt[a + a*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2778, 2968, 3023, 2751, 2649, 206}"
544,1,123,0,0.2006239,"\int \frac{(c+d \sin (e+f x))^2}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^2/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{4 d (3 c-d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 d^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 a f}","-\frac{4 d (3 c-d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 d^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 a f}",1,"-((Sqrt[2]*(c - d)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (4*(3*c - d)*d*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*a*f)","A",4,4,27,0.1481,1,"{2761, 2751, 2649, 206}"
545,1,79,0,0.0699759,"\int \frac{c+d \sin (e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\sqrt{2} (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 d \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}","-\frac{\sqrt{2} (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}-\frac{2 d \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}",1,"-((Sqrt[2]*(c - d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (2*d*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]])","A",3,3,25,0.1200,1,"{2751, 2649, 206}"
546,1,47,0,0.0227762,"\int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[1/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f))","A",2,2,14,0.1429,1,"{2649, 206}"
547,1,123,0,0.2140569,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]","\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)}","\frac{2 \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f)) + (2*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)","A",5,5,27,0.1852,1,"{2780, 2649, 206, 2773, 208}"
548,1,175,0,0.4247369,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2),x]","\frac{d \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^2}+\frac{\sqrt{d} (3 c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^2 (c+d)^{3/2}}","\frac{d \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^2}+\frac{\sqrt{d} (3 c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^2 (c+d)^{3/2}}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^2*f)) + (Sqrt[d]*(3*c + d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^2*(c + d)^(3/2)*f) + (d*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",6,6,27,0.2222,1,"{2779, 2985, 2649, 206, 2773, 208}"
549,1,247,0,0.7321427,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3),x]","\frac{d (7 c+d) \cos (e+f x)}{4 f \left(c^2-d^2\right)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{d \cos (e+f x)}{2 f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{\sqrt{d} \left(15 c^2+10 c d+7 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{a} f (c-d)^3 (c+d)^{5/2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^3}","\frac{d (7 c+d) \cos (e+f x)}{4 f \left(c^2-d^2\right)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac{d \cos (e+f x)}{2 f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{\sqrt{d} \left(15 c^2+10 c d+7 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 \sqrt{a} f (c-d)^3 (c+d)^{5/2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{\sqrt{a} f (c-d)^3}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^3*f)) + (Sqrt[d]*(15*c^2 + 10*c*d + 7*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[a]*(c - d)^3*(c + d)^(5/2)*f) + (d*Cos[e + f*x])/(2*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + (d*(7*c + d)*Cos[e + f*x])/(4*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",7,7,27,0.2593,1,"{2779, 2984, 2985, 2649, 206, 2773, 208}"
550,1,192,0,0.4627575,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^(3/2),x]","\frac{d^2 (3 c-7 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{6 a^2 f}-\frac{(c+11 d) (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{d \left(3 c^2-24 c d+13 d^2\right) \cos (e+f x)}{3 a f \sqrt{a \sin (e+f x)+a}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}","\frac{d^2 (3 c-7 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{6 a^2 f}-\frac{(c+11 d) (c-d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{d \left(3 c^2-24 c d+13 d^2\right) \cos (e+f x)}{3 a f \sqrt{a \sin (e+f x)+a}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-((c - d)^2*(c + 11*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) + (d*(3*c^2 - 24*c*d + 13*d^2)*Cos[e + f*x])/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*c - 7*d)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(6*a^2*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",6,6,27,0.2222,1,"{2765, 2968, 3023, 2751, 2649, 206}"
551,1,138,0,0.2159458,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{(c-d) (c+7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (c-5 d) \cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a}}","-\frac{(c-d) (c+7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (c-5 d) \cos (e+f x)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"-((c - d)*(c + 7*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) + ((c - 5*d)*d*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",4,4,27,0.1481,1,"{2760, 2751, 2649, 206}"
552,1,87,0,0.0727975,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{(c+3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}}","-\frac{(c+3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-((c + 3*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((c - d)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",3,3,25,0.1200,1,"{2750, 2649, 206}"
553,1,77,0,0.0405825,"\int \frac{1}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^(-3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]/(2*Sqrt[2]*a^(3/2)*f) - Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",3,3,14,0.2143,1,"{2650, 2649, 206}"
554,1,164,0,0.4165881,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])),x]","-\frac{2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)^2 \sqrt{c+d}}-\frac{(c-5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^2}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2}}","-\frac{2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)^2 \sqrt{c+d}}-\frac{(c-5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^2}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2}}",1,"-((c - 5*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^2*f) - (2*d^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)^2*Sqrt[c + d]*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))","A",6,6,27,0.2222,1,"{2766, 2985, 2649, 206, 2773, 208}"
555,1,243,0,0.7429847,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2),x]","-\frac{d^{3/2} (5 c+3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)^3 (c+d)^{3/2}}-\frac{(c-9 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^3}-\frac{d (c+3 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}","-\frac{d^{3/2} (5 c+3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)^3 (c+d)^{3/2}}-\frac{(c-9 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^3}-\frac{d (c+3 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}",1,"-((c - 9*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^3*f) - (d^(3/2)*(5*c + 3*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)^3*(c + d)^(3/2)*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) - (d*(c + 3*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",7,7,27,0.2593,1,"{2766, 2984, 2985, 2649, 206, 2773, 208}"
556,1,318,0,1.1080884,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3),x]","-\frac{d^{3/2} \left(35 c^2+42 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 a^{3/2} f (c-d)^4 (c+d)^{5/2}}-\frac{(c-13 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^4}-\frac{d (2 c+d) (c+7 d) \cos (e+f x)}{4 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{d (c+2 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}","-\frac{d^{3/2} \left(35 c^2+42 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 a^{3/2} f (c-d)^4 (c+d)^{5/2}}-\frac{(c-13 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^4}-\frac{d (2 c+d) (c+7 d) \cos (e+f x)}{4 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{d (c+2 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}",1,"-((c - 13*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^4*f) - (d^(3/2)*(35*c^2 + 42*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(3/2)*(c - d)^4*(c + d)^(5/2)*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) - (d*(c + 2*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (d*(2*c + d)*(c + 7*d)*Cos[e + f*x])/(4*a*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",8,7,27,0.2593,1,"{2766, 2984, 2985, 2649, 206, 2773, 208}"
557,1,194,0,0.4691943,"\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{3 \left(c^2+6 c d+25 d^2\right) (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{d^2 (c-9 d) \cos (e+f x)}{4 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{(3 c+13 d) (c-d)^2 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a \sin (e+f x)+a)^{5/2}}","-\frac{3 \left(c^2+6 c d+25 d^2\right) (c-d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}+\frac{d^2 (c-9 d) \cos (e+f x)}{4 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{(3 c+13 d) (c-d)^2 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"(-3*(c - d)*(c^2 + 6*c*d + 25*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)^2*(3*c + 13*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) + ((c - 9*d)*d^2*Cos[e + f*x])/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(4*f*(a + a*Sin[e + f*x])^(5/2))","A",6,6,27,0.2222,1,"{2765, 2968, 3019, 2751, 2649, 206}"
558,1,147,0,0.2298357,"\int \frac{(c+d \sin (e+f x))^2}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{\left(3 c^2+10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{3 (c-d) (c+3 d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}","-\frac{\left(3 c^2+10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{3 (c-d) (c+3 d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}",1,"-((3*c^2 + 10*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - (3*(c - d)*(c + 3*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(4*f*(a + a*Sin[e + f*x])^(5/2))","A",4,4,27,0.1481,1,"{2760, 2750, 2649, 206}"
559,1,126,0,0.0979192,"\int \frac{c+d \sin (e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{(3 c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{(3 c+5 d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}","-\frac{(3 c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{(3 c+5 d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"-((3*c + 5*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c + 5*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))","A",4,4,25,0.1600,1,"{2750, 2650, 2649, 206}"
560,1,107,0,0.0594574,"\int \frac{1}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^(-5/2),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{3 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{3 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)) - (3*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))","A",4,3,14,0.2143,1,"{2650, 2649, 206}"
561,1,218,0,0.741456,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])),x]","-\frac{\left(3 c^2-14 c d+43 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^3}+\frac{2 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f (c-d)^3 \sqrt{c+d}}-\frac{(3 c-11 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2}}","-\frac{\left(3 c^2-14 c d+43 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^3}+\frac{2 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f (c-d)^3 \sqrt{c+d}}-\frac{(3 c-11 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2}}",1,"-((3*c^2 - 14*c*d + 43*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^3*f) + (2*d^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^3*Sqrt[c + d]*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c - 11*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2))","A",7,7,27,0.2593,1,"{2766, 2978, 2985, 2649, 206, 2773, 208}"
562,1,313,0,1.099692,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2),x]","-\frac{\left(3 c^2-22 c d+115 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^4}+\frac{d^{5/2} (7 c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac{d (c-7 d) (3 c+5 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{3 (c-5 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}","-\frac{\left(3 c^2-22 c d+115 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^4}+\frac{d^{5/2} (7 c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{5/2} f (c-d)^4 (c+d)^{3/2}}-\frac{d (c-7 d) (3 c+5 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{3 (c-5 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))}",1,"-((3*c^2 - 22*c*d + 115*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^4*f) + (d^(5/2)*(7*c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^4*(c + d)^(3/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])) - (3*(c - 5*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) - ((c - 7*d)*d*(3*c + 5*d)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",8,8,27,0.2963,1,"{2766, 2978, 2984, 2985, 2649, 206, 2773, 208}"
563,1,400,0,1.5150491,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^3} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3),x]","-\frac{3 d (c+3 d) \left(c^2-10 c d-7 d^2\right) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{d \left(3 c^2-20 c d-31 d^2\right) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{3 d^{5/2} \left(21 c^2+30 c d+13 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac{3 \left(c^2-10 c d+73 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^5}-\frac{(3 c-19 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}","-\frac{3 d (c+3 d) \left(c^2-10 c d-7 d^2\right) \cos (e+f x)}{16 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{d \left(3 c^2-20 c d-31 d^2\right) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^2}+\frac{3 d^{5/2} \left(21 c^2+30 c d+13 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{4 a^{5/2} f (c-d)^5 (c+d)^{5/2}}-\frac{3 \left(c^2-10 c d+73 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^5}-\frac{(3 c-19 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^2}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^2}",1,"(-3*(c^2 - 10*c*d + 73*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^5*f) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2) - ((3*c - 19*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) - (d*(3*c^2 - 20*c*d - 31*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))","A",9,8,27,0.2963,1,"{2766, 2978, 2984, 2985, 2649, 206, 2773, 208}"
564,1,203,0,0.4145284,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{5 \sqrt{a} (c+d)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 \sqrt{d} f}","-\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{5 \sqrt{a} (c+d)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 \sqrt{d} f}",1,"(-5*Sqrt[a]*(c + d)^3*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(8*Sqrt[d]*f) - (5*a*(c + d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (5*a*(c + d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])","A",5,3,29,0.1034,1,"{2770, 2775, 205}"
565,1,156,0,0.2903057,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{3 \sqrt{a} (c+d)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 \sqrt{d} f}","-\frac{3 a (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{3 \sqrt{a} (c+d)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 \sqrt{d} f}",1,"(-3*Sqrt[a]*(c + d)^2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*Sqrt[d]*f) - (3*a*(c + d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])","A",4,3,29,0.1034,1,"{2770, 2775, 205}"
566,1,105,0,0.1823734,"\int \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{a \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{a} (c+d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{d} f}","-\frac{a \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{a} (c+d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{d} f}",1,"-((Sqrt[a]*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[d]*f)) - (a*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])","A",3,3,29,0.1034,1,"{2770, 2775, 205}"
567,1,61,0,0.0925582,"\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{d} f}","-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{d} f}",1,"(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[d]*f)","A",2,2,29,0.06897,1,"{2775, 205}"
568,1,45,0,0.092265,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}","-\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}",1,"(-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",1,1,29,0.03448,1,"{2771}"
569,1,95,0,0.1920688,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}","-\frac{4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}",1,"(-2*a*Cos[e + f*x])/(3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a*Cos[e + f*x])/(3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",2,2,29,0.06897,1,"{2772, 2771}"
570,1,142,0,0.2947394,"\int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{16 a \cos (e+f x)}{15 f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a \cos (e+f x)}{15 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a \cos (e+f x)}{5 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}","-\frac{16 a \cos (e+f x)}{15 f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a \cos (e+f x)}{15 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a \cos (e+f x)}{5 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}",1,"(-2*a*Cos[e + f*x])/(5*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a*Cos[e + f*x])/(15*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a*Cos[e + f*x])/(15*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",3,2,29,0.06897,1,"{2772, 2771}"
571,1,285,0,0.5666705,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2),x]","\frac{5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{64 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a \sin (e+f x)+a}}","\frac{5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{64 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a \sin (e+f x)+a}}",1,"(5*a^(3/2)*(c - 15*d)*(c + d)^3*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(64*d^(3/2)*f) + (5*a^2*(c - 15*d)*(c + d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(64*d*f*Sqrt[a + a*Sin[e + f*x]]) + (5*a^2*(c - 15*d)*(c + d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(96*d*f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(c - 15*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(24*d*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(4*d*f*Sqrt[a + a*Sin[e + f*x]])","A",7,5,29,0.1724,1,"{2763, 21, 2770, 2775, 205}"
572,1,228,0,0.4349599,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2),x]","\frac{a^{3/2} (c-11 d) (c+d)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 d f \sqrt{a \sin (e+f x)+a}}","\frac{a^{3/2} (c-11 d) (c+d)^2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 d f \sqrt{a \sin (e+f x)+a}}",1,"(a^(3/2)*(c - 11*d)*(c + d)^2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(8*d^(3/2)*f) + (a^2*(c - 11*d)*(c + d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*d*f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(c - 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*d*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(3*d*f*Sqrt[a + a*Sin[e + f*x]])","A",6,5,29,0.1724,1,"{2763, 21, 2770, 2775, 205}"
573,1,171,0,0.3108607,"\int (a+a \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]],x]","\frac{a^{3/2} (c-7 d) (c+d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-7 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d f \sqrt{a \sin (e+f x)+a}}","\frac{a^{3/2} (c-7 d) (c+d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-7 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d f \sqrt{a \sin (e+f x)+a}}",1,"(a^(3/2)*(c - 7*d)*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*d^(3/2)*f) + (a^2*(c - 7*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*d*f*Sqrt[a + a*Sin[e + f*x]])","A",5,5,29,0.1724,1,"{2763, 21, 2770, 2775, 205}"
574,1,111,0,0.20945,"\int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/Sqrt[c + d*Sin[e + f*x]],x]","\frac{a^{3/2} (c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} f}-\frac{a^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d f \sqrt{a \sin (e+f x)+a}}","\frac{a^{3/2} (c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} f}-\frac{a^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d f \sqrt{a \sin (e+f x)+a}}",1,"(a^(3/2)*(c - 3*d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(3/2)*f) - (a^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2763, 21, 2775, 205}"
575,1,117,0,0.2136534,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} f}","\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{3/2} f}",1,"(-2*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(3/2)*f) + (2*a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2762, 21, 2775, 205}"
576,1,115,0,0.2145422,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 a^2 (c-d) \cos (e+f x)}{3 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+5 d) \cos (e+f x)}{3 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}","\frac{2 a^2 (c-d) \cos (e+f x)}{3 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+5 d) \cos (e+f x)}{3 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}",1,"(2*a^2*(c - d)*Cos[e + f*x])/(3*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*a^2*(c + 5*d)*Cos[e + f*x])/(3*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",3,3,29,0.1034,1,"{2762, 21, 2771}"
577,1,172,0,0.324019,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{4 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}","-\frac{4 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^2 (c+9 d) \cos (e+f x)}{15 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x])/(5*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (2*a^2*(c + 9*d)*Cos[e + f*x])/(15*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c + 9*d)*Cos[e + f*x])/(15*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2762, 21, 2772, 2771}"
578,1,229,0,0.4479416,"\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]","-\frac{16 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}","-\frac{16 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^2*(c + 13*d)*Cos[e + f*x])/(35*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^2*(c + 13*d)*Cos[e + f*x])/(105*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^2*(c + 13*d)*Cos[e + f*x])/(105*d*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",5,4,29,0.1379,1,"{2762, 21, 2772, 2771}"
579,1,377,0,0.8549357,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{a^3 \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d)^2 \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d)^3 \left(3 c^2-34 c d+283 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{128 d^{5/2} f}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}","-\frac{a^3 \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d)^2 \left(3 c^2-34 c d+283 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d)^3 \left(3 c^2-34 c d+283 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{128 d^{5/2} f}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}",1,"-(a^(5/2)*(c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(128*d^(5/2)*f) - (a^3*(c + d)^2*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(128*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(c + d)*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(192*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(240*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*a^3*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(40*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2))/(5*d*f)","A",7,5,29,0.1724,1,"{2763, 2981, 2770, 2775, 205}"
580,1,312,0,0.7261025,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{a^3 \left(3 c^2-26 c d+163 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left(3 c^2-26 c d+163 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d)^2 \left(3 c^2-26 c d+163 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{64 d^{5/2} f}+\frac{a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}{4 d f}","-\frac{a^3 \left(3 c^2-26 c d+163 d^2\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left(3 c^2-26 c d+163 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d)^2 \left(3 c^2-26 c d+163 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{64 d^{5/2} f}+\frac{a^3 (3 c-17 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}{4 d f}",1,"-(a^(5/2)*(c + d)^2*(3*c^2 - 26*c*d + 163*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(64*d^(5/2)*f) - (a^3*(c + d)*(3*c^2 - 26*c*d + 163*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(64*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(3*c^2 - 26*c*d + 163*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(96*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a^3*(3*c - 17*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(24*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))/(4*d*f)","A",6,5,29,0.1724,1,"{2763, 2981, 2770, 2775, 205}"
581,1,241,0,0.5759599,"\int (a+a \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{a^3 \left(c^2-6 c d+25 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d) \left(c^2-6 c d+25 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 d^{5/2} f}+\frac{a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}{3 d f}","-\frac{a^3 \left(c^2-6 c d+25 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d) \left(c^2-6 c d+25 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{8 d^{5/2} f}+\frac{a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}{3 d f}",1,"-(a^(5/2)*(c + d)*(c^2 - 6*c*d + 25*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(8*d^(5/2)*f) - (a^3*(c^2 - 6*c*d + 25*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a^3*(3*c - 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*d*f)","A",5,5,29,0.1724,1,"{2763, 2981, 2770, 2775, 205}"
582,1,178,0,0.433055,"\int \frac{(a+a \sin (e+f x))^{5/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{a^{5/2} \left(3 c^2-10 c d+19 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 d^{5/2} f}+\frac{3 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}{2 d f}","-\frac{a^{5/2} \left(3 c^2-10 c d+19 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 d^{5/2} f}+\frac{3 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}{2 d f}",1,"-(a^(5/2)*(3*c^2 - 10*c*d + 19*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*d^(5/2)*f) + (3*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*d*f)","A",4,4,29,0.1379,1,"{2763, 2981, 2775, 205}"
583,1,180,0,0.4381981,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{a^3 (3 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a^{5/2} (3 c-5 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} f}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) \sqrt{c+d \sin (e+f x)}}","-\frac{a^3 (3 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a^{5/2} (3 c-5 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} f}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) \sqrt{c+d \sin (e+f x)}}",1,"(a^(5/2)*(3*c - 5*d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(5/2)*f) + (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (a^3*(3*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2762, 2981, 2775, 205}"
584,1,183,0,0.4439843,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 a^3 (c-d) (3 c+7 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} f}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}","\frac{2 a^3 (c-d) (3 c+7 d) \cos (e+f x)}{3 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{d^{5/2} f}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}",1,"(-2*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(5/2)*f) + (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*a^3*(c - d)*(3*c + 7*d)*Cos[e + f*x])/(3*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2762, 2980, 2775, 205}"
585,1,189,0,0.484717,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{2 a^3 \left(3 c^2+14 c d+43 d^2\right) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 a^3 (c-d) (3 c+11 d) \cos (e+f x)}{15 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}","-\frac{2 a^3 \left(3 c^2+14 c d+43 d^2\right) \cos (e+f x)}{15 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 a^3 (c-d) (3 c+11 d) \cos (e+f x)}{15 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) + (2*a^3*(c - d)*(3*c + 11*d)*Cos[e + f*x])/(15*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*a^3*(3*c^2 + 14*c*d + 43*d^2)*Cos[e + f*x])/(15*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",3,3,29,0.1034,1,"{2762, 2980, 2771}"
586,1,254,0,0.6253489,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(9/2),x]","-\frac{4 a^3 \left(3 c^2+22 c d+115 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^3 \left(3 c^2+22 c d+115 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}","-\frac{4 a^3 \left(3 c^2+22 c d+115 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a^3 \left(3 c^2+22 c d+115 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}+\frac{6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(7*d*(c + d)*f*(c + d*Sin[e + f*x])^(7/2)) + (6*a^3*(c - d)*(c + 5*d)*Cos[e + f*x])/(35*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (2*a^3*(3*c^2 + 22*c*d + 115*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a^3*(3*c^2 + 22*c*d + 115*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2762, 2980, 2772, 2771}"
587,1,317,0,0.7757089,"\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx","Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]","-\frac{16 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}","-\frac{16 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^3 \left(c^2+10 c d+73 d^2\right) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}}",1,"(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(9*d*(c + d)*f*(c + d*Sin[e + f*x])^(9/2)) + (2*a^3*(c - d)*(3*c + 19*d)*Cos[e + f*x])/(63*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",5,4,29,0.1379,1,"{2762, 2980, 2772, 2771}"
588,1,249,0,0.9287634,"\int \frac{(c+d \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\sqrt{d} \left(15 c^2-10 c d+7 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 \sqrt{a} f}-\frac{d \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{d (7 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}","-\frac{\sqrt{d} \left(15 c^2-10 c d+7 d^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{4 \sqrt{a} f}-\frac{d \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}}-\frac{d (7 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{2} (c-d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"-(Sqrt[d]*(15*c^2 - 10*c*d + 7*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*Sqrt[a]*f) - (Sqrt[2]*(c - d)^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) - ((7*c - d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + a*Sin[e + f*x]]) - (d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])","A",7,7,29,0.2414,1,"{2778, 2983, 2982, 2782, 208, 2775, 205}"
589,1,188,0,0.5978109,"\int \frac{(c+d \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{d \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{d} (3 c-d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}","-\frac{d \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a \sin (e+f x)+a}}-\frac{\sqrt{d} (3 c-d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"-(((3*c - d)*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)) - (Sqrt[2]*(c - d)^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) - (d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])","A",6,6,29,0.2069,1,"{2778, 2982, 2782, 208, 2775, 205}"
590,1,141,0,0.2963267,"\int \frac{\sqrt{c+d \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}","-\frac{2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f}",1,"(-2*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) - (Sqrt[2]*Sqrt[c - d]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)","A",5,5,29,0.1724,1,"{2777, 2775, 205, 2782, 208}"
591,1,79,0,0.1072811,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f \sqrt{c-d}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f \sqrt{c-d}}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f))","A",2,2,29,0.06897,1,"{2782, 208}"
592,1,131,0,0.2428677,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{2 d \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)^{3/2}}","\frac{2 d \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)^{3/2}}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^(3/2)*f)) + (2*d*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,29,0.1379,1,"{2779, 12, 2782, 208}"
593,1,191,0,0.4759024,"\int \frac{1}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{2 d (5 c+d) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 d \cos (e+f x)}{3 f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)^{5/2}}","\frac{2 d (5 c+d) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}+\frac{2 d \cos (e+f x)}{3 f \left(c^2-d^2\right) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{\sqrt{a} f (c-d)^{5/2}}",1,"-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^(5/2)*f)) + (2*d*Cos[e + f*x])/(3*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) + (2*d*(5*c + d)*Cos[e + f*x])/(3*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",5,5,29,0.1724,1,"{2779, 2984, 12, 2782, 208}"
594,1,251,0,0.9027149,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{d^{3/2} (5 c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f}-\frac{(c+9 d) (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 a f \sqrt{a \sin (e+f x)+a}}","-\frac{d^{3/2} (5 c-3 d) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f}-\frac{(c+9 d) (c-d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{3/2}}+\frac{d (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"-(((5*c - 3*d)*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*f)) - ((c - d)^(3/2)*(c + 9*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) + ((c - 3*d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",7,7,29,0.2414,1,"{2765, 2983, 2982, 2782, 208, 2775, 205}"
595,1,194,0,0.5641862,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f}-\frac{\sqrt{c-d} (c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}","-\frac{2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f}-\frac{\sqrt{c-d} (c+5 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"(-2*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*f) - (Sqrt[c - d]*(c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",6,6,29,0.2069,1,"{2765, 2982, 2782, 208, 2775, 205}"
596,1,126,0,0.2205654,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{(c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{c-d}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}","-\frac{(c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{c-d}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}",1,"-((c + d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*Sqrt[c - d]*f) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*(a + a*Sin[e + f*x])^(3/2))","A",4,4,29,0.1379,1,"{2764, 12, 2782, 208}"
597,1,135,0,0.2368372,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{(c-3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (c-d) (a \sin (e+f x)+a)^{3/2}}","-\frac{(c-3 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f (c-d) (a \sin (e+f x)+a)^{3/2}}",1,"-((c - 3*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^(3/2)*f) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))","A",4,4,29,0.1379,1,"{2766, 12, 2782, 208}"
598,1,197,0,0.4942964,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{(c-7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{5/2}}-\frac{d (c+5 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt{c+d \sin (e+f x)}}","-\frac{(c-7 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{5/2}}-\frac{d (c+5 d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} \sqrt{c+d \sin (e+f x)}}",1,"-((c - 7*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^(5/2)*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) - (d*(c + 5*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",5,5,29,0.1724,1,"{2766, 2984, 12, 2782, 208}"
599,1,271,0,0.8547854,"\int \frac{1}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{(c-11 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{7/2}}-\frac{d \left(3 c^2+38 c d+19 d^2\right) \cos (e+f x)}{6 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{d (3 c+7 d) \cos (e+f x)}{6 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}","-\frac{(c-11 d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-d)^{7/2}}-\frac{d \left(3 c^2+38 c d+19 d^2\right) \cos (e+f x)}{6 a f (c-d)^3 (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{d (3 c+7 d) \cos (e+f x)}{6 a f (c-d)^2 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 f (c-d) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}",1,"-((c - 11*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^(7/2)*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c + 7*d)*Cos[e + f*x])/(6*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 38*c*d + 19*d^2)*Cos[e + f*x])/(6*a*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",6,5,29,0.1724,1,"{2766, 2984, 12, 2782, 208}"
600,1,260,0,0.8640621,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{\sqrt{c-d} \left(3 c^2+14 c d+43 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{2 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{5/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{(c-d) (3 c+11 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}}","-\frac{\sqrt{c-d} \left(3 c^2+14 c d+43 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f}-\frac{2 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{5/2} f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}-\frac{(c-d) (3 c+11 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}}",1,"(-2*d^(5/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(5/2)*f) - (Sqrt[c - d]*(3*c^2 + 14*c*d + 43*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*(3*c + 11*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))","A",7,7,29,0.2414,1,"{2765, 2977, 2982, 2782, 208, 2775, 205}"
601,1,184,0,0.5380814,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{3 (c+d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f \sqrt{c-d}}-\frac{(3 c+7 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}}","-\frac{3 (c+d)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f \sqrt{c-d}}-\frac{(3 c+7 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac{(c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"(-3*(c + d)^2*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*Sqrt[c - d]*f) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c + 7*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))","A",5,5,29,0.1724,1,"{2765, 2978, 12, 2782, 208}"
602,1,191,0,0.4883924,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{(3 c-5 d) (c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{3/2}}-\frac{(3 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (c-d) (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}}","-\frac{(3 c-5 d) (c+d) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{3/2}}-\frac{(3 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (c-d) (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}}",1,"-((3*c - 5*d)*(c + d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(3/2)*f) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))","A",5,5,29,0.1724,1,"{2764, 2978, 12, 2782, 208}"
603,1,201,0,0.4941969,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{\left(3 c^2-10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{5/2}}-\frac{3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (c-d) (a \sin (e+f x)+a)^{5/2}}","-\frac{\left(3 c^2-10 c d+19 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{5/2}}-\frac{3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f (c-d) (a \sin (e+f x)+a)^{5/2}}",1,"-((3*c^2 - 10*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(5/2)*f) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)) - (3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2))","A",5,5,29,0.1724,1,"{2766, 2978, 12, 2782, 208}"
604,1,270,0,0.8613449,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{3 \left(c^2-6 c d+25 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{7/2}}-\frac{d (c-7 d) (3 c+7 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{(3 c-13 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt{c+d \sin (e+f x)}}","-\frac{3 \left(c^2-6 c d+25 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{7/2}}-\frac{d (c-7 d) (3 c+7 d) \cos (e+f x)}{16 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{(3 c-13 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} \sqrt{c+d \sin (e+f x)}}",1,"(-3*(c^2 - 6*c*d + 25*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(7/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]) - ((3*c - 13*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) - ((c - 7*d)*d*(3*c + 7*d)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",6,6,29,0.2069,1,"{2766, 2978, 2984, 12, 2782, 208}"
605,1,355,0,1.2640335,"\int \frac{1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{d \left(-57 c^2 d+9 c^3-493 c d^2-299 d^3\right) \cos (e+f x)}{48 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{d \left(9 c^2-54 c d-95 d^2\right) \cos (e+f x)}{48 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\left(3 c^2-26 c d+163 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{9/2}}-\frac{(3 c-17 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}","-\frac{d \left(-57 c^2 d+9 c^3-493 c d^2-299 d^3\right) \cos (e+f x)}{48 a^2 f (c-d)^4 (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{d \left(9 c^2-54 c d-95 d^2\right) \cos (e+f x)}{48 a^2 f (c-d)^3 (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{\left(3 c^2-26 c d+163 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{16 \sqrt{2} a^{5/2} f (c-d)^{9/2}}-\frac{(3 c-17 d) \cos (e+f x)}{16 a f (c-d)^2 (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{4 f (c-d) (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{3/2}}",1,"-((3*c^2 - 26*c*d + 163*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(9/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)) - ((3*c - 17*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) - (d*(9*c^2 - 54*c*d - 95*d^2)*Cos[e + f*x])/(48*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (d*(9*c^3 - 57*c^2*d - 493*c*d^2 - 299*d^3)*Cos[e + f*x])/(48*a^2*(c - d)^4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",7,6,29,0.2069,1,"{2766, 2978, 2984, 12, 2782, 208}"
606,1,129,0,0.1681112,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{1}{2};\frac{1}{2},-n;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{1}{2};\frac{1}{2},-n;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, -n, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n)","A",4,4,25,0.1600,1,"{2788, 140, 139, 138}"
607,1,320,0,0.6625314,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3,x]","-\frac{2^{m+\frac{1}{2}} \left(3 c^2 d m \left(m^2+5 m+6\right)+c^3 \left(m^3+6 m^2+11 m+6\right)+3 c d^2 \left(m^3+4 m^2+4 m+3\right)+d^3 m \left(m^2+3 m+5\right)\right) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1) (m+2) (m+3)}-\frac{d \left(2 c^2 \left(m^2+6 m+8\right)-c d \left(-2 m^2-3 m+5\right)+d^2 (m+4)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}-\frac{d^2 (c (m+5)+d m) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2) (m+3)}-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^2}{f (m+3)}","-\frac{2^{m+\frac{1}{2}} \left(3 c^2 d m \left(m^2+5 m+6\right)+c^3 \left(m^3+6 m^2+11 m+6\right)+3 c d^2 \left(m^3+4 m^2+4 m+3\right)+d^3 m \left(m^2+3 m+5\right)\right) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1) (m+2) (m+3)}-\frac{d \left(2 c^2 \left(m^2+6 m+8\right)-c d \left(-2 m^2-3 m+5\right)+d^2 (m+4)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}-\frac{d^2 (c (m+5)+d m) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2) (m+3)}-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^2}{f (m+3)}",1,"-((d*(d^2*(4 + m) - c*d*(5 - 3*m - 2*m^2) + 2*c^2*(8 + 6*m + m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)*(3 + m))) - (2^(1/2 + m)*(d^3*m*(5 + 3*m + m^2) + 3*c^2*d*m*(6 + 5*m + m^2) + 3*c*d^2*(3 + 4*m + 4*m^2 + m^3) + c^3*(6 + 11*m + 6*m^2 + m^3))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)*(3 + m)) - (d^2*(d*m + c*(5 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)*(3 + m)) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2)/(f*(3 + m))","A",6,6,25,0.2400,1,"{2783, 2968, 3023, 2751, 2652, 2651}"
608,1,193,0,0.2688185,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]","-\frac{2^{m+\frac{1}{2}} \left(c^2 \left(m^2+3 m+2\right)+2 c d m (m+2)+d^2 \left(m^2+m+1\right)\right) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1) (m+2)}+\frac{d (d-2 c (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1) (m+2)}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)}","-\frac{2^{m+\frac{1}{2}} \left(c^2 \left(m^2+3 m+2\right)+2 c d m (m+2)+d^2 \left(m^2+m+1\right)\right) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1) (m+2)}+\frac{d (d-2 c (m+2)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1) (m+2)}-\frac{d^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)}",1,"(d*(d - 2*c*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m))","A",4,4,25,0.1600,1,"{2761, 2751, 2652, 2651}"
609,1,117,0,0.0796591,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]","-\frac{2^{m+\frac{1}{2}} (c m+c+d m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1)}-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}","-\frac{2^{m+\frac{1}{2}} (c m+c+d m) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (m+1)}-\frac{d \cos (e+f x) (a \sin (e+f x)+a)^m}{f (m+1)}",1,"-((d*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*(c + c*m + d*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m))","A",3,3,23,0.1304,1,"{2751, 2652, 2651}"
610,1,74,0,0.0299459,"\int (a+a \sin (e+f x))^m \, dx","Int[(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"-((2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)","A",2,2,12,0.1667,1,"{2652, 2651}"
611,1,100,0,0.128288,"\int \frac{(a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x]),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])","A",3,3,25,0.1200,1,"{2788, 137, 136}"
612,1,100,0,0.1257839,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","Int[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2,x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)^2 \sqrt{1-\sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)^2 \sqrt{1-\sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)^2*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])","A",3,3,25,0.1200,1,"{2788, 137, 136}"
613,1,100,0,0.1262228,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","Int[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3,x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},3;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)^3 \sqrt{1-\sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},3;m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)^3 \sqrt{1-\sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 3, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)^3*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])","A",3,3,25,0.1200,1,"{2788, 137, 136}"
614,1,138,0,0.1869487,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]","\frac{\sqrt{2} (c-d)^2 \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{5}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}","\frac{\sqrt{2} (c-d)^2 \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{5}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}",1,"(Sqrt[2]*(c - d)^2*AppellF1[1/2 + m, 1/2, -5/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",4,4,27,0.1481,1,"{2788, 140, 139, 138}"
615,1,136,0,0.1840058,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{2} (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}","\frac{\sqrt{2} (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}",1,"(Sqrt[2]*(c - d)*AppellF1[1/2 + m, 1/2, -3/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",4,4,27,0.1481,1,"{2788, 140, 139, 138}"
616,1,131,0,0.1648518,"\int (a+a \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{1}{2};\frac{1}{2},-\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, -1/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",4,4,27,0.1481,1,"{2788, 140, 139, 138}"
617,1,131,0,0.1635894,"\int \frac{(a+a \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,27,0.1481,1,"{2788, 140, 139, 138}"
618,1,138,0,0.1822392,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{3}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 3/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,27,0.1481,1,"{2788, 140, 139, 138}"
619,1,138,0,0.1800859,"\int \frac{(a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{5}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{1}{2};\frac{1}{2},\frac{5}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{f (2 m+1) (c-d)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}",1,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 5/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d)^2*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",4,4,27,0.1481,1,"{2788, 140, 139, 138}"
620,1,111,0,0.1060976,"\int (1+\sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx","Int[(1 + Sin[e + f*x])^m*(3 + 5*Sin[e + f*x])^(-1 - m),x]","-\frac{2^{-2 m-1} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{5 \sin (e+f x)+3}\right)^{\frac{1}{2}-m} (5 \sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};-\frac{1-\sin (e+f x)}{5 \sin (e+f x)+3}\right)}{f}","-\frac{4^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{1-\sin (e+f x)}{4 (\sin (e+f x)+1)}\right)}{f (\sin (e+f x)+1)}",1,"-((2^(-1 - 2*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, -((1 - Sin[e + f*x])/(3 + 5*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + 5*Sin[e + f*x]))^(1/2 - m))/(f*(3 + 5*Sin[e + f*x])^m))","A",2,2,27,0.07407,1,"{2788, 132}"
621,1,122,0,0.1157721,"\int (1+\sin (e+f x))^m (3+4 \sin (e+f x))^{-1-m} \, dx","Int[(1 + Sin[e + f*x])^m*(3 + 4*Sin[e + f*x])^(-1 - m),x]","-\frac{2^{m+\frac{1}{2}} 7^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{4 \sin (e+f x)+3}\right)^{\frac{1}{2}-m} (4 \sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};-\frac{1-\sin (e+f x)}{2 (4 \sin (e+f x)+3)}\right)}{f}","-\frac{\left(\frac{7}{2}\right)^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{1-\sin (e+f x)}{7 (\sin (e+f x)+1)}\right)}{f (\sin (e+f x)+1)}",1,"-((2^(1/2 + m)*7^(-1/2 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, -(1 - Sin[e + f*x])/(2*(3 + 4*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + 4*Sin[e + f*x]))^(1/2 - m))/(f*(3 + 4*Sin[e + f*x])^m))","A",2,2,27,0.07407,1,"{2788, 132}"
622,1,28,0,0.0196123,"\int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx","Int[(1 + Sin[e + f*x])^m*(3 + 3*Sin[e + f*x])^(-1 - m),x]","-\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)}","-\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)}",1,"-((3^(-1 - m)*Cos[e + f*x])/(f*(1 + Sin[e + f*x])))","A",2,2,27,0.07407,1,"{22, 2648}"
623,1,122,0,0.1147807,"\int (1+\sin (e+f x))^m (3+2 \sin (e+f x))^{-1-m} \, dx","Int[(1 + Sin[e + f*x])^m*(3 + 2*Sin[e + f*x])^(-1 - m),x]","-\frac{2^{m+\frac{1}{2}} 5^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{2 \sin (e+f x)+3}\right)^{\frac{1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right)}{f}","-\frac{2^{m+\frac{1}{2}} 5^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{2 \sin (e+f x)+3}\right)^{\frac{1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right)}{f}",1,"-((2^(1/2 + m)*5^(-1/2 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/(2*(3 + 2*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + 2*Sin[e + f*x]))^(1/2 - m))/(f*(3 + 2*Sin[e + f*x])^m))","A",2,2,27,0.07407,1,"{2788, 132}"
624,1,106,0,0.1011385,"\int (1+\sin (e+f x))^m (3+\sin (e+f x))^{-1-m} \, dx","Int[(1 + Sin[e + f*x])^m*(3 + Sin[e + f*x])^(-1 - m),x]","-\frac{2^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{\sin (e+f x)+3}\right)^{\frac{1}{2}-m} (\sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{\sin (e+f x)+3}\right)}{f}","-\frac{2^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left(\frac{\sin (e+f x)+1}{\sin (e+f x)+3}\right)^{\frac{1}{2}-m} (\sin (e+f x)+3)^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{\sin (e+f x)+3}\right)}{f}",1,"-((2^(-1/2 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/(3 + Sin[e + f*x])]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + Sin[e + f*x]))^(1/2 - m))/(f*(3 + Sin[e + f*x])^m))","A",2,2,25,0.08000,1,"{2788, 132}"
625,1,65,0,0.0213046,"\int 3^{-1-m} (1+\sin (e+f x))^m \, dx","Int[3^(-1 - m)*(1 + Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{\sin (e+f x)+1}}",1,"-((2^(1/2 + m)*3^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2])/(f*Sqrt[1 + Sin[e + f*x]]))","A",2,2,18,0.1111,1,"{12, 2651}"
626,1,94,0,0.0922346,"\int (3-\sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Int[(3 - Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","-\frac{\cos (e+f x) (3-\sin (e+f x))^{-m-1} \left(\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)^{m+1} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f}","-\frac{\cos (e+f x) (3-\sin (e+f x))^{-m-1} \left(\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)^{m+1} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f}",1,"-((Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (-2*(1 - Sin[e + f*x]))/(1 + Sin[e + f*x])]*(3 - Sin[e + f*x])^(-1 - m)*((3 - Sin[e + f*x])/(1 + Sin[e + f*x]))^(1 + m)*(1 + Sin[e + f*x])^m)/f)","A",2,2,27,0.07407,1,"{2788, 132}"
627,1,114,0,0.099271,"\int (3-2 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Int[(3 - 2*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-2 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-2 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 - 2*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(1 + Sin[e + f*x])^m)/(Sqrt[5]*f*m*(3 - 2*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,27,0.07407,1,"{2788, 132}"
628,1,43,0,0.0541109,"\int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Int[(3 - 3*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)}","\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)}",1,"(Cos[e + f*x]*(3 - 3*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m)/(f*(1 + 2*m))","A",1,1,27,0.03704,1,"{2742}"
629,1,113,0,0.0963059,"\int (3-4 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Int[(3 - 4*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{7} f m (1-\sin (e+f x))}","\frac{2^{m+1} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (4 \sin (e+f x)-3)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{7 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f (\sin (e+f x)+1)}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (-2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(1 + Sin[e + f*x])^m)/(Sqrt[7]*f*m*(3 - 4*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,27,0.07407,1,"{2788, 132}"
630,1,111,0,0.0933549,"\int (3-5 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx","Int[(3 - 5*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m,x]","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-5 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{3-5 \sin (e+f x)}{\sin (e+f x)+1}\right)}{4 f m (1-\sin (e+f x))}","\frac{\cos (e+f x) (3-5 \sin (e+f x))^{-m} (5 \sin (e+f x)-3)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{4 (1-\sin (e+f x))}{\sin (e+f x)+1}\right)}{f (\sin (e+f x)+1)}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((3 - 5*Sin[e + f*x])/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(1 + Sin[e + f*x])^m)/(4*f*m*(3 - 5*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,27,0.07407,1,"{2788, 132}"
631,1,115,0,0.1055331,"\int (3+5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 + 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (5 \sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{5 \sin (e+f x)+3}{4 (\sin (e+f x)+1)}\right)}{4 f m (1-\sin (e+f x))}","-\frac{4^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{a-a \sin (e+f x)}{4 (\sin (e+f x) a+a)}\right)}{f}",1,"-(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + 5*Sin[e + f*x])/(4*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*f*m*(1 - Sin[e + f*x])*(3 + 5*Sin[e + f*x])^m)","A",2,2,29,0.06897,1,"{2788, 132}"
632,1,118,0,0.1134795,"\int (3+4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 + 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (4 \sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (4 \sin (e+f x)+3)}{7 (\sin (e+f x)+1)}\right)}{\sqrt{7} f m (1-\sin (e+f x))}","-\frac{\left(\frac{7}{2}\right)^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{a-a \sin (e+f x)}{7 (\sin (e+f x) a+a)}\right)}{f}",1,"-((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 4*Sin[e + f*x]))/(7*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(Sqrt[7]*f*m*(1 - Sin[e + f*x])*(3 + 4*Sin[e + f*x])^m))","A",2,2,29,0.06897,1,"{2788, 132}"
633,1,39,0,0.0170718,"\int (3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f}","-\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f}",1,"-((Cos[e + f*x]*(3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f)","A",2,2,29,0.06897,1,"{23, 2648}"
634,1,118,0,0.1125554,"\int (3+2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 + 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (2 \sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (2 \sin (e+f x)+3)}{5 (\sin (e+f x)+1)}\right)}{\sqrt{5} f m (1-\sin (e+f x))}","-\frac{\left(\frac{5}{2}\right)^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{a-a \sin (e+f x)}{5 (\sin (e+f x) a+a)}\right)}{f}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 2*Sin[e + f*x]))/(5*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(Sqrt[5]*f*m*(1 - Sin[e + f*x])*(3 + 2*Sin[e + f*x])^m)","A",2,2,29,0.06897,1,"{2788, 132}"
635,1,117,0,0.1072639,"\int (3+\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{\sin (e+f x)+3}{2 (\sin (e+f x)+1)}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}","-\frac{2^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{a-a \sin (e+f x)}{2 (\sin (e+f x) a+a)}\right)}{f}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + Sin[e + f*x])/(2*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(2*Sqrt[2]*f*m*(1 - Sin[e + f*x])*(3 + Sin[e + f*x])^m)","A",2,2,27,0.07407,1,"{2788, 132}"
636,1,81,0,0.0412067,"\int 3^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[3^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}","-\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"-((2^(1/2 + m)*3^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)","A",3,3,20,0.1500,1,"{12, 2652, 2651}"
637,1,118,0,0.1088213,"\int (3-\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 - Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-\sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}","-\frac{\cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 - Sin[e + f*x])/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(2*Sqrt[2]*f*m*(1 - Sin[e + f*x])*(3 - Sin[e + f*x])^m)","A",2,2,29,0.06897,1,"{2788, 132}"
638,1,116,0,0.1047672,"\int (3-2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 - 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-2 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}","-\frac{2^{m+1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};-\frac{5 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 - 2*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(Sqrt[5]*f*m*(3 - 2*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,29,0.06897,1,"{2788, 132}"
639,1,45,0,0.0624405,"\int (3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)}","\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)}",1,"(Cos[e + f*x]*(3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m))","A",1,1,29,0.03448,1,"{2742}"
640,1,115,0,0.1057647,"\int (3-4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 - 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{7} f m (1-\sin (e+f x))}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-4 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{7} f m (1-\sin (e+f x))}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (-2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(Sqrt[7]*f*m*(3 - 4*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,29,0.06897,1,"{2788, 132}"
641,1,113,0,0.101995,"\int (3-5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(3 - 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-5 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{3-5 \sin (e+f x)}{\sin (e+f x)+1}\right)}{4 f m (1-\sin (e+f x))}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (3-5 \sin (e+f x))^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{3-5 \sin (e+f x)}{\sin (e+f x)+1}\right)}{4 f m (1-\sin (e+f x))}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((3 - 5*Sin[e + f*x])/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*f*m*(3 - 5*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,29,0.06897,1,"{2788, 132}"
642,1,113,0,0.0980652,"\int (-3+5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 + 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (5 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{3-5 \sin (e+f x)}{\sin (e+f x)+1}\right)}{4 f m (1-\sin (e+f x))}","-\frac{\cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{4 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"-(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((3 - 5*Sin[e + f*x])/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*f*m*(1 - Sin[e + f*x])*(-3 + 5*Sin[e + f*x])^m)","A",2,2,29,0.06897,1,"{2788, 132}"
643,1,116,0,0.1052069,"\int (-3+4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 + 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (4 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{2 (3-4 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{7} f m (1-\sin (e+f x))}","-\frac{2^{m+1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\frac{7 (a-a \sin (e+f x))}{\sin (e+f x) a+a}\right)}{f}",1,"-((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (-2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(Sqrt[7]*f*m*(1 - Sin[e + f*x])*(-3 + 4*Sin[e + f*x])^m))","A",2,2,29,0.06897,1,"{2788, 132}"
644,1,45,0,0.0630852,"\int (-3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)}","\frac{\cos (e+f x) (3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)}",1,"(Cos[e + f*x]*(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m))","A",1,1,29,0.03448,1,"{2742}"
645,1,117,0,0.100542,"\int (-3+2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 + 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (3-2 \sin (e+f x))}{\sin (e+f x)+1}\right)}{\sqrt{5} f m (1-\sin (e+f x))}",1,"-((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 - 2*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(Sqrt[5]*f*m*(1 - Sin[e + f*x])*(-3 + 2*Sin[e + f*x])^m))","A",2,2,29,0.06897,1,"{2788, 132}"
646,1,116,0,0.0967191,"\int (-3+\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}",1,"-(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 - Sin[e + f*x])/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(2*Sqrt[2]*f*m*(1 - Sin[e + f*x])*(-3 + Sin[e + f*x])^m)","A",2,2,27,0.07407,1,"{2788, 132}"
647,1,81,0,0.0470816,"\int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3)^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{(-3)^{-m-1} 2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}","-\frac{(-3)^{-m-1} 2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f}",1,"-(((-3)^(-1 - m)*2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/f)","A",3,3,20,0.1500,1,"{12, 2652, 2651}"
648,1,119,0,0.1024299,"\int (-3-\sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 - Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{\sin (e+f x)+3}{2 (\sin (e+f x)+1)}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-\sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{\sin (e+f x)+3}{2 (\sin (e+f x)+1)}\right)}{2 \sqrt{2} f m (1-\sin (e+f x))}",1,"-(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + Sin[e + f*x])/(2*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(2*Sqrt[2]*f*m*(-3 - Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,29,0.06897,1,"{2788, 132}"
649,1,119,0,0.1139021,"\int (-3-2 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 - 2*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (2 \sin (e+f x)+3)}{5 (\sin (e+f x)+1)}\right)}{\sqrt{5} f m (1-\sin (e+f x))}","-\frac{\sqrt{-\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-2 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (2 \sin (e+f x)+3)}{5 (\sin (e+f x)+1)}\right)}{\sqrt{5} f m (1-\sin (e+f x))}",1,"-((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 2*Sin[e + f*x]))/(5*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/(Sqrt[5]*f*m*(-3 - 2*Sin[e + f*x])^m*(1 - Sin[e + f*x])))","A",2,2,29,0.06897,1,"{2788, 132}"
650,1,39,0,0.0163482,"\int (-3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\cos (e+f x) (-3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f}","-\frac{\cos (e+f x) (-3 \sin (e+f x)-3)^{-m-1} (a \sin (e+f x)+a)^m}{f}",1,"-((Cos[e + f*x]*(-3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f)","A",2,2,29,0.06897,1,"{23, 2648}"
651,1,117,0,0.1003787,"\int (-3-4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 - 4*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-4 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (4 \sin (e+f x)+3)}{7 (\sin (e+f x)+1)}\right)}{\sqrt{7} f m (1-\sin (e+f x))}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-4 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{2 (4 \sin (e+f x)+3)}{7 (\sin (e+f x)+1)}\right)}{\sqrt{7} f m (1-\sin (e+f x))}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 4*Sin[e + f*x]))/(7*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(Sqrt[7]*f*m*(-3 - 4*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,29,0.06897,1,"{2788, 132}"
652,1,115,0,0.0982541,"\int (-3-5 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(-3 - 5*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-5 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{5 \sin (e+f x)+3}{4 (\sin (e+f x)+1)}\right)}{4 f m (1-\sin (e+f x))}","\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (-5 \sin (e+f x)-3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m;1-m;\frac{5 \sin (e+f x)+3}{4 (\sin (e+f x)+1)}\right)}{4 f m (1-\sin (e+f x))}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + 5*Sin[e + f*x])/(4*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*f*m*(-3 - 5*Sin[e + f*x])^m*(1 - Sin[e + f*x]))","A",2,2,29,0.06897,1,"{2788, 132}"
653,1,116,0,0.1891308,"\int (d \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx","Int[(d*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m,x]","-\frac{\cos (e+f x) \left(\frac{\sin (e+f x)+1}{1-\sin (e+f x)}\right)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-m} \, _2F_1\left(\frac{1}{2}-m,-m;1-m;-\frac{2 \sin (e+f x)}{1-\sin (e+f x)}\right)}{d f m (\sin (e+f x)+1)}","-\frac{\cos (e+f x) \left(\frac{\sin (e+f x)+1}{1-\sin (e+f x)}\right)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{-m} \, _2F_1\left(\frac{1}{2}-m,-m;1-m;-\frac{2 \sin (e+f x)}{1-\sin (e+f x)}\right)}{d f m (\sin (e+f x)+1)}",1,"-((Cos[e + f*x]*Hypergeometric2F1[1/2 - m, -m, 1 - m, (-2*Sin[e + f*x])/(1 - Sin[e + f*x])]*((1 + Sin[e + f*x])/(1 - Sin[e + f*x]))^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(d*f*m*(d*Sin[e + f*x])^m*(1 + Sin[e + f*x])))","A",4,4,27,0.1481,1,"{2787, 2786, 2785, 132}"
654,1,129,0,0.1486581,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m),x]","-\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left(\frac{(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right)}{f (c+d)}","-\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left(\frac{(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right)}{f (c+d)}",1,"-((2^(1/2 + m)*a*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*(1 - Sin[e + f*x]))/(2*(c + d*Sin[e + f*x]))]*(a + a*Sin[e + f*x])^(-1 + m)*(((c + d)*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^(1/2 - m))/((c + d)*f*(c + d*Sin[e + f*x])^m))","A",2,2,29,0.06897,1,"{2788, 132}"
655,1,107,0,0.1225087,"\int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n,x]","-\frac{8 \sqrt{2} a^3 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{8 \sqrt{2} a^3 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(-8*Sqrt[2]*a^3*AppellF1[1/2, -5/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,25,0.1200,1,"{2784, 139, 138}"
656,1,107,0,0.1070542,"\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","-\frac{4 \sqrt{2} a^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{4 \sqrt{2} a^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(-4*Sqrt[2]*a^2*AppellF1[1/2, -3/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,25,0.1200,1,"{2784, 139, 138}"
657,1,105,0,0.0813342,"\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","-\frac{2 \sqrt{2} a \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{2 \sqrt{2} a \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"(-2*Sqrt[2]*a*AppellF1[1/2, -1/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,23,0.1304,1,"{2755, 139, 138}"
658,1,104,0,0.0636288,"\int (c+d \sin (e+f x))^n \, dx","Int[(c + d*Sin[e + f*x])^n,x]","-\frac{\sqrt{2} \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{\sqrt{2} \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"-((Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,12,0.2500,1,"{2665, 139, 138}"
659,1,107,0,0.1065124,"\int \frac{(c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x]),x]","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{\sqrt{2} a f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{\sqrt{2} a f \sqrt{\sin (e+f x)+1}}",1,"-((AppellF1[1/2, 3/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(Sqrt[2]*a*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,25,0.1200,1,"{2784, 139, 138}"
660,1,109,0,0.1064196,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^2,x]","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{2 \sqrt{2} a^2 f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{2 \sqrt{2} a^2 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[1/2, 5/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(2*Sqrt[2]*a^2*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,25,0.1200,1,"{2784, 139, 138}"
661,1,109,0,0.1077324,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^3,x]","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{7}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{4 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{7}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{4 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[1/2, 7/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(4*Sqrt[2]*a^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,25,0.1200,1,"{2784, 139, 138}"
662,1,257,0,0.4813079,"\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^n,x]","-\frac{2 a^3 \left(3 c^2-2 c d (4 n+7)+d^2 \left(16 n^2+56 n+43\right)\right) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}","-\frac{2 a^3 \left(3 c^2-2 c d (4 n+7)+d^2 \left(16 n^2+56 n+43\right)\right) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)}",1,"(2*a^3*(3*c - d*(11 + 4*n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(5 + 2*n)) - (2*a^3*(3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",5,5,27,0.1852,1,"{2763, 2981, 2776, 70, 69}"
663,1,160,0,0.2234727,"\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^n \, dx","Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^n,x]","\frac{2 a^2 (c-d (4 n+5)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}","\frac{2 a^2 (c-d (4 n+5)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}",1,"(-2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(c - d*(5 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",5,5,27,0.1852,1,"{2763, 21, 2776, 70, 69}"
664,1,85,0,0.0991956,"\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx","Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n,x]","-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{a \sin (e+f x)+a}}","-\frac{2 a \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} \, _2F_1\left(\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"(-2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(f*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,27,0.1111,1,"{2776, 70, 69}"
665,1,129,0,0.1687856,"\int \frac{(c+d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^n/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right)}{f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-n,1;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\right)}{f \sqrt{a \sin (e+f x)+a}}",1,"-((AppellF1[1 + n, 1/2, 1, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)*f*(1 + n)*(1 - Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]]))","A",3,3,27,0.1111,0,"{2788, 137, 136}"
666,1,130,0,0.1612945,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(3/2),x]","\frac{d \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},2;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right)}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-n,2;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"(d*AppellF1[1 + n, 1/2, 2, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)^2*f*(1 + n)*(a - a*Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]])","A",3,3,27,0.1111,0,"{2788, 137, 136}"
667,1,137,0,0.1778843,"\int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{d^2 \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left(n+1;\frac{1}{2},3;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right)}{f (n+1) (c-d)^3 \sqrt{a \sin (e+f x)+a} \left(a^2-a^2 \sin (e+f x)\right)}","-\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};-n,3;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\right)}{4 a^2 f \sqrt{a \sin (e+f x)+a}}",1,"-((d^2*AppellF1[1 + n, 1/2, 3, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)^3*f*(1 + n)*Sqrt[a + a*Sin[e + f*x]]*(a^2 - a^2*Sin[e + f*x])))","A",3,3,27,0.1111,0,"{2788, 137, 136}"
668,1,107,0,0.0913667,"\int (a+a \sin (e+f x)) \sqrt[3]{c+d \sin (e+f x)} \, dx","Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/3),x]","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};-\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};-\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*Sqrt[2]*a*AppellF1[1/2, -1/2, -1/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))","A",3,3,25,0.1200,1,"{2755, 139, 138}"
669,1,107,0,0.08664,"\int \frac{a+a \sin (e+f x)}{\sqrt[3]{c+d \sin (e+f x)}} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(1/3),x]","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left(\frac{1}{2};-\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1} \sqrt[3]{c+d \sin (e+f x)}}","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left(\frac{1}{2};-\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f \sqrt{\sin (e+f x)+1} \sqrt[3]{c+d \sin (e+f x)}}",1,"(-2*Sqrt[2]*a*AppellF1[1/2, -1/2, 1/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*((c + d*Sin[e + f*x])/(c + d))^(1/3))/(f*Sqrt[1 + Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1/3))","A",3,3,25,0.1200,1,"{2755, 139, 138}"
670,1,112,0,0.0963662,"\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx","Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(4/3),x]","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left(\frac{1}{2};-\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f (c+d) \sqrt{\sin (e+f x)+1} \sqrt[3]{c+d \sin (e+f x)}}","-\frac{2 \sqrt{2} a \cos (e+f x) \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left(\frac{1}{2};-\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f (c+d) \sqrt{\sin (e+f x)+1} \sqrt[3]{c+d \sin (e+f x)}}",1,"(-2*Sqrt[2]*a*AppellF1[1/2, -1/2, 4/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*((c + d*Sin[e + f*x])/(c + d))^(1/3))/((c + d)*f*Sqrt[1 + Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1/3))","A",3,3,25,0.1200,1,"{2755, 139, 138}"
671,1,171,0,0.2113775,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx","Int[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^3,x]","-\frac{\left(4 a d \left(4 c^2+d^2\right)+3 b \left(c^3+4 c d^2\right)\right) \cos (e+f x)}{6 f}-\frac{d \left(20 a c d+6 b c^2+9 b d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right)-\frac{(4 a d+3 b c) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}","-\frac{\left(4 a d \left(4 c^2+d^2\right)+3 b \left(c^3+4 c d^2\right)\right) \cos (e+f x)}{6 f}-\frac{d \left(20 a c d+6 b c^2+9 b d^2\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right)-\frac{(4 a d+3 b c) \cos (e+f x) (c+d \sin (e+f x))^2}{12 f}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}",1,"((8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*x)/8 - ((4*a*d*(4*c^2 + d^2) + 3*b*(c^3 + 4*c*d^2))*Cos[e + f*x])/(6*f) - (d*(6*b*c^2 + 20*a*c*d + 9*b*d^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((3*b*c + 4*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(12*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f)","A",3,2,23,0.08696,1,"{2753, 2734}"
672,1,106,0,0.1007072,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx","Int[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]","-\frac{2 \left(3 a c d+b \left(c^2+d^2\right)\right) \cos (e+f x)}{3 f}+\frac{1}{2} x \left(a \left(2 c^2+d^2\right)+2 b c d\right)-\frac{d (3 a d+2 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}","-\frac{2 \left(3 a c d+b \left(c^2+d^2\right)\right) \cos (e+f x)}{3 f}+\frac{1}{2} x \left(a \left(2 c^2+d^2\right)+2 b c d\right)-\frac{d (3 a d+2 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}",1,"((2*b*c*d + a*(2*c^2 + d^2))*x)/2 - (2*(3*a*c*d + b*(c^2 + d^2))*Cos[e + f*x])/(3*f) - (d*(2*b*c + 3*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f)","A",2,2,23,0.08696,1,"{2753, 2734}"
673,1,53,0,0.022245,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Int[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{(a d+b c) \cos (e+f x)}{f}+\frac{1}{2} x (2 a c+b d)-\frac{b d \sin (e+f x) \cos (e+f x)}{2 f}","-\frac{(a d+b c) \cos (e+f x)}{f}+\frac{1}{2} x (2 a c+b d)-\frac{b d \sin (e+f x) \cos (e+f x)}{2 f}",1,"((2*a*c + b*d)*x)/2 - ((b*c + a*d)*Cos[e + f*x])/f - (b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,21,0.04762,1,"{2734}"
674,1,16,0,0.0080893,"\int (a+b \sin (e+f x)) \, dx","Int[a + b*Sin[e + f*x],x]","a x-\frac{b \cos (e+f x)}{f}","a x-\frac{b \cos (e+f x)}{f}",1,"a*x - (b*Cos[e + f*x])/f","A",2,1,10,0.1000,1,"{2638}"
675,1,65,0,0.0882793,"\int \frac{a+b \sin (e+f x)}{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x]),x]","\frac{b x}{d}-\frac{2 (b c-a d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d f \sqrt{c^2-d^2}}","\frac{b x}{d}-\frac{2 (b c-a d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d f \sqrt{c^2-d^2}}",1,"(b*x)/d - (2*(b*c - a*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d*Sqrt[c^2 - d^2]*f)","A",4,4,23,0.1739,1,"{2735, 2660, 618, 204}"
676,1,98,0,0.0964343,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx","Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^2,x]","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2}}-\frac{(b c-a d) \cos (e+f x)}{f \left(c^2-d^2\right) (c+d \sin (e+f x))}","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2}}-\frac{(b c-a d) \cos (e+f x)}{f \left(c^2-d^2\right) (c+d \sin (e+f x))}",1,"(2*(a*c - b*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(3/2)*f) - ((b*c - a*d)*Cos[e + f*x])/((c^2 - d^2)*f*(c + d*Sin[e + f*x]))","A",5,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
677,1,164,0,0.1975443,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx","Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^3,x]","-\frac{\left(3 b c d-a \left(2 c^2+d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2}}+\frac{\left(3 a c d-b \left(c^2+2 d^2\right)\right) \cos (e+f x)}{2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}-\frac{(b c-a d) \cos (e+f x)}{2 f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}","-\frac{\left(3 b c d-a \left(2 c^2+d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2}}+\frac{\left(3 a c d-b \left(c^2+2 d^2\right)\right) \cos (e+f x)}{2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}-\frac{(b c-a d) \cos (e+f x)}{2 f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}",1,"-(((3*b*c*d - a*(2*c^2 + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(5/2)*f)) - ((b*c - a*d)*Cos[e + f*x])/(2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((3*a*c*d - b*(c^2 + 2*d^2))*Cos[e + f*x])/(2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))","A",6,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
678,1,314,0,0.5461762,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3 \, dx","Int[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3,x]","-\frac{\left(20 a^2 d^2 \left(4 c^2+d^2\right)+30 a b c d \left(c^2+4 d^2\right)+b^2 \left(-\left(-52 c^2 d^2+3 c^4-16 d^4\right)\right)\right) \cos (e+f x)}{30 d f}-\frac{\left(100 a^2 c d^2+30 a b d \left(2 c^2+3 d^2\right)+b^2 \left(-\left(6 c^3-71 c d^2\right)\right)\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} x \left(4 a^2 \left(2 c^3+3 c d^2\right)+6 a b d \left(4 c^2+d^2\right)+b^2 c \left(4 c^2+9 d^2\right)\right)-\frac{\left(4 d^2 \left(5 a^2+4 b^2\right)-3 b c (b c-10 a d)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac{b (b c-10 a d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{b^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}","-\frac{\left(20 a^2 d^2 \left(4 c^2+d^2\right)+30 a b c d \left(c^2+4 d^2\right)+b^2 \left(-\left(-52 c^2 d^2+3 c^4-16 d^4\right)\right)\right) \cos (e+f x)}{30 d f}-\frac{\left(100 a^2 c d^2+30 a b d \left(2 c^2+3 d^2\right)+b^2 \left(-\left(6 c^3-71 c d^2\right)\right)\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} x \left(4 a^2 \left(2 c^3+3 c d^2\right)+6 a b d \left(4 c^2+d^2\right)+b^2 c \left(4 c^2+9 d^2\right)\right)-\frac{\left(4 d^2 \left(5 a^2+4 b^2\right)-3 b c (b c-10 a d)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d f}+\frac{b (b c-10 a d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d f}-\frac{b^2 \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}",1,"((6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*x)/8 - ((20*a^2*d^2*(4*c^2 + d^2) + 30*a*b*c*d*(c^2 + 4*d^2) - b^2*(3*c^4 - 52*c^2*d^2 - 16*d^4))*Cos[e + f*x])/(30*d*f) - ((100*a^2*c*d^2 + 30*a*b*d*(2*c^2 + 3*d^2) - b^2*(6*c^3 - 71*c*d^2))*Cos[e + f*x]*Sin[e + f*x])/(120*f) - ((4*(5*a^2 + 4*b^2)*d^2 - 3*b*c*(b*c - 10*a*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*d*f) + (b*(b*c - 10*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d*f) - (b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f)","A",4,3,25,0.1200,1,"{2791, 2753, 2734}"
679,1,217,0,0.2811436,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx","Int[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2,x]","-\frac{\left(8 a^2 b c d+a^3 \left(-d^2\right)+4 a b^2 \left(3 c^2+2 d^2\right)+8 b^3 c d\right) \cos (e+f x)}{6 b f}+\frac{1}{8} x \left(4 a^2 \left(2 c^2+d^2\right)+16 a b c d+b^2 \left(4 c^2+3 d^2\right)\right)-\frac{\left(2 a d (8 b c-a d)+3 b^2 \left(4 c^2+3 d^2\right)\right) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f}","-\frac{\left(8 a^2 b c d+a^3 \left(-d^2\right)+4 a b^2 \left(3 c^2+2 d^2\right)+8 b^3 c d\right) \cos (e+f x)}{6 b f}+\frac{1}{8} x \left(4 a^2 \left(2 c^2+d^2\right)+16 a b c d+b^2 \left(4 c^2+3 d^2\right)\right)-\frac{\left(2 a d (8 b c-a d)+3 b^2 \left(4 c^2+3 d^2\right)\right) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f}",1,"((16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*x)/8 - ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2*(3*c^2 + 2*d^2))*Cos[e + f*x])/(6*b*f) - ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (d*(8*b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*b*f)","A",3,3,25,0.1200,1,"{2791, 2753, 2734}"
680,1,107,0,0.0936561,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx","Int[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x]),x]","-\frac{2 \left(a^2 d+3 a b c+b^2 d\right) \cos (e+f x)}{3 f}+\frac{1}{2} x \left(2 a^2 c+2 a b d+b^2 c\right)-\frac{b (2 a d+3 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}","-\frac{2 \left(a^2 d+3 a b c+b^2 d\right) \cos (e+f x)}{3 f}+\frac{1}{2} x \left(2 a^2 c+2 a b d+b^2 c\right)-\frac{b (2 a d+3 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}",1,"((2*a^2*c + b^2*c + 2*a*b*d)*x)/2 - (2*(3*a*b*c + a^2*d + b^2*d)*Cos[e + f*x])/(3*f) - (b*(3*b*c + 2*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (d*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)","A",2,2,23,0.08696,1,"{2753, 2734}"
681,1,50,0,0.0161034,"\int (a+b \sin (e+f x))^2 \, dx","Int[(a + b*Sin[e + f*x])^2,x]","\frac{1}{2} x \left(2 a^2+b^2\right)-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}","\frac{1}{2} x \left(2 a^2+b^2\right)-\frac{2 a b \cos (e+f x)}{f}-\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}",1,"((2*a^2 + b^2)*x)/2 - (2*a*b*Cos[e + f*x])/f - (b^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",1,1,12,0.08333,1,"{2644}"
682,1,93,0,0.1817085,"\int \frac{(a+b \sin (e+f x))^2}{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x]),x]","\frac{2 (b c-a d)^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \sqrt{c^2-d^2}}-\frac{b x (b c-2 a d)}{d^2}-\frac{b^2 \cos (e+f x)}{d f}","\frac{2 (b c-a d)^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \sqrt{c^2-d^2}}-\frac{b x (b c-2 a d)}{d^2}-\frac{b^2 \cos (e+f x)}{d f}",1,"-((b*(b*c - 2*a*d)*x)/d^2) + (2*(b*c - a*d)^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^2*Sqrt[c^2 - d^2]*f) - (b^2*Cos[e + f*x])/(d*f)","A",5,5,25,0.2000,1,"{2746, 2735, 2660, 618, 204}"
683,1,129,0,0.2308199,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^2} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2,x]","-\frac{2 (b c-a d) \left(a c d+b \left(c^2-2 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \left(c^2-d^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{d f \left(c^2-d^2\right) (c+d \sin (e+f x))}+\frac{b^2 x}{d^2}","-\frac{2 (b c-a d) \left(a c d+b \left(c^2-2 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2 f \left(c^2-d^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{d f \left(c^2-d^2\right) (c+d \sin (e+f x))}+\frac{b^2 x}{d^2}",1,"(b^2*x)/d^2 - (2*(b*c - a*d)*(a*c*d + b*(c^2 - 2*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^2*(c^2 - d^2)^(3/2)*f) + ((b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))","A",5,5,25,0.2000,1,"{2790, 2735, 2660, 618, 204}"
684,1,196,0,0.2793834,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^3} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3,x]","-\frac{\left(a^2 \left(-\left(2 c^2+d^2\right)\right)+6 a b c d-b^2 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{2 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}-\frac{\left(3 a c d+b \left(c^2-4 d^2\right)\right) (b c-a d) \cos (e+f x)}{2 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}","-\frac{\left(a^2 \left(-\left(2 c^2+d^2\right)\right)+6 a b c d-b^2 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{2 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}-\frac{\left(3 a c d+b \left(c^2-4 d^2\right)\right) (b c-a d) \cos (e+f x)}{2 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}",1,"-(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(5/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) - ((b*c - a*d)*(3*a*c*d + b*(c^2 - 4*d^2))*Cos[e + f*x])/(2*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2790, 2754, 12, 2660, 618, 204}"
685,1,305,0,0.5631262,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]","-\frac{\left(a^2 \left(-\left(2 c^3+3 c d^2\right)\right)+2 a b d \left(4 c^2+d^2\right)-b^2 c \left(c^2+4 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{7/2}}+\frac{\left(a^2 d^2 \left(11 c^2+4 d^2\right)-a b \left(4 c^3 d+26 c d^3\right)+b^2 \left(-\left(-10 c^2 d^2+c^4-6 d^4\right)\right)\right) \cos (e+f x)}{6 d f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^3}-\frac{\left(5 a c d+b \left(c^2-6 d^2\right)\right) (b c-a d) \cos (e+f x)}{6 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}","-\frac{\left(a^2 \left(-\left(2 c^3+3 c d^2\right)\right)+2 a b d \left(4 c^2+d^2\right)-b^2 c \left(c^2+4 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{7/2}}+\frac{\left(a^2 d^2 \left(11 c^2+4 d^2\right)-a b \left(4 c^3 d+26 c d^3\right)+b^2 \left(-\left(-10 c^2 d^2+c^4-6 d^4\right)\right)\right) \cos (e+f x)}{6 d f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^3}-\frac{\left(5 a c d+b \left(c^2-6 d^2\right)\right) (b c-a d) \cos (e+f x)}{6 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}",1,"-(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3) - ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Cos[e + f*x])/(6*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) + ((a^2*d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(6*d*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x]))","A",7,6,25,0.2400,1,"{2790, 2754, 12, 2660, 618, 204}"
686,1,493,0,0.9475135,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx","Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]","-\frac{\left(90 a^2 b c d^2 \left(c^2+4 d^2\right)+40 a^3 d^3 \left(4 c^2+d^2\right)-6 a b^2 d \left(-52 c^2 d^2+3 c^4-16 d^4\right)+b^3 \left(17 c^3 d^2+2 c^5+96 c d^4\right)\right) \cos (e+f x)}{60 d^2 f}+\frac{b \left(-90 a^2 d^2+18 a b c d+b^2 \left(-\left(2 c^2+25 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}-\frac{\left(90 a^2 b c d^2+40 a^3 d^3-a b^2 \left(18 c^2 d-96 d^3\right)+b^3 \left(2 c^3+21 c d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac{\left(90 a^2 b d^2 \left(2 c^2+3 d^2\right)+200 a^3 c d^3-6 a b^2 d \left(6 c^3-71 c d^2\right)+b^3 \left(36 c^2 d^2+4 c^4+75 d^4\right)\right) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac{1}{16} x \left(18 a^2 b d \left(4 c^2+d^2\right)+8 a^3 \left(2 c^3+3 c d^2\right)+6 a b^2 c \left(4 c^2+9 d^2\right)+b^3 d \left(18 c^2+5 d^2\right)\right)+\frac{b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}","\frac{(a d+b c) \left(a^2 d^2+8 a b c d+b^2 \left(c^2+6 d^2\right)\right) \cos ^3(e+f x)}{3 f}-\frac{\left(3 a^2 b c \left(c^2+3 d^2\right)+a^3 \left(3 c^2 d+d^3\right)+3 a b^2 d \left(3 c^2+d^2\right)+b^3 c \left(c^2+3 d^2\right)\right) \cos (e+f x)}{f}-\frac{3 b d \left(a^2 d^2+3 a b c d+b^2 c^2\right) \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{\left(18 a^2 b d \left(4 c^2+d^2\right)+24 a^3 c d^2+6 a b^2 c \left(4 c^2+9 d^2\right)+b^3 d \left(18 c^2+5 d^2\right)\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left(18 a^2 b d \left(4 c^2+d^2\right)+8 a^3 \left(2 c^3+3 c d^2\right)+6 a b^2 c \left(4 c^2+9 d^2\right)+b^3 d \left(18 c^2+5 d^2\right)\right)-\frac{3 b^2 d^2 (a d+b c) \cos ^5(e+f x)}{5 f}-\frac{b^3 d^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{5 b^3 d^3 \sin ^3(e+f x) \cos (e+f x)}{24 f}",1,"((18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*x)/16 - ((40*a^3*d^3*(4*c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*Cos[e + f*x])/(60*d^2*f) - ((200*a^3*c*d^3 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) - 6*a*b^2*d*(6*c^3 - 71*c*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*Cos[e + f*x]*Sin[e + f*x])/(240*d*f) - ((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^2*f) + (b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d^2*f) + (b^2*(2*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d^2*f) - (b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(6*d*f)","A",5,4,25,0.1600,1,"{2793, 3023, 2753, 2734}"
687,1,315,0,0.459742,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^2 \, dx","Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2,x]","-\frac{\left(4 a^2 b^2 \left(20 c^2+13 d^2\right)+30 a^3 b c d-3 a^4 d^2+120 a b^3 c d+4 b^4 \left(5 c^2+4 d^2\right)\right) \cos (e+f x)}{30 b f}-\frac{\left(60 a^2 b c d-6 a^3 d^2+a b^2 \left(100 c^2+71 d^2\right)+90 b^3 c d\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} x \left(24 a^2 b c d+4 a^3 \left(2 c^2+d^2\right)+3 a b^2 \left(4 c^2+3 d^2\right)+6 b^3 c d\right)-\frac{\left(3 a d (10 b c-a d)+4 b^2 \left(5 c^2+4 d^2\right)\right) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac{d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}","-\frac{\left(4 a^2 b^2 \left(20 c^2+13 d^2\right)+30 a^3 b c d-3 a^4 d^2+120 a b^3 c d+4 b^4 \left(5 c^2+4 d^2\right)\right) \cos (e+f x)}{30 b f}-\frac{\left(60 a^2 b c d-6 a^3 d^2+a b^2 \left(100 c^2+71 d^2\right)+90 b^3 c d\right) \sin (e+f x) \cos (e+f x)}{120 f}+\frac{1}{8} x \left(24 a^2 b c d+4 a^3 \left(2 c^2+d^2\right)+3 a b^2 \left(4 c^2+3 d^2\right)+6 b^3 c d\right)-\frac{\left(3 a d (10 b c-a d)+4 b^2 \left(5 c^2+4 d^2\right)\right) \cos (e+f x) (a+b \sin (e+f x))^2}{60 b f}-\frac{d (10 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^3}{20 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^4}{5 b f}",1,"((24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*x)/8 - ((30*a^3*b*c*d + 120*a*b^3*c*d - 3*a^4*d^2 + 4*b^4*(5*c^2 + 4*d^2) + 4*a^2*b^2*(20*c^2 + 13*d^2))*Cos[e + f*x])/(30*b*f) - ((60*a^2*b*c*d + 90*b^3*c*d - 6*a^3*d^2 + a*b^2*(100*c^2 + 71*d^2))*Cos[e + f*x]*Sin[e + f*x])/(120*f) - ((3*a*d*(10*b*c - a*d) + 4*b^2*(5*c^2 + 4*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(60*b*f) - (d*(10*b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(20*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^4)/(5*b*f)","A",4,3,25,0.1200,1,"{2791, 2753, 2734}"
688,1,171,0,0.1974852,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x)) \, dx","Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x]),x]","-\frac{\left(16 a^2 b c+3 a^3 d+12 a b^2 d+4 b^3 c\right) \cos (e+f x)}{6 f}-\frac{b \left(6 a^2 d+20 a b c+9 b^2 d\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(12 a^2 b d+8 a^3 c+12 a b^2 c+3 b^3 d\right)-\frac{(3 a d+4 b c) \cos (e+f x) (a+b \sin (e+f x))^2}{12 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^3}{4 f}","-\frac{\left(16 a^2 b c+3 a^3 d+12 a b^2 d+4 b^3 c\right) \cos (e+f x)}{6 f}-\frac{b \left(6 a^2 d+20 a b c+9 b^2 d\right) \sin (e+f x) \cos (e+f x)}{24 f}+\frac{1}{8} x \left(12 a^2 b d+8 a^3 c+12 a b^2 c+3 b^3 d\right)-\frac{(3 a d+4 b c) \cos (e+f x) (a+b \sin (e+f x))^2}{12 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^3}{4 f}",1,"((8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*x)/8 - ((16*a^2*b*c + 4*b^3*c + 3*a^3*d + 12*a*b^2*d)*Cos[e + f*x])/(6*f) - (b*(20*a*b*c + 6*a^2*d + 9*b^2*d)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((4*b*c + 3*a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*f) - (d*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*f)","A",3,2,23,0.08696,1,"{2753, 2734}"
689,1,90,0,0.0687045,"\int (a+b \sin (e+f x))^3 \, dx","Int[(a + b*Sin[e + f*x])^3,x]","-\frac{2 b \left(4 a^2+b^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}","-\frac{2 b \left(4 a^2+b^2\right) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left(2 a^2+3 b^2\right)-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}",1,"(a*(2*a^2 + 3*b^2)*x)/2 - (2*b*(4*a^2 + b^2)*Cos[e + f*x])/(3*f) - (5*a*b^2*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (b*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)","A",2,2,12,0.1667,1,"{2656, 2734}"
690,1,156,0,0.3783296,"\int \frac{(a+b \sin (e+f x))^3}{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x]),x]","-\frac{b x \left(-6 a^2 d^2+6 a b c d+b^2 \left(-\left(2 c^2+d^2\right)\right)\right)}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{2 (b c-a d)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \sqrt{c^2-d^2}}","-\frac{b x \left(-6 a^2 d^2+6 a b c d+b^2 \left(-\left(2 c^2+d^2\right)\right)\right)}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{2 (b c-a d)^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \sqrt{c^2-d^2}}",1,"-(b*(6*a*b*c*d - 6*a^2*d^2 - b^2*(2*c^2 + d^2))*x)/(2*d^3) - (2*(b*c - a*d)^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*Sqrt[c^2 - d^2]*f) + (b^2*(2*b*c - 5*a*d)*Cos[e + f*x])/(2*d^2*f) - (b^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(2*d*f)","A",6,6,25,0.2400,1,"{2793, 3023, 2735, 2660, 618, 204}"
691,1,208,0,0.4953645,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^2} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2,x]","\frac{b \left(-a^2 d^2+2 a b c d+b^2 \left(-\left(2 c^2-d^2\right)\right)\right) \cos (e+f x)}{d^2 f \left(c^2-d^2\right)}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{2 (b c-a d)^2 \left(a c d+2 b c^2-3 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \left(c^2-d^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left(c^2-d^2\right) (c+d \sin (e+f x))}","\frac{b \left(-a^2 d^2+2 a b c d+b^2 \left(-\left(2 c^2-d^2\right)\right)\right) \cos (e+f x)}{d^2 f \left(c^2-d^2\right)}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{2 (b c-a d)^2 \left(a c d+2 b c^2-3 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \left(c^2-d^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left(c^2-d^2\right) (c+d \sin (e+f x))}",1,"-((b^2*(2*b*c - 3*a*d)*x)/d^3) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(3/2)*f) + (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x])/(d^2*(c^2 - d^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2792, 3023, 2735, 2660, 618, 204}"
692,1,255,0,0.6311054,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3,x]","-\frac{\left(9 a^2 b c d^4-a^3 d^3 \left(2 c^2+d^2\right)-3 a b^2 d^3 \left(c^2+2 d^2\right)+b^3 \left(-5 c^3 d^2+2 c^5+6 c d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \left(c^2-d^2\right)^{5/2}}+\frac{(b c-a d)^2 \left(3 a c d+2 b c^2-5 b d^2\right) \cos (e+f x)}{2 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}+\frac{b^3 x}{d^3}","-\frac{\left(9 a^2 b c d^4-a^3 d^3 \left(2 c^2+d^2\right)-3 a b^2 d^3 \left(c^2+2 d^2\right)+b^3 \left(-5 c^3 d^2+2 c^5+6 c d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{d^3 f \left(c^2-d^2\right)^{5/2}}+\frac{(b c-a d)^2 \left(3 a c d+2 b c^2-5 b d^2\right) \cos (e+f x)}{2 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^2}+\frac{b^3 x}{d^3}",1,"(b^3*x)/d^3 - ((9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6*c*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(5/2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*b*d^2)*Cos[e + f*x])/(2*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2792, 3021, 2735, 2660, 618, 204}"
693,1,325,0,0.7249193,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4,x]","-\frac{(a c-b d) \left(a^2 \left(-\left(2 c^2+3 d^2\right)\right)+10 a b c d-b^2 \left(3 c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{7/2}}-\frac{\left(a^2 d^2 \left(11 c^2+4 d^2\right)+5 a b c d \left(c^2-7 d^2\right)+b^2 \left(-5 c^2 d^2+2 c^4+18 d^4\right)\right) (b c-a d) \cos (e+f x)}{6 d^2 f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))}+\frac{\left(5 a c d+2 b c^2-7 b d^2\right) (b c-a d)^2 \cos (e+f x)}{6 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^3}","-\frac{(a c-b d) \left(a^2 \left(-\left(2 c^2+3 d^2\right)\right)+10 a b c d-b^2 \left(3 c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{7/2}}-\frac{\left(a^2 d^2 \left(11 c^2+4 d^2\right)+5 a b c d \left(c^2-7 d^2\right)+b^2 \left(-5 c^2 d^2+2 c^4+18 d^4\right)\right) (b c-a d) \cos (e+f x)}{6 d^2 f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))}+\frac{\left(5 a c d+2 b c^2-7 b d^2\right) (b c-a d)^2 \cos (e+f x)}{6 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^3}",1,"-(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3) + ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*Cos[e + f*x])/(6*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) - ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*Cos[e + f*x])/(6*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x]))","A",7,7,25,0.2800,1,"{2792, 3021, 2754, 12, 2660, 618, 204}"
694,1,54,0,0.0816141,"\int \frac{\frac{b B}{a}+B \sin (x)}{a+b \sin (x)} \, dx","Int[((b*B)/a + B*Sin[x])/(a + b*Sin[x]),x]","\frac{B x}{b}-\frac{2 B \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a b}","\frac{B x}{b}-\frac{2 B \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a b}",1,"(B*x)/b - (2*Sqrt[a^2 - b^2]*B*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b)","A",4,4,20,0.2000,1,"{2735, 2660, 618, 204}"
695,1,6,0,0.0013596,"\int \frac{\frac{a B}{b}+B \sin (x)}{a+b \sin (x)} \, dx","Int[((a*B)/b + B*Sin[x])/(a + b*Sin[x]),x]","\frac{B x}{b}","\frac{B x}{b}",1,"(B*x)/b","A",2,2,20,0.1000,1,"{21, 8}"
696,1,12,0,0.0288894,"\int \frac{a+b \sin (x)}{(b+a \sin (x))^2} \, dx","Int[(a + b*Sin[x])/(b + a*Sin[x])^2,x]","-\frac{\cos (x)}{a \sin (x)+b}","-\frac{\cos (x)}{a \sin (x)+b}",1,"-(Cos[x]/(b + a*Sin[x]))","A",2,2,15,0.1333,1,"{2754, 8}"
697,1,34,0,0.0327097,"\int \frac{2-\sin (x)}{2+\sin (x)} \, dx","Int[(2 - Sin[x])/(2 + Sin[x]),x]","\frac{4 x}{\sqrt{3}}-x+\frac{8 \tan ^{-1}\left(\frac{\cos (x)}{\sin (x)+\sqrt{3}+2}\right)}{\sqrt{3}}","\frac{4 x}{\sqrt{3}}-x+\frac{8 \tan ^{-1}\left(\frac{\cos (x)}{\sin (x)+\sqrt{3}+2}\right)}{\sqrt{3}}",1,"-x + (4*x)/Sqrt[3] + (8*ArcTan[Cos[x]/(2 + Sqrt[3] + Sin[x])])/Sqrt[3]","A",2,2,13,0.1538,1,"{2735, 2657}"
698,1,235,0,0.6478155,"\int \frac{(c+d \sin (e+f x))^4}{a+b \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x]),x]","\frac{d^2 \left(-3 a^2 d^2+12 a b c d+b^2 \left(-\left(17 c^2+2 d^2\right)\right)\right) \cos (e+f x)}{3 b^3 f}+\frac{d x \left(8 a^2 b c d^2-2 a^3 d^3-a b^2 d \left(12 c^2+d^2\right)+4 b^3 c \left(2 c^2+d^2\right)\right)}{2 b^4}+\frac{2 (b c-a d)^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \sqrt{a^2-b^2}}-\frac{d^3 (8 b c-3 a d) \sin (e+f x) \cos (e+f x)}{6 b^2 f}-\frac{d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}","\frac{d^2 \left(-3 a^2 d^2+12 a b c d+b^2 \left(-\left(17 c^2+2 d^2\right)\right)\right) \cos (e+f x)}{3 b^3 f}+\frac{d x \left(8 a^2 b c d^2-2 a^3 d^3-a b^2 d \left(12 c^2+d^2\right)+4 b^3 c \left(2 c^2+d^2\right)\right)}{2 b^4}+\frac{2 (b c-a d)^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \sqrt{a^2-b^2}}-\frac{d^3 (8 b c-3 a d) \sin (e+f x) \cos (e+f x)}{6 b^2 f}-\frac{d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}",1,"(d*(8*a^2*b*c*d^2 - 2*a^3*d^3 + 4*b^3*c*(2*c^2 + d^2) - a*b^2*d*(12*c^2 + d^2))*x)/(2*b^4) + (2*(b*c - a*d)^4*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*f) + (d^2*(12*a*b*c*d - 3*a^2*d^2 - b^2*(17*c^2 + 2*d^2))*Cos[e + f*x])/(3*b^3*f) - (d^3*(8*b*c - 3*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*b^2*f) - (d^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*b*f)","A",7,7,25,0.2800,1,"{2793, 3033, 3023, 2735, 2660, 618, 204}"
699,1,156,0,0.3630932,"\int \frac{(c+d \sin (e+f x))^3}{a+b \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + b*Sin[e + f*x]),x]","-\frac{d x \left(-2 a^2 d^2+6 a b c d+b^2 \left(-\left(6 c^2+d^2\right)\right)\right)}{2 b^3}+\frac{2 (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \sqrt{a^2-b^2}}-\frac{d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac{d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}","-\frac{d x \left(-2 a^2 d^2+6 a b c d+b^2 \left(-\left(6 c^2+d^2\right)\right)\right)}{2 b^3}+\frac{2 (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \sqrt{a^2-b^2}}-\frac{d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac{d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f}",1,"-(d*(6*a*b*c*d - 2*a^2*d^2 - b^2*(6*c^2 + d^2))*x)/(2*b^3) + (2*(b*c - a*d)^3*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*f) - (d^2*(5*b*c - 2*a*d)*Cos[e + f*x])/(2*b^2*f) - (d^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(2*b*f)","A",6,6,25,0.2400,1,"{2793, 3023, 2735, 2660, 618, 204}"
700,1,93,0,0.1670605,"\int \frac{(c+d \sin (e+f x))^2}{a+b \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + b*Sin[e + f*x]),x]","\frac{2 (b c-a d)^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 f \sqrt{a^2-b^2}}+\frac{d x (2 b c-a d)}{b^2}-\frac{d^2 \cos (e+f x)}{b f}","\frac{2 (b c-a d)^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 f \sqrt{a^2-b^2}}+\frac{d x (2 b c-a d)}{b^2}-\frac{d^2 \cos (e+f x)}{b f}",1,"(d*(2*b*c - a*d)*x)/b^2 + (2*(b*c - a*d)^2*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*f) - (d^2*Cos[e + f*x])/(b*f)","A",5,5,25,0.2000,1,"{2746, 2735, 2660, 618, 204}"
701,1,65,0,0.0719839,"\int \frac{c+d \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{2 (b c-a d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b f \sqrt{a^2-b^2}}+\frac{d x}{b}","\frac{2 (b c-a d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b f \sqrt{a^2-b^2}}+\frac{d x}{b}",1,"(d*x)/b + (2*(b*c - a*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]*f)","A",4,4,23,0.1739,1,"{2735, 2660, 618, 204}"
702,1,47,0,0.0341297,"\int \frac{1}{a+b \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}",1,"(2*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*f)","A",3,3,12,0.2500,1,"{2660, 618, 204}"
703,1,117,0,0.1533675,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Int[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)}-\frac{2 d \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)}","\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)}-\frac{2 d \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)}",1,"(2*b*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)*f) - (2*d*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)*Sqrt[c^2 - d^2]*f)","A",7,4,25,0.1600,1,"{2747, 2660, 618, 204}"
704,1,185,0,0.4682318,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Int[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)^2}+\frac{2 d \left(a c d-b \left(2 c^2-d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^2}-\frac{d^2 \cos (e+f x)}{f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))}","\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)^2}+\frac{2 d \left(a c d-b \left(2 c^2-d^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^2}-\frac{d^2 \cos (e+f x)}{f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))}",1,"(2*b^2*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)^2*f) + (2*d*(a*c*d - b*(2*c^2 - d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*(c^2 - d^2)^(3/2)*f) - (d^2*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))","A",8,5,25,0.2000,1,"{2802, 3001, 2660, 618, 204}"
705,1,284,0,1.0797865,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx","Int[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]","\frac{d \left(-a^2 d^2 \left(2 c^2+d^2\right)+6 a b c^3 d-b^2 \left(-5 c^2 d^2+6 c^4+2 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^3}+\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)^3}-\frac{d^2 \left(-3 a c d+5 b c^2-2 b d^2\right) \cos (e+f x)}{2 f \left(c^2-d^2\right)^2 (b c-a d)^2 (c+d \sin (e+f x))}-\frac{d^2 \cos (e+f x)}{2 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^2}","\frac{d \left(-a^2 d^2 \left(2 c^2+d^2\right)+6 a b c^3 d-b^2 \left(-5 c^2 d^2+6 c^4+2 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^3}+\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2} (b c-a d)^3}-\frac{d^2 \left(-3 a c d+5 b c^2-2 b d^2\right) \cos (e+f x)}{2 f \left(c^2-d^2\right)^2 (b c-a d)^2 (c+d \sin (e+f x))}-\frac{d^2 \cos (e+f x)}{2 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^2}",1,"(2*b^3*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)^3*f) + (d*(6*a*b*c^3*d - a^2*d^2*(2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^3*(c^2 - d^2)^(5/2)*f) - (d^2*Cos[e + f*x])/(2*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) - (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*Cos[e + f*x])/(2*(b*c - a*d)^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))","A",9,6,25,0.2400,1,"{2802, 3055, 3001, 2660, 618, 204}"
706,1,306,0,0.9407127,"\int \frac{(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^2,x]","\frac{d (2 b c-a d) \left(-3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-2 d^2\right)\right)\right) \cos (e+f x)}{b^3 f \left(a^2-b^2\right)}+\frac{d^2 \left(-3 a^2 d^2+4 a b c d+b^2 \left(-\left(2 c^2-d^2\right)\right)\right) \sin (e+f x) \cos (e+f x)}{2 b^2 f \left(a^2-b^2\right)}-\frac{d^2 x \left(-6 a^2 d^2+16 a b c d+b^2 \left(-\left(12 c^2+d^2\right)\right)\right)}{2 b^4}+\frac{2 \left(3 a^2 d+a b c-4 b^2 d\right) (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}","\frac{d (2 b c-a d) \left(-3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-2 d^2\right)\right)\right) \cos (e+f x)}{b^3 f \left(a^2-b^2\right)}+\frac{d^2 \left(-3 a^2 d^2+4 a b c d+b^2 \left(-\left(2 c^2-d^2\right)\right)\right) \sin (e+f x) \cos (e+f x)}{2 b^2 f \left(a^2-b^2\right)}-\frac{d^2 x \left(-6 a^2 d^2+16 a b c d+b^2 \left(-\left(12 c^2+d^2\right)\right)\right)}{2 b^4}+\frac{2 \left(3 a^2 d+a b c-4 b^2 d\right) (b c-a d)^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}",1,"-(d^2*(16*a*b*c*d - 6*a^2*d^2 - b^2*(12*c^2 + d^2))*x)/(2*b^4) + (2*(b*c - a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*f) + (d*(2*b*c - a*d)*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(b^3*(a^2 - b^2)*f) + (d^2*(4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))","A",7,7,25,0.2800,1,"{2792, 3033, 3023, 2735, 2660, 618, 204}"
707,1,205,0,0.4590155,"\int \frac{(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + b*Sin[e + f*x])^2,x]","\frac{d \left(-2 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-d^2\right)\right)\right) \cos (e+f x)}{b^2 f \left(a^2-b^2\right)}+\frac{2 (b c-a d)^2 \left(2 a^2 d+a b c-3 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{d^2 x (3 b c-2 a d)}{b^3}","\frac{d \left(-2 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-d^2\right)\right)\right) \cos (e+f x)}{b^2 f \left(a^2-b^2\right)}+\frac{2 (b c-a d)^2 \left(2 a^2 d+a b c-3 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{d^2 x (3 b c-2 a d)}{b^3}",1,"(d^2*(3*b*c - 2*a*d)*x)/b^3 + (2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*f) + (d*(2*a*b*c*d - 2*a^2*d^2 - b^2*(c^2 - d^2))*Cos[e + f*x])/(b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2792, 3023, 2735, 2660, 618, 204}"
708,1,129,0,0.2179059,"\int \frac{(c+d \sin (e+f x))^2}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + b*Sin[e + f*x])^2,x]","\frac{2 (b c-a d) \left(a^2 d+a b c-2 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{d^2 x}{b^2}","\frac{2 (b c-a d) \left(a^2 d+a b c-2 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{d^2 x}{b^2}",1,"(d^2*x)/b^2 + (2*(b*c - a*d)*(a*b*c + a^2*d - 2*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*f) + ((b*c - a*d)^2*Cos[e + f*x])/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))","A",5,5,25,0.2000,1,"{2790, 2735, 2660, 618, 204}"
709,1,97,0,0.0923569,"\int \frac{c+d \sin (e+f x)}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])/(a + b*Sin[e + f*x])^2,x]","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d) \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{(b c-a d) \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}",1,"(2*(a*c - b*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*f) + ((b*c - a*d)*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))","A",5,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
710,1,83,0,0.0595759,"\int \frac{1}{(a+b \sin (e+f x))^2} \, dx","Int[(a + b*Sin[e + f*x])^(-2),x]","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{b \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{b \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}",1,"(2*a*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*f) + (b*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))","A",5,5,12,0.4167,1,"{2664, 12, 2660, 618, 204}"
711,1,181,0,0.4378594,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx","Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])),x]","\frac{2 b \left(-2 a^2 d+a b c+b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^2}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}+\frac{2 d^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)^2}","\frac{2 b \left(-2 a^2 d+a b c+b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^2}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}+\frac{2 d^2 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)^2}",1,"(2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^2*f) + (2*d^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]*f) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x]))","A",8,5,25,0.2000,1,"{2802, 3001, 2660, 618, 204}"
712,1,290,0,1.1971454,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, dx","Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2),x]","\frac{d \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right) \cos (e+f x)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))}+\frac{2 b^2 \left(-3 a^2 d+a b c+2 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^3}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac{2 d^2 \left(-a c d+3 b c^2-2 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^3}","\frac{d \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right) \cos (e+f x)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))}+\frac{2 b^2 \left(-3 a^2 d+a b c+2 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^3}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac{2 d^2 \left(-a c d+3 b c^2-2 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^3}",1,"(2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^3*(c^2 - d^2)^(3/2)*f) + (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x]))","A",9,6,25,0.2400,1,"{2802, 3055, 3001, 2660, 618, 204}"
713,1,458,0,2.4386605,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx","Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]","-\frac{d^2 \left(-a^2 d^2 \left(2 c^2+d^2\right)+2 a b c d \left(4 c^2-d^2\right)-3 b^2 \left(-5 c^2 d^2+4 c^4+2 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^4}-\frac{\left(-a^2 b d^3 \left(7 c^2-4 d^2\right)+3 a^3 c d^4-3 a b^2 c d^4-b^3 \left(-11 c^2 d^3+2 c^4 d+6 d^5\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 (c+d \sin (e+f x))}+\frac{d \left(a^2 d^2+b^2 \left(2 c^2-3 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^2}+\frac{2 b^3 \left(-4 a^2 d+a b c+3 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^4}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}","-\frac{d^2 \left(-a^2 d^2 \left(2 c^2+d^2\right)+2 a b c d \left(4 c^2-d^2\right)-3 b^2 \left(-5 c^2 d^2+4 c^4+2 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^4}-\frac{\left(-a^2 b d^3 \left(7 c^2-4 d^2\right)+3 a^3 c d^4-3 a b^2 c d^4-b^3 \left(-11 c^2 d^3+2 c^4 d+6 d^5\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 (c+d \sin (e+f x))}+\frac{d \left(a^2 d^2+b^2 \left(2 c^2-3 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^2}+\frac{2 b^3 \left(-4 a^2 d+a b c+3 b^2 d\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2} (b c-a d)^4}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}",1,"(2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(5/2)*f) + (d*(a^2*d^2 + b^2*(2*c^2 - 3*d^2))*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - ((3*a^3*c*d^4 - 3*a*b^2*c*d^4 - a^2*b*d^3*(7*c^2 - 4*d^2) - b^3*(2*c^4*d - 11*c^2*d^3 + 6*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))","A",10,6,25,0.2400,1,"{2802, 3055, 3001, 2660, 618, 204}"
714,1,534,0,2.1551766,"\int \frac{(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]","-\frac{d \left(-a^3 b^2 d^2 \left(16 c^2-21 d^2\right)-a^2 b^3 c d \left(4 c^2+55 d^2\right)+30 a^4 b c d^3-12 a^5 d^4+a b^4 \left(43 c^2 d^2+6 c^4-6 d^4\right)-b^5 c d \left(17 c^2-10 d^2\right)\right) \cos (e+f x)}{2 b^4 f \left(a^2-b^2\right)^2}+\frac{(b c-a d)^3 \left(a^2 b^2 \left(2 c^2-29 d^2\right)+6 a^3 b c d+12 a^4 d^2-12 a b^3 c d+b^4 \left(c^2+20 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 f \left(a^2-b^2\right)^{5/2}}+\frac{d^2 \left(a^2 b^2 d \left(c^2+10 d^2\right)+7 a^3 b c d^2-6 a^4 d^3-a b^3 c \left(3 c^2+16 d^2\right)+b^4 d \left(8 c^2-d^2\right)\right) \sin (e+f x) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)^2}-\frac{d^3 x \left(-12 a^2 d^2+30 a b c d+b^2 \left(-\left(20 c^2+d^2\right)\right)\right)}{2 b^5}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left(4 a^2 d+3 a b c-7 b^2 d\right) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}","-\frac{d \left(-a^3 b^2 d^2 \left(16 c^2-21 d^2\right)-a^2 b^3 c d \left(4 c^2+55 d^2\right)+30 a^4 b c d^3-12 a^5 d^4+a b^4 \left(43 c^2 d^2+6 c^4-6 d^4\right)-b^5 c d \left(17 c^2-10 d^2\right)\right) \cos (e+f x)}{2 b^4 f \left(a^2-b^2\right)^2}+\frac{(b c-a d)^3 \left(a^2 b^2 \left(2 c^2-29 d^2\right)+6 a^3 b c d+12 a^4 d^2-12 a b^3 c d+b^4 \left(c^2+20 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 f \left(a^2-b^2\right)^{5/2}}+\frac{d^2 \left(a^2 b^2 d \left(c^2+10 d^2\right)+7 a^3 b c d^2-6 a^4 d^3-a b^3 c \left(3 c^2+16 d^2\right)+b^4 d \left(8 c^2-d^2\right)\right) \sin (e+f x) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)^2}-\frac{d^3 x \left(-12 a^2 d^2+30 a b c d+b^2 \left(-\left(20 c^2+d^2\right)\right)\right)}{2 b^5}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left(4 a^2 d+3 a b c-7 b^2 d\right) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}",1,"-(d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*x)/(2*b^5) + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)*f) - (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(2*b^4*(a^2 - b^2)^2*f) + (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)","A",8,8,25,0.3200,1,"{2792, 3047, 3033, 3023, 2735, 2660, 618, 204}"
715,1,318,0,0.9718691,"\int \frac{(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^3,x]","\frac{d^2 \left(-3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-2 d^2\right)\right)\right) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)}+\frac{(b c-a d)^2 \left(a^2 b^2 \left(2 c^2-15 d^2\right)+4 a^3 b c d+6 a^4 d^2-10 a b^3 c d+b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{3 (b c-a d)^3 \left(a^2 d+a b c-2 b^2 d\right) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{d^3 x (4 b c-3 a d)}{b^4}","\frac{d^2 \left(-3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-2 d^2\right)\right)\right) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)}+\frac{(b c-a d)^2 \left(a^2 b^2 \left(2 c^2-15 d^2\right)+4 a^3 b c d+6 a^4 d^2-10 a b^3 c d+b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{3 (b c-a d)^3 \left(a^2 d+a b c-2 b^2 d\right) \cos (e+f x)}{2 b^3 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{d^3 x (4 b c-3 a d)}{b^4}",1,"(d^3*(4*b*c - 3*a*d)*x)/b^4 + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*f) + (d^2*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(2*b^3*(a^2 - b^2)*f) + (3*(b*c - a*d)^3*(a*b*c + a^2*d - 2*b^2*d)*Cos[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)","A",7,7,25,0.2800,1,"{2792, 3031, 3023, 2735, 2660, 618, 204}"
716,1,248,0,0.8064638,"\int \frac{(c+d \sin (e+f x))^3}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^3/(a + b*Sin[e + f*x])^3,x]","\frac{(b c-a d) \left(a^2 b^2 \left(2 c^2-5 d^2\right)+2 a^3 b c d+2 a^4 d^2-8 a b^3 c d+b^4 \left(c^2+6 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left(2 a^2 d+3 a b c-5 b^2 d\right) \cos (e+f x)}{2 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{d^3 x}{b^3}","\frac{(b c-a d) \left(a^2 b^2 \left(2 c^2-5 d^2\right)+2 a^3 b c d+2 a^4 d^2-8 a b^3 c d+b^4 \left(c^2+6 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{(b c-a d)^2 \left(2 a^2 d+3 a b c-5 b^2 d\right) \cos (e+f x)}{2 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{d^3 x}{b^3}",1,"(d^3*x)/b^3 + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*f) + ((b*c - a*d)^2*(3*a*b*c + 2*a^2*d - 5*b^2*d)*Cos[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)","A",6,6,25,0.2400,1,"{2792, 3021, 2735, 2660, 618, 204}"
717,1,196,0,0.2851291,"\int \frac{(c+d \sin (e+f x))^2}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^2/(a + b*Sin[e + f*x])^3,x]","-\frac{\left(a^2 \left(-\left(2 c^2+d^2\right)\right)+6 a b c d-b^2 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{\left(a^2 d+3 a b c-4 b^2 d\right) (b c-a d) \cos (e+f x)}{2 b f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}","-\frac{\left(a^2 \left(-\left(2 c^2+d^2\right)\right)+6 a b c d-b^2 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x)}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{\left(a^2 d+3 a b c-4 b^2 d\right) (b c-a d) \cos (e+f x)}{2 b f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}",1,"-(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((b*c - a*d)*(3*a*b*c + a^2*d - 4*b^2*d)*Cos[e + f*x])/(2*b*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x]))","A",6,6,25,0.2400,1,"{2790, 2754, 12, 2660, 618, 204}"
718,1,162,0,0.1741766,"\int \frac{c+d \sin (e+f x)}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])/(a + b*Sin[e + f*x])^3,x]","\frac{\left(2 a^2 c-3 a b d+b^2 c\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{\left(a^2 (-d)+3 a b c-2 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x)}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}","\frac{\left(2 a^2 c-3 a b d+b^2 c\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{\left(a^2 (-d)+3 a b c-2 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x)}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}",1,"((2*a^2*c + b^2*c - 3*a*b*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*f) + ((b*c - a*d)*Cos[e + f*x])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((3*a*b*c - a^2*d - 2*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x]))","A",6,5,23,0.2174,1,"{2754, 12, 2660, 618, 204}"
719,1,131,0,0.1136175,"\int \frac{1}{(a+b \sin (e+f x))^3} \, dx","Int[(a + b*Sin[e + f*x])^(-3),x]","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{3 a b \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{b \cos (e+f x)}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2}}+\frac{3 a b \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{b \cos (e+f x)}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}",1,"((2*a^2 + b^2)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*f) + (b*Cos[e + f*x])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (3*a*b*Cos[e + f*x])/(2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x]))","A",6,6,12,0.5000,1,"{2664, 2754, 12, 2660, 618, 204}"
720,1,285,0,1.0369585,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))} \, dx","Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])),x]","-\frac{b \left(-a^2 b^2 \left(2 c^2-5 d^2\right)+6 a^3 b c d-6 a^4 d^2-b^4 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^3}+\frac{b^2 \left(-5 a^2 d+3 a b c+2 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2}-\frac{2 d^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)^3}","-\frac{b \left(-a^2 b^2 \left(2 c^2-5 d^2\right)+6 a^3 b c d-6 a^4 d^2-b^4 \left(c^2+2 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^3}+\frac{b^2 \left(-5 a^2 d+3 a b c+2 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2}-\frac{2 d^3 \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \sqrt{c^2-d^2} (b c-a d)^3}",1,"-((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^3*Sqrt[c^2 - d^2]*f) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x]))","A",9,6,25,0.2400,1,"{2802, 3055, 3001, 2660, 618, 204}"
721,1,454,0,2.4387318,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx","Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2),x]","-\frac{b^2 \left(-a^2 b^2 \left(2 c^2-15 d^2\right)+8 a^3 b c d-12 a^4 d^2-2 a b^3 c d-b^4 \left(c^2+6 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^4}-\frac{d \left(a^2 b^2 d \left(7 c^2-11 d^2\right)+2 a^4 d^3-3 a b^3 c \left(c^2-d^2\right)-2 b^4 d \left(2 c^2-3 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 (c+d \sin (e+f x))}+\frac{3 b^2 \left(-2 a^2 d+a b c+b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac{2 d^3 \left(-a c d+4 b c^2-3 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^4}","-\frac{b^2 \left(-a^2 b^2 \left(2 c^2-15 d^2\right)+8 a^3 b c d-12 a^4 d^2-2 a b^3 c d-b^4 \left(c^2+6 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^4}-\frac{d \left(a^2 b^2 d \left(7 c^2-11 d^2\right)+2 a^4 d^3-3 a b^3 c \left(c^2-d^2\right)-2 b^4 d \left(2 c^2-3 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 (c+d \sin (e+f x))}+\frac{3 b^2 \left(-2 a^2 d+a b c+b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))}-\frac{2 d^3 \left(-a c d+4 b c^2-3 b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{3/2} (b c-a d)^4}",1,"-((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(3/2)*f) - (d*(2*a^4*d^3 + a^2*b^2*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*d^2) - 3*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])) + (3*b^2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x]))","A",10,6,25,0.2400,1,"{2802, 3055, 3001, 2660, 618, 204}"
722,1,669,0,3.3089853,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx","Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]","-\frac{b^3 \left(-a^2 b^2 \left(2 c^2-29 d^2\right)+10 a^3 b c d-20 a^4 d^2-4 a b^3 c d-b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^5}-\frac{d^3 \left(a^2 d^2 \left(2 c^2+d^2\right)-a b \left(10 c^3 d-4 c d^3\right)+b^2 \left(-29 c^2 d^2+20 c^4+12 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^5}+\frac{3 d \left(-a^2 b^3 d \left(-12 c^2 d^2+3 c^4+7 d^4\right)-2 a^3 b^2 c d^4-a^4 b \left(3 c^2 d^3-2 d^5\right)+a^5 c d^4+a b^4 c \left(-2 c^2 d^2+c^4+2 d^4\right)+b^5 d \left(-7 c^2 d^2+2 c^4+4 d^4\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right)^2 (b c-a d)^4 (c+d \sin (e+f x))}-\frac{d \left(2 a^2 b^2 d \left(4 c^2-5 d^2\right)+a^4 d^3-3 a b^3 c \left(c^2-d^2\right)-b^4 d \left(5 c^2-6 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 (c+d \sin (e+f x))^2}+\frac{b^2 \left(-7 a^2 d+3 a b c+4 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}","-\frac{b^3 \left(-a^2 b^2 \left(2 c^2-29 d^2\right)+10 a^3 b c d-20 a^4 d^2-4 a b^3 c d-b^4 \left(c^2+12 d^2\right)\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (e+f x)\right)+b}{\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{5/2} (b c-a d)^5}-\frac{d^3 \left(a^2 d^2 \left(2 c^2+d^2\right)-a b \left(10 c^3 d-4 c d^3\right)+b^2 \left(-29 c^2 d^2+20 c^4+12 d^4\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{1}{2} (e+f x)\right)+d}{\sqrt{c^2-d^2}}\right)}{f \left(c^2-d^2\right)^{5/2} (b c-a d)^5}+\frac{3 d \left(-a^2 b^3 d \left(-12 c^2 d^2+3 c^4+7 d^4\right)-2 a^3 b^2 c d^4-a^4 b \left(3 c^2 d^3-2 d^5\right)+a^5 c d^4+a b^4 c \left(-2 c^2 d^2+c^4+2 d^4\right)+b^5 d \left(-7 c^2 d^2+2 c^4+4 d^4\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right)^2 (b c-a d)^4 (c+d \sin (e+f x))}-\frac{d \left(2 a^2 b^2 d \left(4 c^2-5 d^2\right)+a^4 d^3-3 a b^3 c \left(c^2-d^2\right)-b^4 d \left(5 c^2-6 d^2\right)\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 (c+d \sin (e+f x))^2}+\frac{b^2 \left(-7 a^2 d+3 a b c+4 b^2 d\right) \cos (e+f x)}{2 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}",1,"-((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) - (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) + (3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))","A",11,6,25,0.2400,1,"{2802, 3055, 3001, 2660, 618, 204}"
723,1,298,0,0.4805868,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 \left(56 a c d+15 b c^2+25 b d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 f}-\frac{2 \left(c^2-d^2\right) \left(56 a c d+15 b c^2+25 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}-\frac{2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}","-\frac{2 \left(56 a c d+15 b c^2+25 b d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 f}-\frac{2 \left(c^2-d^2\right) \left(56 a c d+15 b c^2+25 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}-\frac{2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}",1,"(-2*(15*b*c^2 + 56*a*c*d + 25*b*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*f) - (2*(5*b*c + 7*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*f) - (2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + (2*(15*b*c^3 + 161*a*c^2*d + 145*b*c*d^2 + 63*a*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(15*b*c^2 + 56*a*c*d + 25*b*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d*f*Sqrt[c + d*Sin[e + f*x]])","A",8,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
724,1,235,0,0.3588189,"\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 \left(c^2-d^2\right) (5 a d+3 b c) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(20 a c d+3 b \left(c^2+3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (5 a d+3 b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 f}-\frac{2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}","-\frac{2 \left(c^2-d^2\right) (5 a d+3 b c) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(20 a c d+3 b \left(c^2+3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (5 a d+3 b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 f}-\frac{2 b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}",1,"(-2*(3*b*c + 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*f) - (2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (2*(20*a*c*d + 3*b*(c^2 + 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(3*b*c + 5*a*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d*f*Sqrt[c + d*Sin[e + f*x]])","A",7,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
725,1,181,0,0.2130584,"\int (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 (3 a d+b c) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f}","\frac{2 (3 a d+b c) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f}",1,"(-2*b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + (2*(b*c + 3*a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*b*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,25,0.2400,1,"{2753, 2752, 2663, 2661, 2655, 2653}"
726,1,140,0,0.1240151,"\int \frac{a+b \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])/Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 b \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}","\frac{2 b \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}",1,"(2*b*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])","A",5,5,25,0.2000,1,"{2752, 2663, 2661, 2655, 2653}"
727,1,195,0,0.2156713,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 (b c-a d) \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}","-\frac{2 (b c-a d) \cos (e+f x)}{f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d f \sqrt{c+d \sin (e+f x)}}",1,"(-2*(b*c - a*d)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*b*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
728,1,285,0,0.3945325,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 \left(4 a c d-b \left(c^2+3 d^2\right)\right) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \cos (e+f x)}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(4 a c d-b \left(c^2+3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{2 \left(4 a c d-b \left(c^2+3 d^2\right)\right) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 (b c-a d) \cos (e+f x)}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(4 a c d-b \left(c^2+3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*(b*c - a*d)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(4*a*c*d - b*(c^2 + 3*d^2))*Cos[e + f*x])/(3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(4*a*c*d - b*(c^2 + 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
729,1,369,0,0.5263894,"\int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{2 \left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) \cos (e+f x)}{15 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-8 a c d+3 b c^2+5 b d^2\right) \cos (e+f x)}{15 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \cos (e+f x)}{5 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}+\frac{2 \left(-8 a c d+3 b c^2+5 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{2 \left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) \cos (e+f x)}{15 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-8 a c d+3 b c^2+5 b d^2\right) \cos (e+f x)}{15 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \cos (e+f x)}{5 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}+\frac{2 \left(-8 a c d+3 b c^2+5 b d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*(b*c - a*d)*Cos[e + f*x])/(5*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) - (2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*Cos[e + f*x])/(15*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*Cos[e + f*x])/(15*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,6,25,0.2400,1,"{2754, 2752, 2663, 2661, 2655, 2653}"
730,1,451,0,0.9453096,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{4 \left(84 a^2 c d^2+15 a b d \left(3 c^2+5 d^2\right)+b^2 \left(-\left(5 c^3-57 c d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 \left(c^2-d^2\right) \left(-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(21 a^2 d^2 \left(23 c^2+9 d^2\right)+30 a b d \left(3 c^3+29 c d^2\right)+b^2 \left(-\left(-279 c^2 d^2+10 c^4-147 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(7 d^2 \left(9 a^2+7 b^2\right)-10 b c (b c-9 a d)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac{4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}","-\frac{4 \left(84 a^2 c d^2+15 a b d \left(3 c^2+5 d^2\right)+b^2 \left(-\left(5 c^3-57 c d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d f}+\frac{4 \left(c^2-d^2\right) \left(-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(21 a^2 d^2 \left(23 c^2+9 d^2\right)+30 a b d \left(3 c^3+29 c d^2\right)+b^2 \left(-\left(-279 c^2 d^2+10 c^4-147 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(7 d^2 \left(9 a^2+7 b^2\right)-10 b c (b c-9 a d)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac{4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}",1,"(-4*(84*a^2*c*d^2 + 15*a*b*d*(3*c^2 + 5*d^2) - b^2*(5*c^3 - 57*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d*f) - (2*(7*(9*a^2 + 7*b^2)*d^2 - 10*b*c*(b*c - 9*a*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d*f) + (4*b*(b*c - 9*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d*f) + (2*(21*a^2*d^2*(23*c^2 + 9*d^2) + 30*a*b*d*(3*c^3 + 29*c*d^2) - b^2*(10*c^4 - 279*c^2*d^2 - 147*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*(c^2 - d^2)*(5*b^2*c^3 - 45*a*b*c^2*d - 84*a^2*c*d^2 - 57*b^2*c*d^2 - 75*a*b*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",9,7,27,0.2593,1,"{2791, 2753, 2752, 2663, 2661, 2655, 2653}"
731,1,347,0,0.633734,"\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 \left(c^2-d^2\right) \left(35 a^2 d^2+42 a b c d+b^2 \left(-\left(6 c^2-25 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \left(70 a^2 c d^2+21 a b d \left(c^2+3 d^2\right)+b^2 \left(-\left(3 c^3-41 c d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(5 d^2 \left(7 a^2+5 b^2\right)-6 b c (b c-7 a d)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}","-\frac{2 \left(c^2-d^2\right) \left(35 a^2 d^2+42 a b c d+b^2 \left(-\left(6 c^2-25 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \left(70 a^2 c d^2+21 a b d \left(c^2+3 d^2\right)+b^2 \left(-\left(3 c^3-41 c d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left(5 d^2 \left(7 a^2+5 b^2\right)-6 b c (b c-7 a d)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d f}+\frac{4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}",1,"(-2*(5*(7*a^2 + 5*b^2)*d^2 - 6*b*c*(b*c - 7*a*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f) + (4*b*(b*c - 7*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) + (4*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) - b^2*(3*c^3 - 41*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(42*a*b*c*d + 35*a^2*d^2 - b^2*(6*c^2 - 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2791, 2753, 2752, 2663, 2661, 2655, 2653}"
732,1,254,0,0.4114918,"\int (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 \left(3 d^2 \left(5 a^2+3 b^2\right)-2 b c (b c-5 a d)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 b \left(c^2-d^2\right) (b c-5 a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 b (b c-5 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}","\frac{2 \left(3 d^2 \left(5 a^2+3 b^2\right)-2 b c (b c-5 a d)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 b \left(c^2-d^2\right) (b c-5 a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 b (b c-5 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d f}-\frac{2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}",1,"(4*b*(b*c - 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d*f) + (2*(3*(5*a^2 + 3*b^2)*d^2 - 2*b*c*(b*c - 5*a*d))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*b*(b*c - 5*a*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2791, 2753, 2752, 2663, 2661, 2655, 2653}"
733,1,203,0,0.2848397,"\int \frac{(a+b \sin (e+f x))^2}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^2/Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 \left(d^2 \left(3 a^2+b^2\right)+2 b c (b c-3 a d)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}","\frac{2 \left(d^2 \left(3 a^2+b^2\right)+2 b c (b c-3 a d)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}",1,"(-2*b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) - (4*b*(b*c - 3*a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*((3*a^2 + b^2)*d^2 + 2*b*c*(b*c - 3*a*d))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2791, 2752, 2663, 2661, 2655, 2653}"
734,1,228,0,0.307907,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 \left(d^2 \left(a^2-b^2\right)-2 a b c d+2 b^2 c^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \sqrt{c+d \sin (e+f x)}}","\frac{2 \left(d^2 \left(a^2-b^2\right)-2 a b c d+2 b^2 c^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{d^2 f \sqrt{c+d \sin (e+f x)}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(2*b^2*c^2 - 2*a*b*c*d + (a^2 - b^2)*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*b*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d^2*f*Sqrt[c + d*Sin[e + f*x]])","A",6,6,27,0.2222,1,"{2790, 2752, 2663, 2661, 2655, 2653}"
735,1,329,0,0.4991524,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 \left(-a^2 d^2+2 a b c d+b^2 \left(2 c^2-3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}-\frac{4 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d) \cos (e+f x)}{3 d f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{2 \left(-a^2 d^2+2 a b c d+b^2 \left(2 c^2-3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}-\frac{4 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d) \cos (e+f x)}{3 d f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^2 f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (4*(b*c - a*d)*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(3*d*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (4*(b*c - a*d)*(2*a*c*d + b*(c^2 - 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(2*a*b*c*d - a^2*d^2 + b^2*(2*c^2 - 3*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2790, 2754, 2752, 2663, 2661, 2655, 2653}"
736,1,460,0,0.8626031,"\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]","\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)-a b \left(6 c^3 d+58 c d^3\right)+b^2 \left(-\left(-19 c^2 d^2+2 c^4-15 d^4\right)\right)\right) \cos (e+f x)}{15 d f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)-a b \left(6 c^3 d+58 c d^3\right)+b^2 \left(-\left(-19 c^2 d^2+2 c^4-15 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{5 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}-\frac{4 \left(4 a c d+b \left(c^2-5 d^2\right)\right) (b c-a d) \cos (e+f x)}{15 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}+\frac{4 \left(4 a c d+b \left(c^2-5 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}","\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)-a b \left(6 c^3 d+58 c d^3\right)+b^2 \left(-\left(-19 c^2 d^2+2 c^4-15 d^4\right)\right)\right) \cos (e+f x)}{15 d f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)-a b \left(6 c^3 d+58 c d^3\right)+b^2 \left(-\left(-19 c^2 d^2+2 c^4-15 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x)}{5 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}-\frac{4 \left(4 a c d+b \left(c^2-5 d^2\right)\right) (b c-a d) \cos (e+f x)}{15 d f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}+\frac{4 \left(4 a c d+b \left(c^2-5 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^2 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x])/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) - (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*Cos[e + f*x])/(15*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*Cos[e + f*x])/(15*d*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,7,27,0.2593,1,"{2790, 2754, 2752, 2663, 2661, 2655, 2653}"
737,1,642,0,1.4005747,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 b \left(-297 a^2 d^2+66 a b c d+b^2 \left(-\left(8 c^2+81 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{2 \left(1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left(10 c^2-49 d^2\right)+5 b^3 \left(8 c^3+67 c d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac{2 \left(495 a^2 b d^2 \left(3 c^2+5 d^2\right)+1848 a^3 c d^3-66 a b^2 d \left(5 c^3-57 c d^2\right)+5 b^3 \left(57 c^2 d^2+8 c^4+135 d^4\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3465 d^2 f}-\frac{2 \left(c^2-d^2\right) \left(495 a^2 b d^2 \left(3 c^2+5 d^2\right)+1848 a^3 c d^3-66 a b^2 d \left(5 c^3-57 c d^2\right)+5 b^3 \left(57 c^2 d^2+8 c^4+135 d^4\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3465 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(495 a^2 b c d^2 \left(3 c^2+29 d^2\right)+231 a^3 d^3 \left(23 c^2+9 d^2\right)-33 a b^2 d \left(-279 c^2 d^2+10 c^4-147 d^4\right)+5 b^3 \left(51 c^3 d^2+8 c^5+741 c d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3465 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}","\frac{2 b \left(-297 a^2 d^2+66 a b c d+b^2 \left(-\left(8 c^2+81 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{2 \left(1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left(10 c^2-49 d^2\right)+5 b^3 \left(8 c^3+67 c d^2\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac{2 \left(495 a^2 b d^2 \left(3 c^2+5 d^2\right)+1848 a^3 c d^3-66 a b^2 d \left(5 c^3-57 c d^2\right)+5 b^3 \left(57 c^2 d^2+8 c^4+135 d^4\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3465 d^2 f}-\frac{2 \left(c^2-d^2\right) \left(495 a^2 b d^2 \left(3 c^2+5 d^2\right)+1848 a^3 c d^3-66 a b^2 d \left(5 c^3-57 c d^2\right)+5 b^3 \left(57 c^2 d^2+8 c^4+135 d^4\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3465 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(495 a^2 b c d^2 \left(3 c^2+29 d^2\right)+231 a^3 d^3 \left(23 c^2+9 d^2\right)-33 a b^2 d \left(-279 c^2 d^2+10 c^4-147 d^4\right)+5 b^3 \left(51 c^3 d^2+8 c^5+741 c d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3465 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}",1,"(-2*(1848*a^3*c*d^3 + 495*a^2*b*d^2*(3*c^2 + 5*d^2) - 66*a*b^2*d*(5*c^3 - 57*c*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3465*d^2*f) - (2*(1485*a^2*b*c*d^2 + 693*a^3*d^3 - 33*a*b^2*d*(10*c^2 - 49*d^2) + 5*b^3*(8*c^3 + 67*c*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(3465*d^2*f) + (2*b*(66*a*b*c*d - 297*a^2*d^2 - b^2*(8*c^2 + 81*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8*b^2*(b*c - 6*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(7/2))/(11*d*f) + (2*(231*a^3*d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2) - 33*a*b^2*d*(10*c^4 - 279*c^2*d^2 - 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3465*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(1848*a^3*c*d^3 + 495*a^2*b*d^2*(3*c^2 + 5*d^2) - 66*a*b^2*d*(5*c^3 - 57*c*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3465*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",10,8,27,0.2963,1,"{2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
738,1,496,0,1.025585,"\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 b \left(-189 a^2 d^2+54 a b c d+b^2 \left(-\left(8 c^2+49 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{2 \left(189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left(6 c^2-25 d^2\right)+b^3 \left(8 c^3+39 c d^2\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{2 \left(c^2-d^2\right) \left(189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left(6 c^2-25 d^2\right)+b^3 \left(8 c^3+39 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(189 a^2 b d^2 \left(c^2+3 d^2\right)+420 a^3 c d^3-a b^2 \left(54 c^3 d-738 c d^3\right)+b^3 \left(33 c^2 d^2+8 c^4+147 d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}","\frac{2 b \left(-189 a^2 d^2+54 a b c d+b^2 \left(-\left(8 c^2+49 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{2 \left(189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left(6 c^2-25 d^2\right)+b^3 \left(8 c^3+39 c d^2\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{2 \left(c^2-d^2\right) \left(189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left(6 c^2-25 d^2\right)+b^3 \left(8 c^3+39 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(189 a^2 b d^2 \left(c^2+3 d^2\right)+420 a^3 c d^3-a b^2 \left(54 c^3 d-738 c d^3\right)+b^3 \left(33 c^2 d^2+8 c^4+147 d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}",1,"(-2*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f) + (2*b*(54*a*b*c*d - 189*a^2*d^2 - b^2*(8*c^2 + 49*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d^2*f) + (8*b^2*(b*c - 5*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d*f) + (2*(420*a^3*c*d^3 + 189*a^2*b*d^2*(c^2 + 3*d^2) - a*b^2*(54*c^3*d - 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",9,8,27,0.2963,1,"{2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
739,1,375,0,0.6777718,"\int (a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 b \left(-105 a^2 d^2+42 a b c d+b^2 \left(-\left(8 c^2+25 d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{2 b \left(c^2-d^2\right) \left(-105 a^2 d^2+42 a b c d+b^2 \left(-\left(8 c^2+25 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left(2 c^2-9 d^2\right)+b^3 \left(8 c^3+19 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}","\frac{2 b \left(-105 a^2 d^2+42 a b c d+b^2 \left(-\left(8 c^2+25 d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{2 b \left(c^2-d^2\right) \left(-105 a^2 d^2+42 a b c d+b^2 \left(-\left(8 c^2+25 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left(2 c^2-9 d^2\right)+b^3 \left(8 c^3+19 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}",1,"(2*b*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f) + (8*b^2*(b*c - 4*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d*f) + (2*(105*a^2*b*c*d^2 + 105*a^3*d^3 - 21*a*b^2*d*(2*c^2 - 9*d^2) + b^3*(8*c^3 + 19*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*b*(c^2 - d^2)*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
740,1,302,0,0.4790753,"\int \frac{(a+b \sin (e+f x))^3}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{2 \left(45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left(2 c^2+d^2\right)+b^3 \left(8 c^3+7 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \left(-45 a^2 d^2+30 a b c d+b^2 \left(-\left(8 c^2+9 d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}","-\frac{2 \left(45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left(2 c^2+d^2\right)+b^3 \left(8 c^3+7 c d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \left(-45 a^2 d^2+30 a b c d+b^2 \left(-\left(8 c^2+9 d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}",1,"(8*b^2*(b*c - 3*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(5*d*f) - (2*b*(30*a*b*c*d - 45*a^2*d^2 - b^2*(8*c^2 + 9*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(45*a^2*b*c*d^2 - 15*a^3*d^3 - 15*a*b^2*d*(2*c^2 + d^2) + b^3*(8*c^3 + 7*c*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2793, 3023, 2752, 2663, 2661, 2655, 2653}"
741,1,361,0,0.6218136,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 b \left(-3 a^2 d^2+6 a b c d+b^2 \left(-\left(4 c^2-d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f \left(c^2-d^2\right)}-\frac{2 b \left(-9 a^2 d^2+18 a b c d+b^2 \left(-\left(8 c^2+d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left(2 c^2-d^2\right)+b^3 \left(8 c^3-5 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}","\frac{2 b \left(-3 a^2 d^2+6 a b c d+b^2 \left(-\left(4 c^2-d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f \left(c^2-d^2\right)}-\frac{2 b \left(-9 a^2 d^2+18 a b c d+b^2 \left(-\left(8 c^2+d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left(2 c^2-d^2\right)+b^3 \left(8 c^3-5 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*b*(6*a*b*c*d - 3*a^2*d^2 - b^2*(4*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c^2 - d^2)*f) - (2*(9*a^2*b*c*d^2 - 3*a^3*d^3 - 9*a*b^2*d*(2*c^2 - d^2) + b^3*(8*c^3 - 5*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*b*(18*a*b*c*d - 9*a^2*d^2 - b^2*(8*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2792, 3023, 2752, 2663, 2661, 2655, 2653}"
742,1,391,0,0.7479298,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 \left(-a^2 d^2+2 a b c d+b^2 \left(8 c^2-9 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(-3 a^2 b d^2 \left(c^2+3 d^2\right)+4 a^3 c d^3-6 a b^2 c d \left(c^2-3 d^2\right)+b^3 \left(-15 c^2 d^2+8 c^4+3 d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(a c d+b \left(c^2-2 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{3 d^2 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}","-\frac{2 \left(-a^2 d^2+2 a b c d+b^2 \left(8 c^2-9 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(-3 a^2 b d^2 \left(c^2+3 d^2\right)+4 a^3 c d^3-6 a b^2 c d \left(c^2-3 d^2\right)+b^3 \left(-15 c^2 d^2+8 c^4+3 d^4\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 d^3 f \left(c^2-d^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(a c d+b \left(c^2-2 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{3 d^2 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (8*(b*c - a*d)^2*(a*c*d + b*(c^2 - 2*d^2))*Cos[e + f*x])/(3*d^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(4*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4 - 15*c^2*d^2 + 3*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(b*c - a*d)*(2*a*b*c*d - a^2*d^2 + b^2*(8*c^2 - 9*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2792, 3021, 2752, 2663, 2661, 2655, 2653}"
743,1,532,0,1.1091203,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]","-\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(-21 c^2 d^2+8 c^4+45 d^4\right)\right) (b c-a d) \cos (e+f x)}{15 d^2 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-3 a^2 b d^2 \left(3 c^2+5 d^2\right)+8 a^3 c d^3-6 a b^2 c d \left(c^2-5 d^2\right)+b^3 \left(-\left(-15 c^2 d^2+8 c^4+15 d^4\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(-21 c^2 d^2+8 c^4+45 d^4\right)\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{15 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}","-\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(-21 c^2 d^2+8 c^4+45 d^4\right)\right) (b c-a d) \cos (e+f x)}{15 d^2 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(-3 a^2 b d^2 \left(3 c^2+5 d^2\right)+8 a^3 c d^3-6 a b^2 c d \left(c^2-5 d^2\right)+b^3 \left(-\left(-15 c^2 d^2+8 c^4+15 d^4\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \left(c^2-d^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left(a^2 d^2 \left(23 c^2+9 d^2\right)+2 a b d \left(7 c^3-39 c d^2\right)+b^2 \left(-21 c^2 d^2+8 c^4+45 d^4\right)\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{15 d^3 f \left(c^2-d^2\right)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(2 a c d+b \left(c^2-3 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{15 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{5/2}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*(b*c - a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 5*d^2) - 3*a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2792, 3021, 2754, 2752, 2663, 2661, 2655, 2653}"
744,1,716,0,1.4360523,"\int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx","Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(9/2),x]","-\frac{2 \left(a^2 d^2 \left(71 c^2+25 d^2\right)+a b \left(26 c^3 d-218 c d^3\right)+b^2 \left(-17 c^2 d^2+8 c^4+105 d^4\right)\right) (b c-a d) \cos (e+f x)}{105 d^2 f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(-9 a^2 b d^2 \left(102 c^2 d^2+5 c^4+21 d^4\right)+16 a^3 c d^3 \left(11 c^2+13 d^2\right)-6 a b^2 c d \left(-62 c^2 d^2+3 c^4-133 d^4\right)+b^3 \left(-\left(-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right)\right)\right) \cos (e+f x)}{105 d^2 f \left(c^2-d^2\right)^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(a^2 d^2 \left(71 c^2+25 d^2\right)+a b \left(26 c^3 d-218 c d^3\right)+b^2 \left(-17 c^2 d^2+8 c^4+105 d^4\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(-9 a^2 b d^2 \left(102 c^2 d^2+5 c^4+21 d^4\right)+16 a^3 c d^3 \left(11 c^2+13 d^2\right)-6 a b^2 c d \left(-62 c^2 d^2+3 c^4-133 d^4\right)+b^3 \left(-\left(-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \left(c^2-d^2\right)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(3 a c d+b \left(c^2-4 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{7/2}}","-\frac{2 \left(a^2 d^2 \left(71 c^2+25 d^2\right)+a b \left(26 c^3 d-218 c d^3\right)+b^2 \left(-17 c^2 d^2+8 c^4+105 d^4\right)\right) (b c-a d) \cos (e+f x)}{105 d^2 f \left(c^2-d^2\right)^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(-9 a^2 b d^2 \left(102 c^2 d^2+5 c^4+21 d^4\right)+16 a^3 c d^3 \left(11 c^2+13 d^2\right)-6 a b^2 c d \left(-62 c^2 d^2+3 c^4-133 d^4\right)+b^3 \left(-\left(-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right)\right)\right) \cos (e+f x)}{105 d^2 f \left(c^2-d^2\right)^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(a^2 d^2 \left(71 c^2+25 d^2\right)+a b \left(26 c^3 d-218 c d^3\right)+b^2 \left(-17 c^2 d^2+8 c^4+105 d^4\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \left(c^2-d^2\right)^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(-9 a^2 b d^2 \left(102 c^2 d^2+5 c^4+21 d^4\right)+16 a^3 c d^3 \left(11 c^2+13 d^2\right)-6 a b^2 c d \left(-62 c^2 d^2+3 c^4-133 d^4\right)+b^3 \left(-\left(-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{105 d^3 f \left(c^2-d^2\right)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left(3 a c d+b \left(c^2-4 d^2\right)\right) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left(c^2-d^2\right)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{7/2}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(7*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(7/2)) + (8*(b*c - a*d)^2*(3*a*c*d + b*(c^2 - 4*d^2))*Cos[e + f*x])/(35*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(5/2)) - (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*Cos[e + f*x])/(105*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*Cos[e + f*x])/(105*d^2*(c^2 - d^2)^4*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*(c^2 - d^2)^4*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",9,8,27,0.2963,1,"{2792, 3021, 2754, 2752, 2663, 2661, 2655, 2653}"
745,1,296,0,1.0882864,"\int \frac{(c+d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x]),x]","-\frac{2 d \left(-3 a^2 d^2+6 a b c d+b^2 \left(-\left(2 c^2+d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 d (7 b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f (a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b f}","-\frac{2 d \left(-3 a^2 d^2+6 a b c d+b^2 \left(-\left(2 c^2+d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 d (7 b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f (a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b f}",1,"(-2*d^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + (2*d*(7*b*c - 3*a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*b^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*d*(6*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*b^3*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(b*c - a*d)^3*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^3*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2793, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
746,1,229,0,0.491392,"\int \frac{(c+d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x]),x]","\frac{2 d (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{2 d (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \sqrt{c+d \sin (e+f x)}}+\frac{2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(2*d*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*d*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(b*c - a*d)^2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",8,8,27,0.2963,1,"{2804, 2655, 2653, 2803, 2663, 2661, 2807, 2805}"
747,1,153,0,0.3288457,"\int \frac{\sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x]),x]","\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \sqrt{c+d \sin (e+f x)}}","\frac{2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \sqrt{c+d \sin (e+f x)}}",1,"(2*d*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(b*c - a*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",5,5,27,0.1852,1,"{2803, 2663, 2661, 2807, 2805}"
748,1,75,0,0.213989,"\int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) \sqrt{c+d \sin (e+f x)}}","\frac{2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) \sqrt{c+d \sin (e+f x)}}",1,"(2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])","A",2,2,27,0.07407,1,"{2807, 2805}"
749,1,220,0,0.6303551,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{2 d^2 \cos (e+f x)}{f \left(c^2-d^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(c^2-d^2\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) (b c-a d) \sqrt{c+d \sin (e+f x)}}","-\frac{2 d^2 \cos (e+f x)}{f \left(c^2-d^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(c^2-d^2\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) (b c-a d) \sqrt{c+d \sin (e+f x)}}",1,"(-2*d^2*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*d*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((b*c - a*d)*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*b*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])","A",7,7,27,0.2593,1,"{2802, 3059, 2655, 2653, 12, 2807, 2805}"
750,1,399,0,1.6529378,"\int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{2 b^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \left(-4 a c d+7 b c^2-3 b d^2\right) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{3 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(c^2-d^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left(-4 a c d+7 b c^2-3 b d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(c^2-d^2\right)^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{2 b^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a+b) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \left(-4 a c d+7 b c^2-3 b d^2\right) \cos (e+f x)}{3 f \left(c^2-d^2\right)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{3 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(c^2-d^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left(-4 a c d+7 b c^2-3 b d^2\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(c^2-d^2\right)^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (2*d^2*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*Cos[e + f*x])/(3*(b*c - a*d)^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*d*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*(b*c - a*d)^2*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*d*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*(b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*b^2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
751,1,534,0,2.0400434,"\int \frac{(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^2,x]","\frac{d \left(-5 a^2 d^2+6 a b c d+b^2 \left(-\left(3 c^2-2 d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 f \left(a^2-b^2\right)}-\frac{\left(2 a^2 b^2 d^2 \left(c^2+8 d^2\right)+24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left(c^2+3 d^2\right)+b^4 \left(16 c^2 d^2+3 c^4+2 d^4\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^4 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{\left(29 a^2 b c d^2-15 a^3 d^3-a b^2 \left(9 c^2 d-12 d^3\right)+b^3 \left(3 c^3-20 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^3 f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{\left(5 a^2 d+2 a b c-7 b^2 d\right) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^4 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}","\frac{d \left(-5 a^2 d^2+6 a b c d+b^2 \left(-\left(3 c^2-2 d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b^2 f \left(a^2-b^2\right)}-\frac{\left(2 a^2 b^2 d^2 \left(c^2+8 d^2\right)+24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left(c^2+3 d^2\right)+b^4 \left(16 c^2 d^2+3 c^4+2 d^4\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^4 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{\left(29 a^2 b c d^2-15 a^3 d^3-a b^2 \left(9 c^2 d-12 d^3\right)+b^3 \left(3 c^3-20 c d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 b^3 f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{\left(5 a^2 d+2 a b c-7 b^2 d\right) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^4 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"(d*(6*a*b*c*d - 5*a^2*d^2 - b^2*(3*c^2 - 2*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])) + ((29*a^2*b*c*d^2 - 15*a^3*d^3 + b^3*(3*c^3 - 20*c*d^2) - a*b^2*(9*c^2*d - 12*d^3))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*b^3*(a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((24*a^3*b*c*d^3 - 15*a^4*d^4 - 12*a*b^3*c*d*(c^2 + 3*d^2) + 2*a^2*b^2*d^2*(c^2 + 8*d^2) + b^4*(3*c^4 + 16*c^2*d^2 + 2*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*b^4*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^3*(2*a*b*c + 5*a^2*d - 7*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b^4*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2792, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
752,1,390,0,1.2610559,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^2,x]","\frac{\left(3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2+4 d^2\right)\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{\left(-3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-2 d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{\left(3 a^2 d+2 a b c-5 b^2 d\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}","\frac{\left(3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2+4 d^2\right)\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{\left(-3 a^2 d^2+2 a b c d+b^2 \left(-\left(c^2-2 d^2\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{\left(3 a^2 d+2 a b c-5 b^2 d\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^3 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"((b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])) - ((2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b^2*(a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((b*c - a*d)*(2*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 + 4*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^3*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^2*(2*a*b*c + 3*a^2*d - 5*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b^3*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2792, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
753,1,351,0,1.0369625,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^2} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^2,x]","\frac{\left(a^2 d^2+2 a b c d+b^2 \left(-\left(c^2+2 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{(b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d) \left(a^2 d+2 a b c-3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}","\frac{\left(a^2 d^2+2 a b c d+b^2 \left(-\left(c^2+2 d^2\right)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{(b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d) \left(a^2 d+2 a b c-3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b^2 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((2*a*b*c*d + a^2*d^2 - b^2*(c^2 + 2*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b^2*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2799, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
754,1,307,0,0.8672694,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^2} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^2,x]","\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}-\frac{(b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(a^2 (-d)+2 a b c-b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}","\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}-\frac{(b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(a^2 (-d)+2 a b c-b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{b f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}}",1,"(b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x])) + (EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*a*b*c - a^2*d - b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2796, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
755,1,325,0,0.9681206,"\int \frac{1}{(a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}-\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{b \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(-3 a^2 d+2 a b c+b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}","\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))}-\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \sqrt{c+d \sin (e+f x)}}+\frac{b \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(-3 a^2 d+2 a b c+b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}",1,"(b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])) + (b*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*a*b*c - 3*a^2*d + b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*(a + b)^2*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])","A",9,9,27,0.3333,1,"{2802, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
756,1,449,0,1.6274356,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{d \left(2 a^2 d^2+b^2 \left(c^2-3 d^2\right)\right) \cos (e+f x)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(2 a^2 d^2+b^2 \left(c^2-3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left(-5 a^2 d+2 a b c+3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}","\frac{d \left(2 a^2 d^2+b^2 \left(c^2-3 d^2\right)\right) \cos (e+f x)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(2 a^2 d^2+b^2 \left(c^2-3 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left(-5 a^2 d+2 a b c+3 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}",1,"(d*(2*a^2*d^2 + b^2*(c^2 - 3*d^2))*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) + ((2*a^2*d^2 + b^2*(c^2 - 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (b*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]]) + (b*(2*a*b*c - 5*a^2*d + 3*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*(a + b)^2*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
757,1,661,0,2.7985951,"\int \frac{1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{\left(-4 a^2 b d^3 \left(5 c^2-3 d^2\right)+8 a^3 c d^4-8 a b^2 c d^4-b^3 \left(-26 c^2 d^3+3 c^4 d+15 d^5\right)\right) \cos (e+f x)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{d \left(2 a^2 d^2+b^2 \left(3 c^2-5 d^2\right)\right) \cos (e+f x)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left(2 a^2 d^2+b^2 \left(3 c^2-5 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(-4 a^2 b d^2 \left(5 c^2-3 d^2\right)+8 a^3 c d^3-8 a b^2 c d^3+b^3 \left(-\left(-26 c^2 d^2+3 c^4+15 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac{b^2 \left(-7 a^2 d+2 a b c+5 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}","-\frac{\left(-4 a^2 b d^3 \left(5 c^2-3 d^2\right)+8 a^3 c d^4-8 a b^2 c d^4-b^3 \left(-26 c^2 d^3+3 c^4 d+15 d^5\right)\right) \cos (e+f x)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{d \left(2 a^2 d^2+b^2 \left(3 c^2-5 d^2\right)\right) \cos (e+f x)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left(2 a^2 d^2+b^2 \left(3 c^2-5 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(-4 a^2 b d^2 \left(5 c^2-3 d^2\right)+8 a^3 c d^3-8 a b^2 c d^3+b^3 \left(-\left(-26 c^2 d^2+3 c^4+15 d^4\right)\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right)^2 (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac{b^2 \left(-7 a^2 d+2 a b c+5 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{f (a-b) (a+b)^2 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}",1,"(d*(2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - ((8*a^3*c*d^4 - 8*a*b^2*c*d^4 - 4*a^2*b*d^3*(5*c^2 - 3*d^2) - b^3*(3*c^4*d - 26*c^2*d^3 + 15*d^5))*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^3*c*d^3 - 8*a*b^2*c*d^3 - 4*a^2*b*d^2*(5*c^2 - 3*d^2) - b^3*(3*c^4 - 26*c^2*d^2 + 15*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(2*a*b*c - 7*a^2*d + 5*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*(a + b)^2*(b*c - a*d)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",11,10,27,0.3704,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
758,1,816,0,3.179124,"\int \frac{(c+d \sin (e+f x))^{9/2}}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3,x]","\frac{\left(35 d^2 a^4+20 b c d a^3+2 b^2 \left(4 c^2-43 d^2\right) a^2-44 b^3 c d a+b^4 \left(4 c^2+63 d^2\right)\right) \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} (b c-a d)^3}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\cos (e+f x) (c+d \sin (e+f x))^{5/2} (b c-a d)^2}{2 b \left(a^2-b^2\right) f (a+b \sin (e+f x))^2}+\frac{\left(7 d a^2+6 b c a-13 b^2 d\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{4 b^2 \left(a^2-b^2\right)^2 f (a+b \sin (e+f x))}+\frac{d \left(-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left(9 c^2+61 d^2\right) a^2-18 b^3 c \left(c^2+5 d^2\right) a+b^4 d \left(45 c^2-8 d^2\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left(a^2-b^2\right)^2 f}-\frac{\left(-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left(26 c^2+223 d^2\right) a^4-12 b^3 c d^2 \left(4 c^2+29 d^2\right) a^3-b^4 d \left(33 c^4+70 d^2 c^2+128 d^4\right) a^2+6 b^5 c \left(3 c^4+38 d^2 c^2+48 d^4\right) a-b^6 d \left(57 c^4+136 d^2 c^2+8 d^4\right)\right) F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{12 b^5 \left(a^2-b^2\right)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left(-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left(3 c^2-13 d^2\right) a^3-b^3 c d \left(21 c^2+361 d^2\right) a^2+9 b^4 \left(2 c^4+17 d^2 c^2-8 d^4\right) a-b^5 c d \left(51 c^2-104 d^2\right)\right) E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left(a^2-b^2\right)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{\left(35 d^2 a^4+20 b c d a^3+2 b^2 \left(4 c^2-43 d^2\right) a^2-44 b^3 c d a+b^4 \left(4 c^2+63 d^2\right)\right) \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} (b c-a d)^3}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\cos (e+f x) (c+d \sin (e+f x))^{5/2} (b c-a d)^2}{2 b \left(a^2-b^2\right) f (a+b \sin (e+f x))^2}+\frac{\left(7 d a^2+6 b c a-13 b^2 d\right) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{4 b^2 \left(a^2-b^2\right)^2 f (a+b \sin (e+f x))}+\frac{d \left(-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left(9 c^2+61 d^2\right) a^2-18 b^3 c \left(c^2+5 d^2\right) a+b^4 d \left(45 c^2-8 d^2\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{12 b^3 \left(a^2-b^2\right)^2 f}-\frac{\left(-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left(26 c^2+223 d^2\right) a^4-12 b^3 c d^2 \left(4 c^2+29 d^2\right) a^3-b^4 d \left(33 c^4+70 d^2 c^2+128 d^4\right) a^2+6 b^5 c \left(3 c^4+38 d^2 c^2+48 d^4\right) a-b^6 d \left(57 c^4+136 d^2 c^2+8 d^4\right)\right) F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{12 b^5 \left(a^2-b^2\right)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left(-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left(3 c^2-13 d^2\right) a^3-b^3 c d \left(21 c^2+361 d^2\right) a^2+9 b^4 \left(2 c^4+17 d^2 c^2-8 d^4\right) a-b^5 c d \left(51 c^2-104 d^2\right)\right) E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right) \sqrt{c+d \sin (e+f x)}}{12 b^4 \left(a^2-b^2\right)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(d*(36*a^3*b*c*d^2 - 35*a^4*d^3 + b^4*d*(45*c^2 - 8*d^2) - 18*a*b^3*c*(c^2 + 5*d^2) + a^2*b^2*d*(9*c^2 + 61*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(12*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(6*a*b*c + 7*a^2*d - 13*b^2*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((185*a^4*b*c*d^3 - 105*a^5*d^4 - b^5*c*d*(51*c^2 - 104*d^2) - 15*a^3*b^2*d^2*(3*c^2 - 13*d^2) - a^2*b^3*c*d*(21*c^2 + 361*d^2) + 9*a*b^4*(2*c^4 + 17*c^2*d^2 - 8*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(12*b^4*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((150*a^5*b*c*d^4 - 105*a^6*d^5 - 12*a^3*b^3*c*d^2*(4*c^2 + 29*d^2) + a^4*b^2*d^3*(26*c^2 + 223*d^2) - b^6*d*(57*c^4 + 136*c^2*d^2 + 8*d^4) + 6*a*b^5*c*(3*c^4 + 38*c^2*d^2 + 48*d^4) - a^2*b^4*d*(33*c^4 + 70*c^2*d^2 + 128*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(12*b^5*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^3*(20*a^3*b*c*d - 44*a*b^3*c*d + 35*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 43*d^2) + b^4*(4*c^2 + 63*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^5*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",11,11,27,0.4074,1,"{2792, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
759,1,605,0,2.2011818,"\int \frac{(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^3,x]","\frac{3 \left(a^2 b^2 d \left(c^2-11 d^2\right)+4 a^3 b c d^2+5 a^4 d^3-2 a b^3 c \left(c^2+5 d^2\right)+b^4 d \left(5 c^2+8 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^4 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(a^2 b^2 d \left(5 c^2+29 d^2\right)+8 a^3 b c d^2-15 a^4 d^3-2 a b^3 c \left(3 c^2+13 d^2\right)+b^4 d \left(13 c^2-8 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(2 a^2 b^2 \left(4 c^2-19 d^2\right)+12 a^3 b c d+15 a^4 d^2-36 a b^3 c d+b^4 \left(4 c^2+35 d^2\right)\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{\left(5 a^2 d+6 a b c-11 b^2 d\right) (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}","\frac{3 \left(a^2 b^2 d \left(c^2-11 d^2\right)+4 a^3 b c d^2+5 a^4 d^3-2 a b^3 c \left(c^2+5 d^2\right)+b^4 d \left(5 c^2+8 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^4 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left(a^2 b^2 d \left(5 c^2+29 d^2\right)+8 a^3 b c d^2-15 a^4 d^3-2 a b^3 c \left(3 c^2+13 d^2\right)+b^4 d \left(13 c^2-8 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left(2 a^2 b^2 \left(4 c^2-19 d^2\right)+12 a^3 b c d+15 a^4 d^2-36 a b^3 c d+b^4 \left(4 c^2+35 d^2\right)\right) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{\left(5 a^2 d+6 a b c-11 b^2 d\right) (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b^2 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}",1,"((b*c - a*d)^2*(6*a*b*c + 5*a^2*d - 11*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) - ((8*a^3*b*c*d^2 - 15*a^4*d^3 + b^4*d*(13*c^2 - 8*d^2) - 2*a*b^3*c*(3*c^2 + 13*d^2) + a^2*b^2*d*(5*c^2 + 29*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b^3*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (3*(b*c - a*d)*(4*a^3*b*c*d^2 + 5*a^4*d^3 + a^2*b^2*d*(c^2 - 11*d^2) - 2*a*b^3*c*(c^2 + 5*d^2) + b^4*d*(5*c^2 + 8*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^4*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^2*(12*a^3*b*c*d - 36*a*b^3*c*d + 15*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 19*d^2) + b^4*(4*c^2 + 35*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^4*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2792, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
760,1,549,0,2.0756483,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^3,x]","\frac{\left(a^2 b^2 d \left(7 c^2-5 d^2\right)+4 a^3 b c d^2+3 a^4 d^3-2 a b^3 c \left(3 c^2+11 d^2\right)+b^4 d \left(11 c^2+8 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(2 a^2 b^2 \left(4 c^2-3 d^2\right)+4 a^3 b c d+3 a^4 d^2-28 a b^3 c d+b^4 \left(4 c^2+15 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{3 \left(a^2 d+2 a b c-3 b^2 d\right) (b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{3 \left(a^2 d+2 a b c-3 b^2 d\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","\frac{\left(a^2 b^2 d \left(7 c^2-5 d^2\right)+4 a^3 b c d^2+3 a^4 d^3-2 a b^3 c \left(3 c^2+11 d^2\right)+b^4 d \left(11 c^2+8 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(2 a^2 b^2 \left(4 c^2-3 d^2\right)+4 a^3 b c d+3 a^4 d^2-28 a b^3 c d+b^4 \left(4 c^2+15 d^2\right)\right) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^3 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{3 \left(a^2 d+2 a b c-3 b^2 d\right) (b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{3 \left(a^2 d+2 a b c-3 b^2 d\right) (b c-a d) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b^2*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*a^3*b*c*d^2 + 3*a^4*d^3 + a^2*b^2*d*(7*c^2 - 5*d^2) + b^4*d*(11*c^2 + 8*d^2) - 2*a*b^3*c*(3*c^2 + 11*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^3*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)*(4*a^3*b*c*d - 28*a*b^3*c*d + 3*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^3*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2792, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
761,1,472,0,1.8093834,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^3} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^3,x]","-\frac{\left(-2 a^2 b^2 \left(4 c^2+5 d^2\right)+4 a^3 b c d+a^4 d^2+20 a b^3 c d-b^4 \left(4 c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{\left(a^2 (-d)+6 a b c-5 b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{(b c-a d) \left(a^2 d+6 a b c-7 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(a^2 (-d)+6 a b c-5 b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{\left(-2 a^2 b^2 \left(4 c^2+5 d^2\right)+4 a^3 b c d+a^4 d^2+20 a b^3 c d-b^4 \left(4 c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{\left(a^2 (-d)+6 a b c-5 b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{(b c-a d) \left(a^2 d+6 a b c-7 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b^2 f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(a^2 (-d)+6 a b c-5 b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f \left(a^2-b^2\right)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((6*a*b*c - a^2*d - 5*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((6*a*b*c - a^2*d - 5*b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((b*c - a*d)*(6*a*b*c + a^2*d - 7*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^2*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((4*a^3*b*c*d + 20*a*b^3*c*d + a^4*d^2 - b^4*(4*c^2 + 3*d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^2*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2799, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
762,1,487,0,1.6047191,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^3} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^3,x]","-\frac{\left(-2 a^2 b^2 \left(4 c^2+5 d^2\right)+12 a^3 b c d-3 a^4 d^2+12 a b^3 c d-b^4 \left(4 c^2-d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f (a-b)^2 (a+b)^3 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left(-5 a^2 d+6 a b c-b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (b c-a d) (a+b \sin (e+f x))}+\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{3 \left(a^2 (-d)+2 a b c-b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(-5 a^2 d+6 a b c-b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{\left(-2 a^2 b^2 \left(4 c^2+5 d^2\right)+12 a^3 b c d-3 a^4 d^2+12 a b^3 c d-b^4 \left(4 c^2-d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f (a-b)^2 (a+b)^3 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left(-5 a^2 d+6 a b c-b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (b c-a d) (a+b \sin (e+f x))}+\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (a+b \sin (e+f x))^2}-\frac{3 \left(a^2 (-d)+2 a b c-b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 b f \left(a^2-b^2\right)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left(-5 a^2 d+6 a b c-b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (b*(6*a*b*c - 5*a^2*d - b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*(a + b*Sin[e + f*x])) + ((6*a*b*c - 5*a^2*d - b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (3*(2*a*b*c - a^2*d - b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((12*a^3*b*c*d + 12*a*b^3*c*d - 3*a^4*d^2 - b^4*(4*c^2 - d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b*(a + b)^3*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2796, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
763,1,503,0,1.6752296,"\int \frac{1}{(a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + b*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]),x]","-\frac{\left(-2 a^2 b^2 \left(4 c^2-3 d^2\right)+20 a^3 b c d-15 a^4 d^2+4 a b^3 c d-b^4 \left(4 c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f (a-b)^2 (a+b)^3 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{3 b^2 \left(-3 a^2 d+2 a b c+b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2}-\frac{\left(-7 a^2 d+6 a b c+b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{3 b \left(-3 a^2 d+2 a b c+b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}","-\frac{\left(-2 a^2 b^2 \left(4 c^2-3 d^2\right)+20 a^3 b c d-15 a^4 d^2+4 a b^3 c d-b^4 \left(4 c^2+3 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f (a-b)^2 (a+b)^3 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{3 b^2 \left(-3 a^2 d+2 a b c+b^2 d\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x))}+\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2}-\frac{\left(-7 a^2 d+6 a b c+b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{3 b \left(-3 a^2 d+2 a b c+b^2 d\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}",1,"(b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (3*b^2*(2*a*b*c - 3*a^2*d + b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])) + (3*b*(2*a*b*c - 3*a^2*d + b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((6*a*b*c - 7*a^2*d + b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]]) - ((20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])","A",10,10,27,0.3704,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
764,1,682,0,2.7684547,"\int \frac{1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{d \left(a^2 b^2 d \left(13 c^2-29 d^2\right)+8 a^4 d^3-6 a b^3 c \left(c^2-d^2\right)-b^4 d \left(7 c^2-15 d^2\right)\right) \cos (e+f x)}{4 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left(a^2 b^2 d \left(13 c^2-29 d^2\right)+8 a^4 d^3-6 a b^3 c \left(c^2-d^2\right)-b^4 d \left(7 c^2-15 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b \left(-2 a^2 b^2 \left(4 c^2-19 d^2\right)+28 a^3 b c d-35 a^4 d^2-4 a b^3 c d-b^4 \left(4 c^2+15 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left(-11 a^2 d+6 a b c+5 b^2 d\right) \cos (e+f x)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{b \left(-11 a^2 d+6 a b c+5 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}","-\frac{d \left(a^2 b^2 d \left(13 c^2-29 d^2\right)+8 a^4 d^3-6 a b^3 c \left(c^2-d^2\right)-b^4 d \left(7 c^2-15 d^2\right)\right) \cos (e+f x)}{4 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left(a^2 b^2 d \left(13 c^2-29 d^2\right)+8 a^4 d^3-6 a b^3 c \left(c^2-d^2\right)-b^4 d \left(7 c^2-15 d^2\right)\right) \sqrt{c+d \sin (e+f x)} E\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 \left(c^2-d^2\right) (b c-a d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b \left(-2 a^2 b^2 \left(4 c^2-19 d^2\right)+28 a^3 b c d-35 a^4 d^2-4 a b^3 c d-b^4 \left(4 c^2+15 d^2\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left(\frac{2 b}{a+b};\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \left(-11 a^2 d+6 a b c+5 b^2 d\right) \cos (e+f x)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{2 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{b \left(-11 a^2 d+6 a b c+5 b^2 d\right) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left(\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)|\frac{2 d}{c+d}\right)}{4 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}",1,"-(d*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(6*a*b*c - 11*a^2*d + 5*b^2*d)*Cos[e + f*x])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (b*(6*a*b*c - 11*a^2*d + 5*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (b*(28*a^3*b*c*d - 4*a*b^3*c*d - 35*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 19*d^2) - b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^3*f*Sqrt[c + d*Sin[e + f*x]])","A",11,10,27,0.3704,1,"{2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
765,1,888,0,3.386033,"\int \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} d^2}{3 b f}-\frac{(13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} d}{12 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(\left(33 c^2+16 d^2\right) b^2+14 a c d b-3 a^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac{(a+b)^{3/2} \left(\left(33 c^2+26 d c+16 d^2\right) b^2-6 a d (2 c+d) b+3 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt{c+d} f}-\frac{\left(\left(33 c^2+16 d^2\right) b^2+14 a c d b-3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{c+d} \left(-5 \left(c^3+4 d^2 c\right) b^3-a d \left(15 c^2+4 d^2\right) b^2+5 a^2 c d^2 b-a^3 d^3\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt{a+b} f d}","-\frac{\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} d^2}{3 b f}-\frac{(13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} d}{12 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(\left(33 c^2+16 d^2\right) b^2+14 a c d b-3 a^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac{(a+b)^{3/2} \left(\left(33 c^2+26 d c+16 d^2\right) b^2-6 a d (2 c+d) b+3 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt{c+d} f}-\frac{\left(\left(33 c^2+16 d^2\right) b^2+14 a c d b-3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{c+d} \left(-5 \left(c^3+4 d^2 c\right) b^3-a d \left(15 c^2+4 d^2\right) b^2+5 a^2 c d^2 b-a^3 d^3\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt{a+b} f d}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(24*b^2*(b*c - a*d)*f) - (Sqrt[c + d]*(5*a^2*b*c*d^2 - a^3*d^3 - a*b^2*d*(15*c^2 + 4*d^2) - 5*b^3*(c^3 + 4*c*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(8*b^3*Sqrt[a + b]*d*f) - ((14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*b*f*Sqrt[a + b*Sin[e + f*x]]) - (d*(13*b*c - 3*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*(2*c + d) + b^2*(33*c^2 + 26*c*d + 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(24*b^3*Sqrt[c + d]*f)","A",8,8,29,0.2759,1,"{2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996}"
766,1,784,0,3.5163373,"\int \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{c+d} \left(-a^2 d^2+6 a b c d+b^2 \left(3 c^2+4 d^2\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b^2 d f \sqrt{a+b}}+\frac{(a+b)^{3/2} (-a d+5 b c+2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 b^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+b \sin (e+f x)}}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \sqrt{a+b \sin (e+f x)}}-\frac{(a d+5 b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} (a d+5 b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b f (b c-a d)}","\frac{\sqrt{c+d} \left(-a^2 d^2+6 a b c d+b^2 \left(3 c^2+4 d^2\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b^2 d f \sqrt{a+b}}+\frac{(a+b)^{3/2} (-a d+5 b c+2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 b^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt{a+b \sin (e+f x)}}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \sqrt{a+b \sin (e+f x)}}-\frac{(a d+5 b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} (a d+5 b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b f (b c-a d)}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(5*b*c + a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b*(b*c - a*d)*f) + (Sqrt[c + d]*(6*a*b*c*d - a^2*d^2 + b^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b^2*Sqrt[a + b]*d*f) + ((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*Sqrt[a + b*Sin[e + f*x]]) - ((5*b*c + a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*(5*b*c - a*d + 2*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*b^2*Sqrt[c + d]*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + b*Sin[e + f*x]])","A",8,8,29,0.2759,1,"{2821, 3047, 3061, 3053, 2811, 2998, 2818, 2996}"
767,1,628,0,1.3265702,"\int \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (b c-a d)}+\frac{\sqrt{c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b d f \sqrt{a+b}}","-\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (b c-a d)}+\frac{\sqrt{c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b d f \sqrt{a+b}}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((b*c - a*d)*f) + (Sqrt[c + d]*(b*c + a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*Sqrt[a + b]*d*f) - (b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)","A",7,7,29,0.2414,1,"{2821, 3053, 2811, 12, 2801, 2818, 2996}"
768,1,198,0,0.1138874,"\int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]],x]","\frac{2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d f \sqrt{a+b}}","\frac{2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d f \sqrt{a+b}}",1,"(2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*d*f)","A",1,1,29,0.03448,1,"{2811}"
769,1,409,0,0.4943667,"\int \frac{\sqrt{a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(3/2),x]","\frac{2 (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)}","\frac{2 (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)}",1,"(-2*(a - b)*Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f) + (2*(a - b)*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f)","A",3,3,29,0.1034,1,"{2795, 2818, 2996}"
770,1,489,0,0.8650096,"\int \frac{\sqrt{a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(5/2),x]","\frac{2 d \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} \left(4 a c d-b \left(3 c^2+d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^2}+\frac{2 (a-b) \sqrt{a+b} (3 c+d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}","\frac{2 d \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} \left(4 a c d-b \left(3 c^2+d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^2}+\frac{2 (a-b) \sqrt{a+b} (3 c+d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}",1,"(2*d*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a - b)*Sqrt[a + b]*(4*a*c*d - b*(3*c^2 + d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f) + (2*(a - b)*Sqrt[a + b]*(3*c + d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f)","A",4,4,29,0.1379,1,"{2796, 2998, 2818, 2996}"
771,1,1080,0,5.2380849,"\int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{4 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}{24 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(\left(15 c^3+284 d^2 c\right) b^3+a d \left(337 c^2+156 d^2\right) b^2+57 a^2 c d^2 b-9 a^3 d^3\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{192 b^2 d (b c-a d) f}-\frac{\sqrt{c+d} \left(\left(5 c^4-120 d^2 c^2-48 d^4\right) b^4-60 a c d \left(c^2+4 d^2\right) b^3-6 a^2 d^2 \left(15 c^2+4 d^2\right) b^2+20 a^3 c d^3 b-3 a^4 d^4\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^3 \sqrt{a+b} d^2 f}-\frac{\left(\left(59 c^2+36 d^2\right) b^2+54 a c d b-9 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{96 b f}+\frac{(a+b)^{3/2} \left(\left(15 c^3+118 d c^2+284 d^2 c+72 d^3\right) b^3+3 a d \left(73 c^2+36 d c+28 d^2\right) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^3 d \sqrt{c+d} f}-\frac{\left(\left(15 c^3+284 d^2 c\right) b^3+a d \left(337 c^2+156 d^2\right) b^2+57 a^2 c d^2 b-9 a^3 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}","-\frac{d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{4 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}{24 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(\left(15 c^3+284 d^2 c\right) b^3+a d \left(337 c^2+156 d^2\right) b^2+57 a^2 c d^2 b-9 a^3 d^3\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{192 b^2 d (b c-a d) f}-\frac{\sqrt{c+d} \left(\left(5 c^4-120 d^2 c^2-48 d^4\right) b^4-60 a c d \left(c^2+4 d^2\right) b^3-6 a^2 d^2 \left(15 c^2+4 d^2\right) b^2+20 a^3 c d^3 b-3 a^4 d^4\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^3 \sqrt{a+b} d^2 f}-\frac{\left(\left(59 c^2+36 d^2\right) b^2+54 a c d b-9 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{96 b f}+\frac{(a+b)^{3/2} \left(\left(15 c^3+118 d c^2+284 d^2 c+72 d^3\right) b^3+3 a d \left(73 c^2+36 d c+28 d^2\right) b^2-3 a^2 d^2 (17 c+6 d) b+9 a^3 d^3\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^3 d \sqrt{c+d} f}-\frac{\left(\left(15 c^3+284 d^2 c\right) b^3+a d \left(337 c^2+156 d^2\right) b^2+57 a^2 c d^2 b-9 a^3 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(192*b^2*d*(b*c - a*d)*f) - (Sqrt[c + d]*(20*a^3*b*c*d^3 - 3*a^4*d^4 - 60*a*b^3*c*d*(c^2 + 4*d^2) - 6*a^2*b^2*d^2*(15*c^2 + 4*d^2) + b^4*(5*c^4 - 120*c^2*d^2 - 48*d^4))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(64*b^3*Sqrt[a + b]*d^2*f) - ((57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(192*b*d*f*Sqrt[a + b*Sin[e + f*x]]) - ((54*a*b*c*d - 9*a^2*d^2 + b^2*(59*c^2 + 36*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(96*b*f) - (d*(17*b*c - 3*a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])/(24*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]])/(4*b*f) + ((a + b)^(3/2)*(9*a^3*d^3 - 3*a^2*b*d^2*(17*c + 6*d) + 3*a*b^2*d*(73*c^2 + 36*c*d + 28*d^2) + b^3*(15*c^3 + 118*c^2*d + 284*c*d^2 + 72*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(192*b^3*d*Sqrt[c + d]*f)","A",9,8,29,0.2759,1,"{2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996}"
772,1,870,0,3.3589532,"\int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{\left(-\left(3 c^2+14 d c+16 d^2\right) b^2-6 a d (4 c+d) b+3 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{3/2}}{24 b^2 d \sqrt{c+d} f}+\frac{(c-d) \sqrt{c+d} \left(\left(3 c^2+16 d^2\right) b^2+38 a c d b+3 a^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) \sqrt{a+b}}{24 b d (b c-a d) f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{\left(\left(3 c^2+16 d^2\right) b^2+38 a c d b+3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{c+d} (b c+a d) \left(-\left(c^2-12 d^2\right) b^2+10 a c d b-a^2 d^2\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^2 d^2 f \sqrt{a+b}}","-\frac{\left(-\left(3 c^2+14 d c+16 d^2\right) b^2-6 a d (4 c+d) b+3 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{3/2}}{24 b^2 d \sqrt{c+d} f}+\frac{(c-d) \sqrt{c+d} \left(\left(3 c^2+16 d^2\right) b^2+38 a c d b+3 a^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) \sqrt{a+b}}{24 b d (b c-a d) f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{\left(\left(3 c^2+16 d^2\right) b^2+38 a c d b+3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{c+d} (b c+a d) \left(-\left(c^2-12 d^2\right) b^2+10 a c d b-a^2 d^2\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^2 d^2 f \sqrt{a+b}}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(38*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 16*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(24*b*d*(b*c - a*d)*f) + (Sqrt[c + d]*(b*c + a*d)*(10*a*b*c*d - a^2*d^2 - b^2*(c^2 - 12*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(8*b^2*Sqrt[a + b]*d^2*f) - ((38*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*d*f*Sqrt[a + b*Sin[e + f*x]]) - ((3*b*c + 7*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*f) - ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*(4*c + d) - b^2*(3*c^2 + 14*c*d + 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(24*b^2*d*Sqrt[c + d]*f) - (b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*f)","A",8,8,29,0.2759,1,"{2821, 3049, 3061, 3053, 2811, 2998, 2818, 2996}"
773,1,740,0,2.2943913,"\int (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{c+d} \left(3 a^2 d^2+6 a b c d+b^2 \left(-\left(c^2-4 d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b d^2 f \sqrt{a+b}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 f}-\frac{b (5 a d+b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} (3 a d+b (c+2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 b d f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} (5 a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 d f (b c-a d)}","\frac{\sqrt{c+d} \left(3 a^2 d^2+6 a b c d+b^2 \left(-\left(c^2-4 d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b d^2 f \sqrt{a+b}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 f}-\frac{b (5 a d+b c) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} (3 a d+b (c+2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 b d f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} (5 a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 d f (b c-a d)}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c + 5*a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*d*(b*c - a*d)*f) + (Sqrt[c + d]*(6*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 - 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b*Sqrt[a + b]*d^2*f) - (b*(b*c + 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d*f*Sqrt[a + b*Sin[e + f*x]]) - (b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*f) + ((a + b)^(3/2)*(3*a*d + b*(c + 2*d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*b*d*Sqrt[c + d]*f)","A",7,7,29,0.2414,1,"{2821, 3061, 3053, 2811, 2998, 2818, 2996}"
774,1,644,0,1.5379228,"\int \frac{(a+b \sin (e+f x))^{3/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^(3/2)/Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{a+b} (b (c-d)-2 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f \sqrt{c+d}}-\frac{\sqrt{a+b} (b c-3 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{b (a-b) \sqrt{a+b} \sqrt{c+d} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d f (b c-a d)}","\frac{\sqrt{a+b} (b (c-d)-2 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f \sqrt{c+d}}-\frac{\sqrt{a+b} (b c-3 a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f \sqrt{c+d}}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f \sqrt{c+d \sin (e+f x)}}-\frac{b (a-b) \sqrt{a+b} \sqrt{c+d} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d f (b c-a d)}",1,"-((b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[c + d*Sin[e + f*x]])) - ((a - b)*b*Sqrt[a + b]*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d*(b*c - a*d)*f) + (Sqrt[a + b]*(b*(c - d) - 2*a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f) - (Sqrt[a + b]*(b*c - 3*a*d)*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f)","A",6,6,29,0.2069,1,"{2821, 3053, 2811, 2998, 2818, 2996}"
775,1,600,0,0.9338198,"\int \frac{(a+b \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{2 \sqrt{a+b} (a d+b (c-2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f (c-d) \sqrt{c+d}}+\frac{2 b \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f \sqrt{c+d}}+\frac{2 (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d f (c-d) \sqrt{c+d}}","-\frac{2 \sqrt{a+b} (a d+b (c-2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f (c-d) \sqrt{c+d}}+\frac{2 b \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f \sqrt{c+d}}+\frac{2 (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d f (c-d) \sqrt{c+d}}",1,"(2*(a - b)*Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*d*Sqrt[c + d]*f) - (2*Sqrt[a + b]*(b*(c - 2*d) + a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*d^2*Sqrt[c + d]*f) + (2*b*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f)","A",5,5,29,0.1724,1,"{2798, 2811, 2998, 2818, 2996}"
776,1,497,0,0.9537662,"\int \frac{(a+b \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 (b c-a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} (a (3 c+d)-b (c+3 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}-\frac{8 (a-b) \sqrt{a+b} (a c-b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}","-\frac{2 (b c-a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} (a (3 c+d)-b (c+3 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}-\frac{8 (a-b) \sqrt{a+b} (a c-b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)}",1,"(-2*(b*c - a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (8*(a - b)*Sqrt[a + b]*(a*c - b*d)*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f) + (2*(a - b)*Sqrt[a + b]*(a*(3*c + d) - b*(c + 3*d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f)","A",4,4,29,0.1379,1,"{2799, 2998, 2818, 2996}"
777,1,1295,0,7.8442826,"\int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2),x]","-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{3 b (b c-7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac{\left(-\left(15 c^2-64 d^2\right) b^2+110 a c d b+93 a^2 d^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac{(a+b)^{3/2} \left(-\left(45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right) b^4+2 a d \left(165 c^3+917 d c^2+2392 d^2 c+516 d^3\right) b^3+30 a^2 d^2 \left(64 c^2+23 d c+22 d^2\right) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt{c+d} f}-\frac{\left(-\left(45 c^3-516 c d^2\right) b^3+a d \left(345 c^2+772 d^2\right) b^2+917 a^2 c d^2 b+15 a^3 d^3\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{960 b d f}-\frac{\left(-\left(45 c^4-1692 d^2 c^2-1024 d^4\right) b^4+8 a d \left(45 c^3+791 d^2 c\right) b^3+2 a^2 d^2 \left(1877 c^2+846 d^2\right) b^2+360 a^3 c d^3 b-45 a^4 d^4\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{1920 b d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(45 c^4-1692 d^2 c^2-1024 d^4\right) b^4+8 a d \left(45 c^3+791 d^2 c\right) b^3+2 a^2 d^2 \left(1877 c^2+846 d^2\right) b^2+360 a^3 c d^3 b-45 a^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac{\sqrt{c+d} (b c+a d) \left(-\left(3 c^4+40 d^2 c^2+240 d^4\right) b^4+28 a c d \left(c^2-20 d^2\right) b^3-2 a^2 d^2 \left(89 c^2+20 d^2\right) b^2+28 a^3 c d^3 b-3 a^4 d^4\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt{a+b} d^3 f}","-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{3 b (b c-7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac{\left(-\left(15 c^2-64 d^2\right) b^2+110 a c d b+93 a^2 d^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac{(a+b)^{3/2} \left(-\left(45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right) b^4+2 a d \left(165 c^3+917 d c^2+2392 d^2 c+516 d^3\right) b^3+30 a^2 d^2 \left(64 c^2+23 d c+22 d^2\right) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt{c+d} f}-\frac{\left(-\left(45 c^3-516 c d^2\right) b^3+a d \left(345 c^2+772 d^2\right) b^2+917 a^2 c d^2 b+15 a^3 d^3\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{960 b d f}-\frac{\left(-\left(45 c^4-1692 d^2 c^2-1024 d^4\right) b^4+8 a d \left(45 c^3+791 d^2 c\right) b^3+2 a^2 d^2 \left(1877 c^2+846 d^2\right) b^2+360 a^3 c d^3 b-45 a^4 d^4\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{1920 b d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(45 c^4-1692 d^2 c^2-1024 d^4\right) b^4+8 a d \left(45 c^3+791 d^2 c\right) b^3+2 a^2 d^2 \left(1877 c^2+846 d^2\right) b^2+360 a^3 c d^3 b-45 a^4 d^4\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac{\sqrt{c+d} (b c+a d) \left(-\left(3 c^4+40 d^2 c^2+240 d^4\right) b^4+28 a c d \left(c^2-20 d^2\right) b^3-2 a^2 d^2 \left(89 c^2+20 d^2\right) b^2+28 a^3 c d^3 b-3 a^4 d^4\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt{a+b} d^3 f}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(360*a^3*b*c*d^3 - 45*a^4*d^4 + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) + 8*a*b^3*d*(45*c^3 + 791*c*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(1920*b^2*d^2*(b*c - a*d)*f) - (Sqrt[c + d]*(b*c + a*d)*(28*a^3*b*c*d^3 - 3*a^4*d^4 + 28*a*b^3*c*d*(c^2 - 20*d^2) - 2*a^2*b^2*d^2*(89*c^2 + 20*d^2) - b^4*(3*c^4 + 40*c^2*d^2 + 240*d^4))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(128*b^3*Sqrt[a + b]*d^3*f) - ((360*a^3*b*c*d^3 - 45*a^4*d^4 + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) + 8*a*b^3*d*(45*c^3 + 791*c*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(1920*b*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - ((917*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(345*c^2 + 772*d^2) - b^3*(45*c^3 - 516*c*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(960*b*d*f) + ((a + b)^(3/2)*(45*a^4*d^4 - 30*a^3*b*d^3*(11*c + 3*d) + 30*a^2*b^2*d^2*(64*c^2 + 23*c*d + 22*d^2) + 2*a*b^3*d*(165*c^3 + 917*c^2*d + 2392*c*d^2 + 516*d^3) - b^4*(45*c^4 - 30*c^3*d - 1692*c^2*d^2 - 1544*c*d^3 - 1024*d^4))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(1920*b^3*d^2*Sqrt[c + d]*f) - ((110*a*b*c*d + 93*a^2*d^2 - b^2*(15*c^2 - 64*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(240*d*f) + (3*b*(b*c - 7*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))/(40*d*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2))/(5*d*f)","A",10,8,29,0.2759,1,"{2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996}"
778,1,1071,0,4.9950578,"\int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2),x]","-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{(a+b)^{3/2} \left(\left(9 c^3-6 d c^2-156 d^2 c-72 d^3\right) b^3-a d \left(51 c^2+172 d c+212 d^2\right) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^2 d^2 \sqrt{c+d} f}-\frac{\left(-9 \left(c^2-4 d^2\right) b^2+54 a c d b+59 a^2 d^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}-\frac{\left(-\left(9 c^3-156 c d^2\right) b^3+a d \left(57 c^2+284 d^2\right) b^2+337 a^2 c d^2 b+15 a^3 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(9 c^3-156 c d^2\right) b^3+a d \left(57 c^2+284 d^2\right) b^2+337 a^2 c d^2 b+15 a^3 d^3\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{192 b d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left(3 \left(c^2+4 d^2\right)^2 b^4-20 a c d \left(c^2-12 d^2\right) b^3+30 a^2 d^2 \left(3 c^2+4 d^2\right) b^2+60 a^3 c d^3 b-5 a^4 d^4\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^2 \sqrt{a+b} d^3 f}","-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{(a+b)^{3/2} \left(\left(9 c^3-6 d c^2-156 d^2 c-72 d^3\right) b^3-a d \left(51 c^2+172 d c+212 d^2\right) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{192 b^2 d^2 \sqrt{c+d} f}-\frac{\left(-9 \left(c^2-4 d^2\right) b^2+54 a c d b+59 a^2 d^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}-\frac{\left(-\left(9 c^3-156 c d^2\right) b^3+a d \left(57 c^2+284 d^2\right) b^2+337 a^2 c d^2 b+15 a^3 d^3\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(9 c^3-156 c d^2\right) b^3+a d \left(57 c^2+284 d^2\right) b^2+337 a^2 c d^2 b+15 a^3 d^3\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{192 b d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left(3 \left(c^2+4 d^2\right)^2 b^4-20 a c d \left(c^2-12 d^2\right) b^3+30 a^2 d^2 \left(3 c^2+4 d^2\right) b^2+60 a^3 c d^3 b-5 a^4 d^4\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{64 b^2 \sqrt{a+b} d^3 f}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(192*b*d^2*(b*c - a*d)*f) + (Sqrt[c + d]*(60*a^3*b*c*d^3 - 5*a^4*d^4 - 20*a*b^3*c*d*(c^2 - 12*d^2) + 3*b^4*(c^2 + 4*d^2)^2 + 30*a^2*b^2*d^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(64*b^2*Sqrt[a + b]*d^3*f) - ((337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(192*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - ((54*a*b*c*d + 59*a^2*d^2 - 9*b^2*(c^2 - 4*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(96*d*f) - ((a + b)^(3/2)*(15*a^3*d^3 - 15*a^2*b*d^2*(11*c + 2*d) - a*b^2*d*(51*c^2 + 172*c*d + 212*d^2) + b^3*(9*c^3 - 6*c^2*d - 156*c*d^2 - 72*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(192*b^2*d^2*Sqrt[c + d]*f) + (b*(3*b*c - 17*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(24*d*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))/(4*d*f)","A",9,8,29,0.2759,1,"{2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996}"
779,1,894,0,3.268502,"\int (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]],x]","-\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac{(3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} b}{12 d f}-\frac{\left(-\left(3 c^2-16 d^2\right) b^2+14 a c d b+33 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)} b}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(3 c^2-16 d^2\right) b^2+14 a c d b+33 a^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left(\left(c^3+4 d^2 c\right) b^3-5 a d \left(c^2-4 d^2\right) b^2+15 a^2 c d^2 b+5 a^3 d^3\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 \sqrt{a+b} d^3 f b}+\frac{(a+b)^{3/2} \left(-\left(3 c^2-2 d c-16 d^2\right) b^2+6 a d (2 c+3 d) b+15 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 d^2 \sqrt{c+d} f b}","-\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac{(3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} b}{12 d f}-\frac{\left(-\left(3 c^2-16 d^2\right) b^2+14 a c d b+33 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)} b}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left(-\left(3 c^2-16 d^2\right) b^2+14 a c d b+33 a^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left(\left(c^3+4 d^2 c\right) b^3-5 a d \left(c^2-4 d^2\right) b^2+15 a^2 c d^2 b+5 a^3 d^3\right) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 \sqrt{a+b} d^3 f b}+\frac{(a+b)^{3/2} \left(-\left(3 c^2-2 d c-16 d^2\right) b^2+6 a d (2 c+3 d) b+15 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 d^2 \sqrt{c+d} f b}",1,"(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(24*d^2*(b*c - a*d)*f) + (Sqrt[c + d]*(15*a^2*b*c*d^2 + 5*a^3*d^3 - 5*a*b^2*d*(c^2 - 4*d^2) + b^3*(c^3 + 4*c*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(8*b*Sqrt[a + b]*d^3*f) - (b*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*d^2*f*Sqrt[a + b*Sin[e + f*x]]) + (b*(3*b*c - 13*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*d*f) + ((a + b)^(3/2)*(15*a^2*d^2 + 6*a*b*d*(2*c + 3*d) - b^2*(3*c^2 - 2*c*d - 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(24*b*d^2*Sqrt[c + d]*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*d*f)","A",8,8,29,0.2759,1,"{2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996}"
780,1,745,0,2.1892708,"\int \frac{(a+b \sin (e+f x))^{5/2}}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^(5/2)/Sqrt[c + d*Sin[e + f*x]],x]","-\frac{\sqrt{c+d} \left(-15 a^2 d^2+10 a b c d+b^2 \left(-\left(3 c^2+4 d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 d^3 f \sqrt{a+b}}+\frac{3 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 d f}-\frac{(a+b)^{3/2} (-7 a d+3 b c-2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 d^2 f \sqrt{c+d}}-\frac{3 b \sqrt{a+b} (c-d) \sqrt{c+d} (b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 d^2 f (b c-a d)}","-\frac{\sqrt{c+d} \left(-15 a^2 d^2+10 a b c d+b^2 \left(-\left(3 c^2+4 d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 d^3 f \sqrt{a+b}}+\frac{3 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 d f}-\frac{(a+b)^{3/2} (-7 a d+3 b c-2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 d^2 f \sqrt{c+d}}-\frac{3 b \sqrt{a+b} (c-d) \sqrt{c+d} (b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 d^2 f (b c-a d)}",1,"(-3*b*Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - 3*a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*d^2*(b*c - a*d)*f) - (Sqrt[c + d]*(10*a*b*c*d - 15*a^2*d^2 - b^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*Sqrt[a + b]*d^3*f) + (3*b^2*(b*c - 3*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*d*f) - ((a + b)^(3/2)*(3*b*c - 7*a*d - 2*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*d^2*Sqrt[c + d]*f)","A",7,7,29,0.2414,1,"{2793, 3061, 3053, 2811, 2998, 2818, 2996}"
781,1,780,0,2.468311,"\int \frac{(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2),x]","\frac{b \left(-2 a^2 d^2+4 a b c d+b^2 \left(-\left(3 c^2-d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d^2 f \left(c^2-d^2\right) \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b} \left(-2 a^2 d^2+4 a b c d+b^2 \left(-\left(3 c^2-d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d^2 f \sqrt{c+d} (b c-a d)}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{(a+b)^{3/2} (2 a d-b (3 c+d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f (c+d)^{3/2}}-\frac{b \sqrt{c+d} (3 b c-5 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d^3 f \sqrt{a+b}}","\frac{b \left(-2 a^2 d^2+4 a b c d+b^2 \left(-\left(3 c^2-d^2\right)\right)\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{d^2 f \left(c^2-d^2\right) \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b} \left(-2 a^2 d^2+4 a b c d+b^2 \left(-\left(3 c^2-d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d^2 f \sqrt{c+d} (b c-a d)}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}-\frac{(a+b)^{3/2} (2 a d-b (3 c+d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^2 f (c+d)^{3/2}}-\frac{b \sqrt{c+d} (3 b c-5 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{d^3 f \sqrt{a+b}}",1,"-((Sqrt[a + b]*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(d^2*Sqrt[c + d]*(b*c - a*d)*f)) - (b*Sqrt[c + d]*(3*b*c - 5*a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*d^3*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c^2 - d^2)*f*Sqrt[a + b*Sin[e + f*x]]) - ((a + b)^(3/2)*(2*a*d - b*(3*c + d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*(c + d)^(3/2)*f)","A",7,7,29,0.2414,1,"{2792, 3061, 3053, 2811, 2998, 2818, 2996}"
782,1,737,0,1.7807497,"\int \frac{(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2),x]","-\frac{2 \sqrt{a+b} \left(a^2 d^2 (3 c+d)+a b d \left(3 c^2-4 c d-7 d^2\right)+b^2 \left(-6 c^2 d+3 c^3-2 c d^2+9 d^3\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 d^3 f (c-d)^2 (c+d)^{3/2}}+\frac{2 b^2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^3 f \sqrt{c+d}}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} \left(4 a c d+3 b c^2-7 b d^2\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 d^2 f (c-d)^2 (c+d)^{3/2}}","-\frac{2 \sqrt{a+b} \left(a^2 d^2 (3 c+d)+a b d \left(3 c^2-4 c d-7 d^2\right)+b^2 \left(-6 c^2 d+3 c^3-2 c d^2+9 d^3\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 d^3 f (c-d)^2 (c+d)^{3/2}}+\frac{2 b^2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{d^3 f \sqrt{c+d}}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}+\frac{2 (a-b) \sqrt{a+b} \left(4 a c d+3 b c^2-7 b d^2\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 d^2 f (c-d)^2 (c+d)^{3/2}}",1,"(2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a - b)*Sqrt[a + b]*(3*b*c^2 + 4*a*c*d - 7*b*d^2)*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*d^2*(c + d)^(3/2)*f) - (2*Sqrt[a + b]*(a^2*d^2*(3*c + d) + a*b*d*(3*c^2 - 4*c*d - 7*d^2) + b^2*(3*c^3 - 6*c^2*d - 2*c*d^2 + 9*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*d^3*(c + d)^(3/2)*f) + (2*b^2*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^3*Sqrt[c + d]*f)","A",6,6,29,0.2069,1,"{2792, 3053, 2811, 2998, 2818, 2996}"
783,1,772,0,2.3029413,"\int \frac{(c+d \sin (e+f x))^{5/2}}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/Sqrt[a + b*Sin[e + f*x]],x]","\frac{\sqrt{a+b} \left(3 a^2 d^2-a b d (7 c+3 d)+b^2 \left(8 c^2+9 c d+2 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 b^3 f \sqrt{c+d}}-\frac{\sqrt{c+d} \left(-3 a^2 d^2+10 a b c d+b^2 \left(-\left(15 c^2+4 d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b^3 f \sqrt{a+b}}+\frac{3 d \sqrt{a+b} (c-d) \sqrt{c+d} (3 b c-a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b^2 f (b c-a d)}-\frac{d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 b f}-\frac{3 d (3 b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b f \sqrt{a+b \sin (e+f x)}}","\frac{\sqrt{a+b} \left(3 a^2 d^2-a b d (7 c+3 d)+b^2 \left(8 c^2+9 c d+2 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{4 b^3 f \sqrt{c+d}}-\frac{\sqrt{c+d} \left(-3 a^2 d^2+10 a b c d+b^2 \left(-\left(15 c^2+4 d^2\right)\right)\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b^3 f \sqrt{a+b}}+\frac{3 d \sqrt{a+b} (c-d) \sqrt{c+d} (3 b c-a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{4 b^2 f (b c-a d)}-\frac{d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{2 b f}-\frac{3 d (3 b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 b f \sqrt{a+b \sin (e+f x)}}",1,"(3*Sqrt[a + b]*(c - d)*d*Sqrt[c + d]*(3*b*c - a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b^2*(b*c - a*d)*f) - (Sqrt[c + d]*(10*a*b*c*d - 3*a^2*d^2 - b^2*(15*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b^3*Sqrt[a + b]*f) - (3*d*(3*b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b*f*Sqrt[a + b*Sin[e + f*x]]) - (d^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*b*f) + (Sqrt[a + b]*(3*a^2*d^2 - a*b*d*(7*c + 3*d) + b^2*(8*c^2 + 9*c*d + 2*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*b^3*Sqrt[c + d]*f)","A",7,7,29,0.2414,1,"{2793, 3061, 3053, 2811, 2998, 2818, 2996}"
784,1,644,0,1.5796518,"\int \frac{(c+d \sin (e+f x))^{3/2}}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/Sqrt[a + b*Sin[e + f*x]],x]","-\frac{\sqrt{a+b} (a d-b (2 c+d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b^2 f \sqrt{c+d}}+\frac{\sqrt{c+d} (3 b c-a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b^2 f \sqrt{a+b}}-\frac{d \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{d \sqrt{a+b} (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (b c-a d)}","-\frac{\sqrt{a+b} (a d-b (2 c+d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b^2 f \sqrt{c+d}}+\frac{\sqrt{c+d} (3 b c-a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b^2 f \sqrt{a+b}}-\frac{d \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{d \sqrt{a+b} (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (b c-a d)}",1,"(Sqrt[a + b]*(c - d)*d*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*(b*c - a*d)*f) + (Sqrt[c + d]*(3*b*c - a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^2*Sqrt[a + b]*f) - (d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[a + b]*(a*d - b*(2*c + d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b^2*Sqrt[c + d]*f)","A",6,6,29,0.2069,1,"{2821, 3053, 2811, 2998, 2818, 2996}"
785,1,198,0,0.1095094,"\int \frac{\sqrt{c+d \sin (e+f x)}}{\sqrt{a+b \sin (e+f x)}} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]],x]","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b f \sqrt{c+d}}","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left(\frac{(a+b) d}{b (c+d)};\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b f \sqrt{c+d}}",1,"(2*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)","A",1,1,29,0.03448,1,"{2811}"
786,1,192,0,0.120762,"\int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{c+d} (b c-a d)}","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{c+d} (b c-a d)}",1,"(2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[c + d]*(b*c - a*d)*f)","A",1,1,29,0.03448,1,"{2818}"
787,1,405,0,0.4592126,"\int \frac{1}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)),x]","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)}+\frac{2 d (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)^2}","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)}+\frac{2 d (a-b) \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (c-d) \sqrt{c+d} (b c-a d)^2}",1,"(2*(a - b)*Sqrt[a + b]*d*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f)","A",3,3,29,0.1034,1,"{2801, 2818, 2996}"
788,1,521,0,1.0186042,"\int \frac{1}{\sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)),x]","-\frac{2 d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac{2 \sqrt{a+b} \left(a d (3 c+d)-b \left(3 c^2+3 c d-2 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^2}-\frac{4 d (a-b) \sqrt{a+b} \left(2 a c d-b \left(3 c^2-d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^3}","-\frac{2 d^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(c^2-d^2\right) (b c-a d) (c+d \sin (e+f x))^{3/2}}-\frac{2 \sqrt{a+b} \left(a d (3 c+d)-b \left(3 c^2+3 c d-2 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^2}-\frac{4 d (a-b) \sqrt{a+b} \left(2 a c d-b \left(3 c^2-d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (c-d)^2 (c+d)^{3/2} (b c-a d)^3}",1,"(-2*d^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (4*(a - b)*Sqrt[a + b]*d*(2*a*c*d - b*(3*c^2 - d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f) - (2*Sqrt[a + b]*(a*d*(3*c + d) - b*(3*c^2 + 3*c*d - 2*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f)","A",4,4,29,0.1379,1,"{2802, 2998, 2818, 2996}"
789,1,822,0,2.6517085,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^(3/2),x]","\frac{2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (b c-a d)^2}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}+\frac{d \sqrt{c+d} (5 b c-3 a d) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt{a+b} f}-\frac{\sqrt{a+b} \left(-\left(2 c^2-6 d c-d^2\right) b^2-2 a d (c+3 d) b+3 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt{c+d} f}+\frac{\left(-\left(2 c^2-d^2\right) b^2+4 a c d b-3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}+\frac{(c-d) \sqrt{c+d} \left(2 b^2 c^2-4 a b d c+3 a^2 d^2-b^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt{a+b} f (b c-a d)}","\frac{2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (b c-a d)^2}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}+\frac{d \sqrt{c+d} (5 b c-3 a d) \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^3 \sqrt{a+b} f}-\frac{\sqrt{a+b} \left(-\left(2 c^2-6 d c-d^2\right) b^2-2 a d (c+3 d) b+3 a^2 d^2\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b^3 \sqrt{c+d} f}+\frac{\left(-\left(2 c^2-d^2\right) b^2+4 a c d b-3 a^2 d^2\right) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left(a^2-b^2\right) f \sqrt{a+b \sin (e+f x)}}+\frac{(c-d) \sqrt{c+d} \left(2 b^2 c^2-4 a b d c+3 a^2 d^2-b^2 d^2\right) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b^2 \sqrt{a+b} f (b c-a d)}",1,"((c - d)*Sqrt[c + d]*(2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2 - b^2*d^2)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*b^2*Sqrt[a + b]*(b*c - a*d)*f) + (d*Sqrt[c + d]*(5*b*c - 3*a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^3*Sqrt[a + b]*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[a + b]*(3*a^2*d^2 - 2*a*b*d*(c + 3*d) - b^2*(2*c^2 - 6*c*d - d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*b^3*Sqrt[c + d]*f)","A",7,7,29,0.2414,1,"{2792, 3061, 3053, 2811, 2998, 2818, 2996}"
790,1,600,0,0.9282651,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{3/2}} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^(3/2),x]","\frac{2 \sqrt{a+b} (a d+b (c-2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b^2 f (a-b) \sqrt{c+d}}+\frac{2 d \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b^2 f \sqrt{a+b}}+\frac{2 (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (a-b) \sqrt{a+b}}","\frac{2 \sqrt{a+b} (a d+b (c-2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{b^2 f (a-b) \sqrt{c+d}}+\frac{2 d \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b^2 f \sqrt{a+b}}+\frac{2 (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b f (a-b) \sqrt{a+b}}",1,"(2*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*b*Sqrt[a + b]*f) + (2*d*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^2*Sqrt[a + b]*f) + (2*Sqrt[a + b]*(b*(c - 2*d) + a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*b^2*Sqrt[c + d]*f)","A",5,5,29,0.1724,1,"{2798, 2811, 2998, 2818, 2996}"
791,1,409,0,0.455118,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2),x]","\frac{2 \sqrt{a+b} (c-d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (a-b) \sqrt{c+d} (b c-a d)}+\frac{2 (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (a-b) \sqrt{a+b} (b c-a d)}","\frac{2 \sqrt{a+b} (c-d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (a-b) \sqrt{c+d} (b c-a d)}+\frac{2 (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (a-b) \sqrt{a+b} (b c-a d)}",1,"(2*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)*f) + (2*Sqrt[a + b]*(c - d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d)*f)","A",3,3,29,0.1034,1,"{2795, 2818, 2996}"
792,1,405,0,0.4507656,"\int \frac{1}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (a-b) \sqrt{c+d} (b c-a d)}+\frac{2 b (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (a-b) \sqrt{a+b} (b c-a d)^2}","\frac{2 \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f (a-b) \sqrt{c+d} (b c-a d)}+\frac{2 b (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{f (a-b) \sqrt{a+b} (b c-a d)^2}",1,"(2*b*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d)*f)","A",3,3,29,0.1034,1,"{2801, 2818, 2996}"
793,1,495,0,0.9080642,"\int \frac{1}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{2 \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{a+b} (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 (b (c-2 d)-a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{a+b} (c-d) \sqrt{c+d} (b c-a d)^2}","-\frac{2 \left(a^2 d^2+b^2 \left(c^2-2 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{a+b} (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 (b (c-2 d)-a d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{f \sqrt{a+b} (c-d) \sqrt{c+d} (b c-a d)^2}",1,"(2*b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (2*(a^2*d^2 + b^2*(c^2 - 2*d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f) + (2*(b*(c - 2*d) - a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f)","A",4,4,29,0.1379,1,"{2802, 2998, 2818, 2996}"
794,1,681,0,2.6799424,"\int \frac{1}{(a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{2 d \left(a^2 d^2+b^2 \left(3 c^2-4 d^2\right)\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(a^2 d^2 (3 c+d)-6 a b d \left(c^2-d^2\right)+b^2 \left(-9 c^2 d+3 c^3-6 c d^2+8 d^3\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^3}+\frac{2 \left(-a^2 b d^2 \left(9 c^2-5 d^2\right)+4 a^3 c d^3-4 a b^2 c d^3+b^3 \left(-\left(-15 c^2 d^2+3 c^4+8 d^4\right)\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^4}+\frac{2 b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}","\frac{2 d \left(a^2 d^2+b^2 \left(3 c^2-4 d^2\right)\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 f \left(a^2-b^2\right) \left(c^2-d^2\right) (b c-a d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left(a^2 d^2 (3 c+d)-6 a b d \left(c^2-d^2\right)+b^2 \left(-9 c^2 d+3 c^3-6 c d^2+8 d^3\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^3}+\frac{2 \left(-a^2 b d^2 \left(9 c^2-5 d^2\right)+4 a^3 c d^3-4 a b^2 c d^3+b^3 \left(-\left(-15 c^2 d^2+3 c^4+8 d^4\right)\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} (c-d)^2 (c+d)^{3/2} (b c-a d)^4}+\frac{2 b^2 \cos (e+f x)}{f \left(a^2-b^2\right) (b c-a d) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}",1,"(2*b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) + (2*d*(a^2*d^2 + b^2*(3*c^2 - 4*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(4*a^3*c*d^3 - 4*a*b^2*c*d^3 - a^2*b*d^2*(9*c^2 - 5*d^2) - b^3*(3*c^4 - 15*c^2*d^2 + 8*d^4))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f) + (2*(a^2*d^2*(3*c + d) - 6*a*b*d*(c^2 - d^2) + b^2*(3*c^3 - 9*c^2*d - 6*c*d^2 + 8*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f)","A",5,5,29,0.1724,1,"{2802, 3055, 2998, 2818, 2996}"
795,1,736,0,1.8429518,"\int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^(5/2),x]","\frac{2 \left(3 a^2 b d (c-2 d)+3 a^3 d^2+a b^2 \left(3 c^2-4 c d-2 d^2\right)+b^3 \left(c^2-7 c d+9 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 b^3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d}}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) \sqrt{c+d} \left(3 a^2 d+4 a b c-7 b^2 d\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 b^2 f (a-b)^2 (a+b)^{3/2}}+\frac{2 d^2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b^3 f \sqrt{a+b}}","\frac{2 \left(3 a^2 b d (c-2 d)+3 a^3 d^2+a b^2 \left(3 c^2-4 c d-2 d^2\right)+b^3 \left(c^2-7 c d+9 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 b^3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d}}+\frac{2 (b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 b f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) \sqrt{c+d} \left(3 a^2 d+4 a b c-7 b^2 d\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 b^2 f (a-b)^2 (a+b)^{3/2}}+\frac{2 d^2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left(\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{b^3 f \sqrt{a+b}}",1,"(2*(c - d)*Sqrt[c + d]*(4*a*b*c + 3*a^2*d - 7*b^2*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*b^2*(a + b)^(3/2)*f) + (2*d^2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^3*Sqrt[a + b]*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(3*a^2*b*(c - 2*d)*d + 3*a^3*d^2 + a*b^2*(3*c^2 - 4*c*d - 2*d^2) + b^3*(c^2 - 7*c*d + 9*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*b^3*Sqrt[a + b]*Sqrt[c + d]*f)","A",6,6,29,0.2069,1,"{2792, 3053, 2811, 2998, 2818, 2996}"
796,1,497,0,1.0007118,"\int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^{5/2}} \, dx","Int[(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^(5/2),x]","\frac{2 (b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) (3 a c-a d+b c-3 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)}+\frac{8 (c-d) \sqrt{c+d} (a c-b d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)}","\frac{2 (b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) (3 a c-a d+b c-3 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)}+\frac{8 (c-d) \sqrt{c+d} (a c-b d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)}",1,"(8*(c - d)*Sqrt[c + d]*(a*c - b*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)*f) + (2*(b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(c - d)*(3*a*c + b*c - a*d - 3*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*(b*c - a*d)*f)","A",4,4,29,0.1379,1,"{2799, 2998, 2818, 2996}"
797,1,489,0,0.8598386,"\int \frac{\sqrt{c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{5/2}} \, dx","Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(5/2),x]","\frac{2 b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) \sqrt{c+d} \left(-3 a^2 d+4 a b c-b^2 d\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)^2}+\frac{2 (3 a+b) (c-d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)}","\frac{2 b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}+\frac{2 (c-d) \sqrt{c+d} \left(-3 a^2 d+4 a b c-b^2 d\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)^2}+\frac{2 (3 a+b) (c-d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)}",1,"(2*(c - d)*Sqrt[c + d]*(4*a*b*c - 3*a^2*d - b^2*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)^2*f) + (2*b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(3*a + b)*(c - d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*(b*c - a*d)*f)","A",4,4,29,0.1379,1,"{2796, 2998, 2818, 2996}"
798,1,516,0,0.9935768,"\int \frac{1}{(a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}} \, dx","Int[1/((a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^{3/2}}+\frac{2 \left(-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)^2}+\frac{4 b (c-d) \sqrt{c+d} \left(-3 a^2 d+2 a b c+b^2 d\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)^3}","\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^{3/2}}+\frac{2 \left(-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f (a-b)^2 \sqrt{a+b} \sqrt{c+d} (b c-a d)^2}+\frac{4 b (c-d) \sqrt{c+d} \left(-3 a^2 d+2 a b c+b^2 d\right) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right)|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right)}{3 f (a-b)^2 (a+b)^{3/2} (b c-a d)^3}",1,"(4*b*(c - d)*Sqrt[c + d]*(2*a*b*c - 3*a^2*d + b^2*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)^3*f) + (2*b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(3*a*b*(c - d) - 3*a^2*d + b^2*(c + 2*d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*(b*c - a*d)^2*f)","A",4,4,29,0.1379,1,"{2802, 2998, 2818, 2996}"
799,1,688,0,1.846568,"\int \frac{1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)),x]","-\frac{2 \left(3 a^2 b d (2 c-3 d)-3 a^3 d^2-3 a b^2 \left(c^2-2 d^2\right)+b^3 \left(c^2-6 c d+8 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} \left(a^2-b^2\right) (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 \left(3 a^2 b^2 d \left(3 c^2-5 d^2\right)+3 a^4 d^3-4 a b^3 c \left(c^2-d^2\right)-b^4 d \left(5 c^2-8 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} \left(a^2-b^2\right) (c-d) \sqrt{c+d} (b c-a d)^4}+\frac{8 b^2 \left(-2 a^2 d+a b c+b^2 d\right) \cos (e+f x)}{3 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 b^2 \cos (e+f x)}{3 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}","-\frac{2 \left(3 a^2 b d (2 c-3 d)-3 a^3 d^2-3 a b^2 \left(c^2-2 d^2\right)+b^3 \left(c^2-6 c d+8 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} \left(a^2-b^2\right) (c-d) \sqrt{c+d} (b c-a d)^3}+\frac{2 \left(3 a^2 b^2 d \left(3 c^2-5 d^2\right)+3 a^4 d^3-4 a b^3 c \left(c^2-d^2\right)-b^4 d \left(5 c^2-8 d^2\right)\right) \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right)}{3 f \sqrt{a+b} \left(a^2-b^2\right) (c-d) \sqrt{c+d} (b c-a d)^4}+\frac{8 b^2 \left(-2 a^2 d+a b c+b^2 d\right) \cos (e+f x)}{3 f \left(a^2-b^2\right)^2 (b c-a d)^2 \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 b^2 \cos (e+f x)}{3 f \left(a^2-b^2\right) (b c-a d) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}",1,"(2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) + (8*b^2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f*x])/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (2*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3*a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^4*f) - (2*(3*a^2*b*(2*c - 3*d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2) + b^3*(c^2 - 6*c*d + 8*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f)","A",5,5,29,0.1724,1,"{2802, 3055, 2998, 2818, 2996}"
800,1,941,0,4.4721667,"\int \frac{1}{(a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2}} \, dx","Int[1/((a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2)),x]","\frac{4 \left(-5 d a^2+2 b c a+3 b^2 d\right) \cos (e+f x) b^2}{3 \left(a^2-b^2\right)^2 (b c-a d)^2 f \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{2 \cos (e+f x) b^2}{3 \left(a^2-b^2\right) (b c-a d) f (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac{8 \left(c d^4 a^5-b \left(3 c^2 d^3-2 d^5\right) a^4-2 b^2 c d^4 a^3-b^3 d \left(3 c^4-12 d^2 c^2+7 d^4\right) a^2+b^4 c \left(c^4-2 d^2 c^2+2 d^4\right) a+b^5 d \left(2 c^4-7 d^2 c^2+4 d^4\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt{a+b} \left(a^2-b^2\right) (c-d)^2 (c+d)^{3/2} (b c-a d)^5 f}-\frac{2 \left(d^3 (3 c+d) a^4-9 b d^2 \left(c^2-d^2\right) a^3+b^2 d \left(9 c^3-18 d c^2-15 d^2 c+16 d^3\right) a^2-3 b^3 \left(c^4-5 d^2 c^2+4 d^4\right) a+b^4 \left(c^4-9 d c^3+16 d^2 c^2+12 d^3 c-16 d^4\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt{a+b} \left(a^2-b^2\right) (c-d)^2 (c+d)^{3/2} (b c-a d)^4 f}-\frac{2 d \left(d^3 a^4+b^2 d \left(11 c^2-13 d^2\right) a^2-4 b^3 c \left(c^2-d^2\right) a-b^4 d \left(7 c^2-8 d^2\right)\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 \left(a^2-b^2\right)^2 (b c-a d)^3 \left(c^2-d^2\right) f (c+d \sin (e+f x))^{3/2}}","\frac{4 \left(-5 d a^2+2 b c a+3 b^2 d\right) \cos (e+f x) b^2}{3 \left(a^2-b^2\right)^2 (b c-a d)^2 f \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{2 \cos (e+f x) b^2}{3 \left(a^2-b^2\right) (b c-a d) f (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2}}-\frac{8 \left(c d^4 a^5-b \left(3 c^2 d^3-2 d^5\right) a^4-2 b^2 c d^4 a^3-b^3 d \left(3 c^4-12 d^2 c^2+7 d^4\right) a^2+b^4 c \left(c^4-2 d^2 c^2+2 d^4\right) a+b^5 d \left(2 c^4-7 d^2 c^2+4 d^4\right)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt{a+b} \left(a^2-b^2\right) (c-d)^2 (c+d)^{3/2} (b c-a d)^5 f}-\frac{2 \left(d^3 (3 c+d) a^4-9 b d^2 \left(c^2-d^2\right) a^3+b^2 d \left(9 c^3-18 d c^2-15 d^2 c+16 d^3\right) a^2-3 b^3 \left(c^4-5 d^2 c^2+4 d^4\right) a+b^4 \left(c^4-9 d c^3+16 d^2 c^2+12 d^3 c-16 d^4\right)\right) F\left(\sin ^{-1}\left(\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right)|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 \sqrt{a+b} \left(a^2-b^2\right) (c-d)^2 (c+d)^{3/2} (b c-a d)^4 f}-\frac{2 d \left(d^3 a^4+b^2 d \left(11 c^2-13 d^2\right) a^2-4 b^3 c \left(c^2-d^2\right) a-b^4 d \left(7 c^2-8 d^2\right)\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{3 \left(a^2-b^2\right)^2 (b c-a d)^3 \left(c^2-d^2\right) f (c+d \sin (e+f x))^{3/2}}",1,"(2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) + (4*b^2*(2*a*b*c - 5*a^2*d + 3*b^2*d)*Cos[e + f*x])/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*d*(a^4*d^3 + a^2*b^2*d*(11*c^2 - 13*d^2) - b^4*d*(7*c^2 - 8*d^2) - 4*a*b^3*c*(c^2 - d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (8*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^5*f) - (2*(a^4*d^3*(3*c + d) - 9*a^3*b*d^2*(c^2 - d^2) + a^2*b^2*d*(9*c^3 - 18*c^2*d - 15*c*d^2 + 16*d^3) + b^4*(c^4 - 9*c^3*d + 16*c^2*d^2 + 12*c*d^3 - 16*d^4) - 3*a*b^3*(c^4 - 5*c^2*d^2 + 4*d^4))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f)","A",6,5,29,0.1724,1,"{2802, 3055, 2998, 2818, 2996}"
801,0,0,0,0.0497791,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","\text{Int}\left((a+b \sin (e+f x))^m (c+d \sin (e+f x))^n,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]","A",0,0,0,0,-1,"{}"
802,1,311,0,0.4390002,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]","-\frac{\sqrt{2} \cos (e+f x) \left(a d (a d-2 b c (m+2))+b^2 \left(c^2 (m+2)+d^2 (m+1)\right)\right) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sin (e+f x)+1}}+\frac{\sqrt{2} d (a+b) \cos (e+f x) (a d-2 b c (m+2)) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sin (e+f x)+1}}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{m+1}}{b f (m+2)}","-\frac{\sqrt{2} \cos (e+f x) \left(a d (a d-2 b c (m+2))+b^2 \left(c^2 (m+2)+d^2 (m+1)\right)\right) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sin (e+f x)+1}}+\frac{\sqrt{2} d (a+b) \cos (e+f x) (a d-2 b c (m+2)) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sin (e+f x)+1}}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{m+1}}{b f (m+2)}",1,"-((d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(1 + m))/(b*f*(2 + m))) + (Sqrt[2]*(a + b)*d*(a*d - 2*b*c*(2 + m))*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Sin[e + f*x])/2, (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]]*((a + b*Sin[e + f*x])/(a + b))^m) - (Sqrt[2]*(a*d*(a*d - 2*b*c*(2 + m)) + b^2*(d^2*(1 + m) + c^2*(2 + m)))*AppellF1[1/2, 1/2, -m, 3/2, (1 - Sin[e + f*x])/2, (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]]*((a + b*Sin[e + f*x])/(a + b))^m)","A",8,5,25,0.2000,1,"{2791, 2756, 2665, 139, 138}"
803,1,229,0,0.2034558,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x)) \, dx","Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]","-\frac{\sqrt{2} (b c-a d) \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b f \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} d (a+b) \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b f \sqrt{\sin (e+f x)+1}}","-\frac{\sqrt{2} (b c-a d) \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b f \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} d (a+b) \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{b f \sqrt{\sin (e+f x)+1}}",1,"-((Sqrt[2]*(a + b)*d*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Sin[e + f*x])/2, (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(b*f*Sqrt[1 + Sin[e + f*x]]*((a + b*Sin[e + f*x])/(a + b))^m)) - (Sqrt[2]*(b*c - a*d)*AppellF1[1/2, 1/2, -m, 3/2, (1 - Sin[e + f*x])/2, (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(b*f*Sqrt[1 + Sin[e + f*x]]*((a + b*Sin[e + f*x])/(a + b))^m)","A",7,4,23,0.1739,1,"{2756, 2665, 139, 138}"
804,1,104,0,0.0661564,"\int (a+b \sin (e+f x))^m \, dx","Int[(a + b*Sin[e + f*x])^m,x]","-\frac{\sqrt{2} \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{f \sqrt{\sin (e+f x)+1}}","-\frac{\sqrt{2} \cos (e+f x) (a+b \sin (e+f x))^m \left(\frac{a+b \sin (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{b (1-\sin (e+f x))}{a+b}\right)}{f \sqrt{\sin (e+f x)+1}}",1,"-((Sqrt[2]*AppellF1[1/2, 1/2, -m, 3/2, (1 - Sin[e + f*x])/2, (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*Sqrt[1 + Sin[e + f*x]]*((a + b*Sin[e + f*x])/(a + b))^m))","A",3,3,12,0.2500,1,"{2665, 139, 138}"
805,0,0,0,0.057435,"\int \frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]),x]","\int \frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)},x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x]","A",0,0,0,0,-1,"{}"
806,0,0,0,0.0563576,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2,x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^2},x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x]","A",0,0,0,0,-1,"{}"
807,0,0,0,0.0549524,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3,x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^3},x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x]","A",0,0,0,0,-1,"{}"
808,0,0,0,0.0776108,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx","Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx","\text{Int}\left((c+d \sin (e+f x))^{5/2} (a+b \sin (e+f x))^m,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
809,0,0,0,0.0755849,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx","Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2),x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx","\text{Int}\left((c+d \sin (e+f x))^{3/2} (a+b \sin (e+f x))^m,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
810,0,0,0,0.0690777,"\int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx","Int[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]],x]","\int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx","\text{Int}\left(\sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^m,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]], x]","A",0,0,0,0,-1,"{}"
811,0,0,0,0.0694018,"\int \frac{(a+b \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]],x]","\int \frac{(a+b \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{\sqrt{c+d \sin (e+f x)}},x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]], x]","A",0,0,0,0,-1,"{}"
812,0,0,0,0.0762709,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2),x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}},x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x]","A",0,0,0,0,-1,"{}"
813,0,0,0,0.0777509,"\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2),x]","\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx","\text{Int}\left(\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}},x\right)",0,"Defer[Int][(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2), x]","A",0,0,0,0,-1,"{}"
814,1,272,0,0.4587908,"\int (d \csc (e+f x))^n (a+a \sin (e+f x))^3 \, dx","Int[(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^3,x]","\frac{a^3 d^4 (11-4 n) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (5-4 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) \left(a^3 \csc (e+f x)+a^3\right) (d \csc (e+f x))^{n-3}}{f (1-n)}","\frac{a^3 d^4 (11-4 n) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (5-4 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) \left(a^3 \csc (e+f x)+a^3\right) (d \csc (e+f x))^{n-3}}{f (1-n)}",1,"(a^3*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(a^3 + a^3*Csc[e + f*x]))/(f*(1 - n)) + (a^3*d^3*(5 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2]) + (a^3*d^4*(11 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*(4 - n)*Sqrt[Cos[e + f*x]^2])","A",8,6,23,0.2609,1,"{3238, 3814, 3997, 3787, 3772, 2643}"
815,1,203,0,0.2569287,"\int (d \csc (e+f x))^n (a+a \sin (e+f x))^2 \, dx","Int[(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^2,x]","\frac{a^2 d^3 (3-2 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{2 a^2 d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}","\frac{a^2 d^3 (3-2 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{2 a^2 d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}",1,"(a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a^2*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (a^2*d^3*(3 - 2*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2])","A",7,5,23,0.2174,1,"{3238, 3788, 3772, 2643, 4046}"
816,1,149,0,0.1497254,"\int (d \csc (e+f x))^n (a+a \sin (e+f x)) \, dx","Int[(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x]),x]","\frac{a d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{f (1-n) \sqrt{\cos ^2(e+f x)}}","\frac{a d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{f (1-n) \sqrt{\cos ^2(e+f x)}}",1,"(a*d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (a*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2])","A",6,4,21,0.1905,1,"{3238, 3787, 3772, 2643}"
817,1,171,0,0.2375423,"\int \frac{(d \csc (e+f x))^n}{a+a \sin (e+f x)} \, dx","Int[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x]),x]","\frac{d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{a f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\sin ^2(e+f x)\right)}{a f \sqrt{\cos ^2(e+f x)}}-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)}","\frac{d n \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{a f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{\cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\sin ^2(e+f x)\right)}{a f \sqrt{\cos ^2(e+f x)}}-\frac{\cot (e+f x) (d \csc (e+f x))^n}{f (a \csc (e+f x)+a)}",1,"-((Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(a + a*Csc[e + f*x]))) + (d*n*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(a*f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (Cos[e + f*x]*(d*Csc[e + f*x])^n*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Sin[e + f*x]^2])/(a*f*Sqrt[Cos[e + f*x]^2])","A",7,5,23,0.2174,1,"{3238, 3820, 3787, 3772, 2643}"
818,1,231,0,0.442028,"\int \frac{(d \csc (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Int[(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x])^2,x]","\frac{2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\sin ^2(e+f x)\right)}{3 a^2 d^2 f \sqrt{\cos ^2(e+f x)}}-\frac{2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac{(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\sin ^2(e+f x)\right)}{3 a^2 d f \sqrt{\cos ^2(e+f x)}}+\frac{\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2}","\frac{2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\sin ^2(e+f x)\right)}{3 a^2 d^2 f \sqrt{\cos ^2(e+f x)}}-\frac{2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac{(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\sin ^2(e+f x)\right)}{3 a^2 d f \sqrt{\cos ^2(e+f x)}}+\frac{\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2}",1,"(-2*n*Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(3*a^2*d^2*f*(1 + Csc[e + f*x])) + (Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(3*d^2*f*(a + a*Csc[e + f*x])^2) + (2*n*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Hypergeometric2F1[1/2, (-2 - n)/2, -n/2, Sin[e + f*x]^2])/(3*a^2*d^2*f*Sqrt[Cos[e + f*x]^2]) - ((1 + 2*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Sin[e + f*x]^2])/(3*a^2*d*f*Sqrt[Cos[e + f*x]^2])","A",8,6,23,0.2609,1,"{3238, 3817, 4020, 3787, 3772, 2643}"
819,1,113,0,0.232322,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x))^m \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \sin ^{-n p}(e+f x) F_1\left(\frac{1}{2};-n p,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) \left(c (d \sin (e+f x))^p\right)^n}{f}","-\frac{2^{m+\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \sin ^{-n p}(e+f x) F_1\left(\frac{1}{2};-n p,\frac{1}{2}-m;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right) \left(c (d \sin (e+f x))^p\right)^n}{f}",1,"-((2^(1/2 + m)*AppellF1[1/2, -(n*p), 1/2 - m, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*Sin[e + f*x]^(n*p)))","A",5,5,27,0.1852,1,"{2826, 2787, 2786, 2785, 133}"
820,1,299,0,0.4952591,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x))^3 \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^3,x]","\frac{a^3 (4 n p+11) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 (4 n p+5) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^3 (2 n p+7) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3)}-\frac{\sin (e+f x) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+3)}","\frac{a^3 (4 n p+11) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 (4 n p+5) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^3 (2 n p+7) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3)}-\frac{\sin (e+f x) \cos (e+f x) \left(a^3 \sin (e+f x)+a^3\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+3)}",1,"-((a^3*(7 + 2*n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p))) + (a^3*(5 + 4*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (a^3*(11 + 4*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p)*Sqrt[Cos[e + f*x]^2]) - (Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(a^3 + a^3*Sin[e + f*x]))/(f*(3 + n*p))","A",7,6,27,0.2222,1,"{2826, 2763, 2968, 3023, 2748, 2643}"
821,1,222,0,0.2492063,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x))^2 \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^2,x]","\frac{2 a^2 \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 (2 n p+3) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^2 \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2)}","\frac{2 a^2 \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 (2 n p+3) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a^2 \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2)}",1,"-((a^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p))) + (a^2*(3 + 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*a^2*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])","A",5,4,27,0.1481,1,"{2826, 2763, 2748, 2643}"
822,1,163,0,0.1188716,"\int \left(c (d \sin (e+f x))^p\right)^n (a+a \sin (e+f x)) \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x]),x]","\frac{a \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}","\frac{a \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}",1,"(a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])","A",4,3,25,0.1200,1,"{2826, 2748, 2643}"
823,1,189,0,0.2299345,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{a+a \sin (e+f x)} \, dx","Int[(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x]),x]","\frac{\cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{1}{2} (n p+2);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f \sqrt{\cos ^2(e+f x)}}-\frac{n p \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f (n p+1) \sqrt{\cos ^2(e+f x)}}-\frac{\cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (a \sin (e+f x)+a)}","\frac{\cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n p}{2};\frac{1}{2} (n p+2);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f \sqrt{\cos ^2(e+f x)}}-\frac{n p \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{a f (n p+1) \sqrt{\cos ^2(e+f x)}}-\frac{\cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (a \sin (e+f x)+a)}",1,"(Cos[e + f*x]*Hypergeometric2F1[1/2, (n*p)/2, (2 + n*p)/2, Sin[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*Sqrt[Cos[e + f*x]^2]) - (n*p*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) - (Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(a + a*Sin[e + f*x]))","A",5,4,27,0.1481,1,"{2826, 2769, 2748, 2643}"
824,1,288,0,0.4865282,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{(a+a \sin (e+f x))^2} \, dx","Int[(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x])^2,x]","\frac{2 \left(1-n^2 p^2\right) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{n p (1-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{2 (1-n p) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (\sin (e+f x)+1)}+\frac{\sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{3 f (a \sin (e+f x)+a)^2}","\frac{2 \left(1-n^2 p^2\right) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{n p (1-2 n p) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{2 (1-n p) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{3 a^2 f (\sin (e+f x)+1)}+\frac{\sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{3 f (a \sin (e+f x)+a)^2}",1,"-(n*p*(1 - 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*(1 - n^2*p^2)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*(1 - n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(1 + Sin[e + f*x])) + (Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*f*(a + a*Sin[e + f*x])^2)","A",6,5,27,0.1852,1,"{2826, 2766, 2978, 2748, 2643}"
825,1,298,0,0.5659339,"\int (d \csc (e+f x))^n (a+b \sin (e+f x))^3 \, dx","Int[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^3,x]","\frac{b d^4 \left(3 a^2 (3-n)+b^2 (2-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d^3 \left(a^2 (2-n)+3 b^2 (1-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)}","\frac{b d^4 \left(3 a^2 (3-n)+b^2 (2-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d^3 \left(a^2 (2-n)+3 b^2 (1-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)}",1,"(a^2*b*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(b + a*Csc[e + f*x]))/(f*(1 - n)) + (a*d^3*(3*b^2*(1 - n) + a^2*(2 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2]) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*(4 - n)*Sqrt[Cos[e + f*x]^2])","A",8,6,23,0.2609,1,"{3238, 3842, 4047, 3772, 2643, 4046}"
826,1,213,0,0.2690876,"\int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx","Int[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^2,x]","\frac{d^3 \left(a^2 (2-n)+b^2 (1-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}","\frac{d^3 \left(a^2 (2-n)+b^2 (1-n)\right) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right)}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}",1,"(a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a*b*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (d^3*(b^2*(1 - n) + a^2*(2 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2])","A",7,5,23,0.2174,1,"{3238, 3788, 3772, 2643, 4046}"
827,1,149,0,0.1484975,"\int (d \csc (e+f x))^n (a+b \sin (e+f x)) \, dx","Int[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x]),x]","\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}","\frac{a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right)}{f (1-n) \sqrt{\cos ^2(e+f x)}}+\frac{b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right)}{f (2-n) \sqrt{\cos ^2(e+f x)}}",1,"(a*d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (b*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2])","A",6,4,21,0.1905,1,"{3238, 3787, 3772, 2643}"
828,1,204,0,0.3989807,"\int \frac{(d \csc (e+f x))^n}{a+b \sin (e+f x)} \, dx","Int[(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x]),x]","\frac{b \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+1} F_1\left(\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)}-\frac{a \cos (e+f x) \sin ^2(e+f x)^{\frac{n+1}{2}} (d \csc (e+f x))^{n+1} F_1\left(\frac{1}{2};\frac{n+1}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)}","\frac{b \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+1} F_1\left(\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)}-\frac{a \cos (e+f x) \sin ^2(e+f x)^{\frac{n+1}{2}} (d \csc (e+f x))^{n+1} F_1\left(\frac{1}{2};\frac{n+1}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d f \left(a^2-b^2\right)}",1,"(b*AppellF1[1/2, n/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*Sin[e + f*x]*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)*d*f) - (a*AppellF1[1/2, (1 + n)/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*(Sin[e + f*x]^2)^((1 + n)/2))/((a^2 - b^2)*d*f)","A",7,5,23,0.2174,1,"{3238, 3869, 2823, 3189, 429}"
829,1,321,0,0.5437212,"\int \frac{(d \csc (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx","Int[(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^2,x]","-\frac{b^2 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n-1}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}-\frac{a^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n+1}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n+1}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}+\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{\frac{n+2}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}","-\frac{b^2 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n-1}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}-\frac{a^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n+1}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n+1}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}+\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{\frac{n+2}{2}} (d \csc (e+f x))^{n+2} F_1\left(\frac{1}{2};\frac{n}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^2 f \left(a^2-b^2\right)^2}",1,"-((b^2*AppellF1[1/2, (-1 + n)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^((-1 + n)/2))/((a^2 - b^2)^2*d^2*f)) - (a^2*AppellF1[1/2, (1 + n)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Sin[e + f*x]*(Sin[e + f*x]^2)^((1 + n)/2))/((a^2 - b^2)^2*d^2*f) + (2*a*b*AppellF1[1/2, n/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*(Sin[e + f*x]^2)^((2 + n)/2))/((a^2 - b^2)^2*d^2*f)","A",10,5,23,0.2174,1,"{3238, 3869, 2824, 3189, 429}"
830,1,432,0,0.7026428,"\int \frac{(d \csc (e+f x))^n}{(a+b \sin (e+f x))^3} \, dx","Int[(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^3,x]","-\frac{3 a b^2 \sin ^4(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n-1}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}+\frac{b^3 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n-2}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}+\frac{3 a^2 b \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}-\frac{a^3 \cos (e+f x) \sin ^2(e+f x)^{\frac{n+3}{2}} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n+1}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}","-\frac{3 a b^2 \sin ^4(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n-1}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}+\frac{b^3 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n-2}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}+\frac{3 a^2 b \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}-\frac{a^3 \cos (e+f x) \sin ^2(e+f x)^{\frac{n+3}{2}} (d \csc (e+f x))^{n+3} F_1\left(\frac{1}{2};\frac{n+1}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{d^3 f \left(a^2-b^2\right)^3}",1,"(-3*a*b^2*AppellF1[1/2, (-1 + n)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*Sin[e + f*x]^4*(Sin[e + f*x]^2)^((-1 + n)/2))/((a^2 - b^2)^3*d^3*f) + (b^3*AppellF1[1/2, (-2 + n)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)^3*d^3*f) + (3*a^2*b*AppellF1[1/2, n/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)^3*d^3*f) - (a^3*AppellF1[1/2, (1 + n)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*(Sin[e + f*x]^2)^((3 + n)/2))/((a^2 - b^2)^3*d^3*f)","A",12,5,23,0.2174,1,"{3238, 3869, 2824, 3189, 429}"
831,0,0,0,0.1124316,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^m \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^m,x]","\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^m \, dx","(d \sin (e+f x))^{-n p} \left(c (d \sin (e+f x))^p\right)^n \text{Int}\left((a+b \sin (e+f x))^m (d \sin (e+f x))^{n p},x\right)",0,"((c*(d*Sin[e + f*x])^p)^n*Defer[Int][(d*Sin[e + f*x])^(n*p)*(a + b*Sin[e + f*x])^m, x])/(d*Sin[e + f*x])^(n*p)","A",0,0,0,0,-1,"{}"
832,1,303,0,0.5482908,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^3 \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^3,x]","\frac{b \left(\frac{3 a^2}{n p+2}+\frac{b^2}{n p+3}\right) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f \sqrt{\cos ^2(e+f x)}}+\frac{a \left(\frac{a^2}{n p+1}+\frac{3 b^2}{n p+2}\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 n p+7) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3)}-\frac{b^2 \sin (e+f x) \cos (e+f x) (a+b \sin (e+f x)) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+3)}","\frac{b \left(3 a^2 (n p+3)+b^2 (n p+2)\right) \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3) \sqrt{\cos ^2(e+f x)}}+\frac{a \left(a^2 (n p+2)+3 b^2 (n p+1)\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{a b^2 (2 n p+7) \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) (n p+3)}-\frac{b^2 \sin (e+f x) \cos (e+f x) (a+b \sin (e+f x)) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+3)}",1,"-((a*b^2*(7 + 2*n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p))) + (a*(a^2/(1 + n*p) + (3*b^2)/(2 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*Sqrt[Cos[e + f*x]^2]) + (b*((3*a^2)/(2 + n*p) + b^2/(3 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*Sqrt[Cos[e + f*x]^2]) - (b^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x]))/(f*(3 + n*p))","A",6,5,27,0.1852,1,"{2826, 2793, 3023, 2748, 2643}"
833,1,221,0,0.2329431,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x))^2 \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^2,x]","\frac{\left(\frac{a^2}{n p+1}+\frac{b^2}{n p+2}\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2)}","\frac{\left(a^2 (n p+2)+b^2 (n p+1)\right) \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) (n p+2) \sqrt{\cos ^2(e+f x)}}+\frac{2 a b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}-\frac{b^2 \sin (e+f x) \cos (e+f x) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2)}",1,"-((b^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p))) + ((a^2/(1 + n*p) + b^2/(2 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*Sqrt[Cos[e + f*x]^2]) + (2*a*b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])","A",5,4,27,0.1481,1,"{2826, 2789, 2643, 3014}"
834,1,163,0,0.1144387,"\int \left(c (d \sin (e+f x))^p\right)^n (a+b \sin (e+f x)) \, dx","Int[(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x]),x]","\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}","\frac{a \sin (e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+1) \sqrt{\cos ^2(e+f x)}}+\frac{b \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(c (d \sin (e+f x))^p\right)^n}{f (n p+2) \sqrt{\cos ^2(e+f x)}}",1,"(a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])","A",4,3,25,0.1200,1,"{2826, 2748, 2643}"
835,1,204,0,0.3381269,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{a+b \sin (e+f x)} \, dx","Int[(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x]),x]","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{a \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}","\frac{b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{a \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),1;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"(b*AppellF1[1/2, -(n*p)/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)*f*(Sin[e + f*x]^2)^((n*p)/2)) - (a*AppellF1[1/2, (1 - n*p)/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cot[e + f*x]*(Sin[e + f*x]^2)^((1 - n*p)/2)*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)*f)","A",6,4,27,0.1481,1,"{2826, 2823, 3189, 429}"
836,1,322,0,0.5080488,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{(a+b \sin (e+f x))^2} \, dx","Int[(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^2,x]","\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-1),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{a^2 \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}","\frac{2 a b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-1),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{a^2 \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),2;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"(2*a*b*AppellF1[1/2, -(n*p)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^2*f*(Sin[e + f*x]^2)^((n*p)/2)) - (b^2*AppellF1[1/2, (-1 - n*p)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((-1 - n*p)/2)*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^2*f) - (a^2*AppellF1[1/2, (1 - n*p)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cot[e + f*x]*(Sin[e + f*x]^2)^((1 - n*p)/2)*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^2*f)","A",11,5,27,0.1852,1,"{2826, 2824, 3189, 429, 16}"
837,1,428,0,0.6518913,"\int \frac{\left(c (d \sin (e+f x))^p\right)^n}{(a+b \sin (e+f x))^3} \, dx","Int[(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^3,x]","\frac{3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{b^3 \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-2),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{3 a b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-1),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{a^3 \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}","\frac{3 a^2 b \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};-\frac{n p}{2},3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}+\frac{b^3 \cos (e+f x) \sin ^2(e+f x)^{-\frac{n p}{2}} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-2),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{3 a b^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (-n p-1)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (-n p-1),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}-\frac{a^3 \cot (e+f x) \sin ^2(e+f x)^{\frac{1}{2} (1-n p)} \left(c (d \sin (e+f x))^p\right)^n F_1\left(\frac{1}{2};\frac{1}{2} (1-n p),3;\frac{3}{2};\cos ^2(e+f x),-\frac{b^2 \cos ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^3}",1,"(3*a^2*b*AppellF1[1/2, -(n*p)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^3*f*(Sin[e + f*x]^2)^((n*p)/2)) + (b^3*AppellF1[1/2, (-2 - n*p)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^3*f*(Sin[e + f*x]^2)^((n*p)/2)) - (3*a*b^2*AppellF1[1/2, (-1 - n*p)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((-1 - n*p)/2)*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^3*f) - (a^3*AppellF1[1/2, (1 - n*p)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cot[e + f*x]*(Sin[e + f*x]^2)^((1 - n*p)/2)*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^3*f)","A",14,5,27,0.1852,1,"{2826, 2824, 3189, 429, 16}"