1,1,384,0,0.9320943,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2),x]","\frac{16 c^2 \left(A \left(4 m^2+32 m+63\right)+C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left(4 m^2+16 m+15\right)}+\frac{64 c^3 \left(A \left(4 m^2+32 m+63\right)+C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left(A \left(4 m^2+32 m+63\right)+C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)}","\frac{16 c^2 \left(A \left(4 m^2+32 m+63\right)+C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left(4 m^2+16 m+15\right)}+\frac{64 c^3 \left(A \left(4 m^2+32 m+63\right)+C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left(A \left(4 m^2+32 m+63\right)+C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)}",1,"(64*c^3*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c^2*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)) + (2*c*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) - (4*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)*(9 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f*(9 + 2*m))","A",5,4,40,0.1000,1,"{3040, 2973, 2740, 2738}"
2,1,285,0,0.7193551,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2),x]","\frac{8 c^2 \left(A \left(4 m^2+24 m+35\right)+C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left(A \left(4 m^2+24 m+35\right)+C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}","\frac{8 c^2 \left(A \left(4 m^2+24 m+35\right)+C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 c \left(A \left(4 m^2+24 m+35\right)+C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}-\frac{4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}",1,"(8*c^2*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) - (4*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f*(7 + 2*m))","A",4,4,40,0.1000,1,"{3040, 2973, 2740, 2738}"
3,1,180,0,0.5696397,"\int (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]]*(A + C*Sin[e + f*x]^2),x]","\frac{2 c (A (2 m+5)-6 C m+C) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+5) \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}{c f (2 m+5)}+\frac{4 c C (2 m+1) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) (2 m+5) \sqrt{c-c \sin (e+f x)}}","\frac{2 c (A (2 m+5)-6 C m+C) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+5) \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}{c f (2 m+5)}+\frac{4 c C (2 m+1) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) (2 m+5) \sqrt{c-c \sin (e+f x)}}",1,"(2*c*(C - 6*C*m + A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (4*c*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(c*f*(5 + 2*m))","A",4,3,40,0.07500,1,"{3040, 2971, 2738}"
4,1,123,0,0.2927377,"\int \frac{(a+a \sin (e+f x))^m \left(A+C \sin ^2(e+f x)\right)}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{(A+C) \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{2 C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}","\frac{(A+C) \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{2 C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"((A + C)*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]]) - (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",4,4,40,0.1000,1,"{3038, 2745, 2667, 68}"
5,1,202,0,0.5859777,"\int \frac{(a+a \sin (e+f x))^m \left(A+C \sin ^2(e+f x)\right)}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{(A (1-2 m)-C (2 m+7)) \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)}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(2 A m+A+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}","\frac{(A (1-2 m)-C (2 m+7)) \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)}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(2 A m+A+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}",1,"((A + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) + ((A + 2*A*m + C*(9 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + ((A*(1 - 2*m) - C*(7 + 2*m))*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)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",5,5,40,0.1250,1,"{3036, 2973, 2745, 2667, 68}"
6,1,207,0,0.628137,"\int \frac{(a+a \sin (e+f x))^m \left(A+C \sin ^2(e+f x)\right)}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{\left(A \left(4 m^2-8 m+3\right)+C \left(4 m^2+24 m+19\right)\right) \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)}{32 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{16 c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}","\frac{\left(A \left(4 m^2-8 m+3\right)+C \left(4 m^2+24 m+19\right)\right) \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)}{32 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A (5-2 m)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{16 c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}",1,"((A + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(8*a*f*(c - c*Sin[e + f*x])^(5/2)) + ((A*(5 - 2*m) - C*(11 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(16*c*f*(c - c*Sin[e + f*x])^(3/2)) + ((A*(3 - 8*m + 4*m^2) + C*(19 + 24*m + 4*m^2))*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)/(32*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",5,5,40,0.1250,1,"{3036, 2972, 2745, 2667, 68}"
7,1,167,0,0.6513491,"\int \frac{A+C \sin ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Int[(A + C*Sin[e + f*x]^2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{(A+C) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{(A+C) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"((A + C)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - ((A - 3*C)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + ((A + C)*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,5,42,0.1190,1,"{3036, 2969, 2737, 2667, 31}"
8,1,257,0,0.6648803,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2),x]","\frac{c 2^{n+\frac{1}{2}} ((m+n+1) (A (m+n+2)+C (-m+n+1))+C (2 m+1) (m-n)) \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) (m+n+1) (m+n+2)}-\frac{C (2 m+1) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f (m+n+1) (m+n+2)}+\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n+1}}{c f (m+n+2)}","\frac{c 2^{n+\frac{1}{2}} ((m+n+1) (A (m+n+2)+C (-m+n+1))+C (2 m+1) (m-n)) \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) (m+n+1) (m+n+2)}-\frac{C (2 m+1) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f (m+n+1) (m+n+2)}+\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n+1}}{c f (m+n+2)}",1,"(2^(1/2 + n)*c*(C*(1 + 2*m)*(m - n) + (1 + m + n)*(C*(1 - m + n) + A*(2 + m + n)))*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)*(1 + m + n)*(2 + m + n)) - (C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*(1 + m + n)*(2 + m + n)) + (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 + n))/(c*f*(2 + m + n))","A",6,6,38,0.1579,1,"{3040, 2973, 2745, 2689, 70, 69}"
9,1,365,0,0.8116411,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (A d (m+n+2)+c (2 C m+C)+C d (-m+n+1)) (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)}{d f (2 m+1) (m+n+2) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt{2} C (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{3}{2};\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (m+n+2) \sqrt{1-\sin (e+f x)}}-\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{n+1}}{d f (m+n+2)}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (d (A (m+n+2)+C (-m+n+1))+c (2 C m+C)) (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)}{d f (2 m+1) (m+n+2) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt{2} C (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{3}{2};\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (m+n+2) \sqrt{1-\sin (e+f x)}}-\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{n+1}}{d f (m+n+2)}",1,"-((C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(2 + m + n))) + (Sqrt[2]*(c*(C + 2*C*m) + C*d*(1 - m + n) + A*d*(2 + m + n))*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)/(d*f*(1 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n) + (Sqrt[2]*C*(d*m - c*(1 + m))*AppellF1[3/2 + m, 1/2, -n, 5/2 + 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])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*d*f*(3 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n)","A",10,6,37,0.1622,1,"{3046, 2987, 2788, 140, 139, 138}"
10,1,392,0,0.9916399,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-2 - m)*(A + C*Sin[e + f*x]^2),x]","-\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) \left(c d (m+1) (A+C)+d^2 (-A m+C m+C)+c^2 (-(2 C m+C))\right) (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)}{d f (m+1) (c-d) (c+d)^2}+\frac{\left(A d^2+c^2 C\right) \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left(c^2-d^2\right)}+\frac{\sqrt{2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^m (c+d \sin (e+f x))^{-m} F_1\left(m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}}","-\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) \left(c d (m+1) (A+C)+d^2 (-A m+C m+C)+c^2 (-(2 C m+C))\right) (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)}{d f (m+1) (c-d) (c+d)^2}+\frac{\left(A d^2+c^2 C\right) \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left(c^2-d^2\right)}+\frac{\sqrt{2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^m (c+d \sin (e+f x))^{-m} F_1\left(m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}}",1,"((c^2*C + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/(d*(c^2 - d^2)*f*(1 + m)) - (2^(1/2 + m)*a*(c*(A + C)*d*(1 + m) + d^2*(C - A*m + C*m) - c^2*(C + 2*C*m))*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)*d*(c + d)^2*f*(1 + m)*(c + d*Sin[e + f*x])^m) + (Sqrt[2]*C*AppellF1[3/2 + m, 1/2, 1 + m, 5/2 + 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])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/(a*(c - d)*d*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*(c + d*Sin[e + f*x])^m)","A",8,7,41,0.1707,1,"{3044, 2987, 2788, 132, 140, 139, 138}"
11,1,384,0,0.9735818,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2),x]","\frac{\sqrt{2} (c-d) \cos (e+f x) (A d (2 m+7)+2 c (2 C m+C)+C d (5-2 m)) (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)}{d f (2 m+1) (2 m+7) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{2 \sqrt{2} C (c-d) (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+7) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)}","\frac{\sqrt{2} (c-d) \cos (e+f x) (d (A (2 m+7)+C (5-2 m))+2 c (2 C m+C)) (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)}{d f (2 m+1) (2 m+7) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{2 \sqrt{2} C (c-d) (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+7) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)}",1,"(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2))/(d*f*(7 + 2*m)) + (Sqrt[2]*(c - d)*(C*d*(5 - 2*m) + A*d*(7 + 2*m) + 2*c*(C + 2*C*m))*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]])/(d*f*(1 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (2*Sqrt[2]*C*(c - d)*(d*m - c*(1 + m))*AppellF1[3/2 + m, 1/2, -3/2, 5/2 + 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])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",10,6,39,0.1538,1,"{3046, 2987, 2788, 140, 139, 138}"
12,1,374,0,0.8628759,"\int (a+a \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \left(A+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]]*(A + C*Sin[e + f*x]^2),x]","\frac{\sqrt{2} \cos (e+f x) (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) (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)}{d f (2 m+1) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{2 \sqrt{2} C (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}","\frac{\sqrt{2} \cos (e+f x) (d (A (2 m+5)+C (3-2 m))+2 c (2 C m+C)) (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)}{d f (2 m+1) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{2 \sqrt{2} C (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}",1,"(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2))/(d*f*(5 + 2*m)) + (Sqrt[2]*(C*d*(3 - 2*m) + A*d*(5 + 2*m) + 2*c*(C + 2*C*m))*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]])/(d*f*(1 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (2*Sqrt[2]*C*(d*m - c*(1 + m))*AppellF1[3/2 + m, 1/2, -1/2, 5/2 + 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])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",10,6,39,0.1538,1,"{3046, 2987, 2788, 140, 139, 138}"
13,1,365,0,0.8699809,"\int \frac{(a+a \sin (e+f x))^m \left(A+C \sin ^2(e+f x)\right)}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{2} \cos (e+f x) (d (A (2 m+3)-2 C m+C)+2 c (2 C m+C)) (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)}{d f (2 m+1) (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{2 \sqrt{2} C (c m+c-d m) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)}}{d f (2 m+3)}","\frac{\sqrt{2} \cos (e+f x) (d (A (2 m+3)-2 C m+C)+2 c (2 C m+C)) (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)}{d f (2 m+1) (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{2 \sqrt{2} C (c m+c-d m) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)}}{d f (2 m+3)}",1,"(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(3 + 2*m)) + (Sqrt[2]*(2*c*(C + 2*C*m) + d*(C - 2*C*m + A*(3 + 2*m)))*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)])/(d*f*(1 + 2*m)*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (2*Sqrt[2]*C*(c + c*m - d*m)*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + 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])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*f*(3 + 2*m)^2*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",10,6,39,0.1538,1,"{3046, 2987, 2788, 140, 139, 138}"
14,1,413,0,0.9576464,"\int \frac{(a+a \sin (e+f x))^m \left(A+C \sin ^2(e+f x)\right)}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(3/2),x]","\frac{\sqrt{2} \cos (e+f x) \left(c d (A+C)-d^2 (4 A m+A-C)-2 c^2 (2 C m+C)\right) (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)}{d f (2 m+1) \left(c^2-d^2\right) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) \left(d^2 (2 A m+A-C)+2 c^2 C (m+1)\right) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) \left(c^2-d^2\right) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(A d^2+c^2 C\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}","\frac{\sqrt{2} \cos (e+f x) \left(c d (A+C)-d^2 (4 A m+A-C)-2 c^2 (2 C m+C)\right) (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)}{d f (2 m+1) \left(c^2-d^2\right) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) \left(d^2 (2 A m+A-C)+2 c^2 C (m+1)\right) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) \left(c^2-d^2\right) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(A d^2+c^2 C\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}",1,"(2*(c^2*C + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(c*(A + C)*d - d^2*(A - C + 4*A*m) - 2*c^2*(C + 2*C*m))*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)])/(d*(c^2 - d^2)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(2*c^2*C*(1 + m) + d^2*(A - C + 2*A*m))*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + 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])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*(c^2 - d^2)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",10,6,39,0.1538,1,"{3044, 2987, 2788, 140, 139, 138}"
15,1,424,0,1.0352339,"\int \frac{(a+a \sin (e+f x))^m \left(A+C \sin ^2(e+f x)\right)}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(5/2),x]","\frac{\sqrt{2} \cos (e+f x) \left(3 c d (A+C)+d^2 (-4 A m+A+3 C)-2 c^2 (2 C m+C)\right) (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)}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) \left(2 c^2 C (m+1)-d^2 (-2 A m+A+3 C)\right) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(A d^2+c^2 C\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}","\frac{\sqrt{2} \cos (e+f x) \left(3 c d (A+C)+d^2 (-4 A m+A+3 C)-2 c^2 (2 C m+C)\right) (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)}{3 d f (2 m+1) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) \left(2 c^2 C (m+1)-d^2 (-2 A m+A+3 C)\right) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \left(A d^2+c^2 C\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}",1,"(2*(c^2*C + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (Sqrt[2]*(3*c*(A + C)*d + d^2*(A + 3*C - 4*A*m) - 2*c^2*(C + 2*C*m))*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)])/(3*(c - d)^2*d*(c + d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(2*c^2*C*(1 + m) - d^2*(A + 3*C - 2*A*m))*AppellF1[3/2 + m, 1/2, 3/2, 5/2 + 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])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(3*a*(c - d)^2*d*(c + d)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",10,6,39,0.1538,1,"{3044, 2987, 2788, 140, 139, 138}"
16,1,174,0,0.6862024,"\int \frac{A+B \sin (e+f x)+C \sin ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Int[(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{(A+B+C) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-B-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A-B+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{(A+B+C) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-B-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A-B+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"((A + B + C)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - ((A - B - 3*C)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + ((A - B + C)*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,5,50,0.1000,1,"{3035, 2969, 2737, 2667, 31}"
17,1,269,0,0.7453117,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\frac{c 2^{n+\frac{1}{2}} \cos (e+f x) ((m+n+1) (A (m+n+2)+C (-m+n+1))+(m-n) (B (m+n+2)+2 C m+C)) (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) (m+n+1) (m+n+2)}-\frac{(B (m+n+2)+2 C m+C) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f (m+n+1) (m+n+2)}+\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n+1}}{c f (m+n+2)}","\frac{c 2^{n+\frac{1}{2}} \cos (e+f x) ((m+n+1) (A (m+n+2)+C (-m+n+1))+(m-n) (B (m+n+2)+2 C m+C)) (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) (m+n+1) (m+n+2)}-\frac{(B (m+n+2)+2 C m+C) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f (m+n+1) (m+n+2)}+\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n+1}}{c f (m+n+2)}",1,"(2^(1/2 + n)*c*((1 + m + n)*(C*(1 - m + n) + A*(2 + m + n)) + (m - n)*(C + 2*C*m + B*(2 + m + n)))*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)*(1 + m + n)*(2 + m + n)) - ((C + 2*C*m + B*(2 + m + n))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*(1 + m + n)*(2 + m + n)) + (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 + n))/(c*f*(2 + m + n))","A",6,6,46,0.1304,1,"{3039, 2973, 2745, 2689, 70, 69}"
18,1,435,0,0.8918011,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","-\frac{16 c^2 \left(-A \left(4 m^2+32 m+63\right)+B \left(-4 m^2-8 m+45\right)-C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left(4 m^2+16 m+15\right)}-\frac{64 c^3 \left(-A \left(4 m^2+32 m+63\right)+B \left(-4 m^2-8 m+45\right)-C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}-\frac{2 c \left(-A \left(4 m^2+32 m+63\right)+B \left(-4 m^2-8 m+45\right)-C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}-\frac{2 (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}","-\frac{16 c^2 \left(-A \left(4 m^2+32 m+63\right)+B \left(-4 m^2-8 m+45\right)-C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left(4 m^2+16 m+15\right)}-\frac{64 c^3 \left(-A \left(4 m^2+32 m+63\right)+B \left(-4 m^2-8 m+45\right)-C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}-\frac{2 c \left(-A \left(4 m^2+32 m+63\right)+B \left(-4 m^2-8 m+45\right)-C \left(4 m^2-16 m+39\right)\right) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}-\frac{2 (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}",1,"(-64*c^3*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) - (16*c^2*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) - (2*(9*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)*(9 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f*(9 + 2*m))","A",5,4,48,0.08333,1,"{3039, 2973, 2740, 2738}"
19,1,322,0,0.7112235,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","-\frac{8 c^2 \left(-A \left(4 m^2+24 m+35\right)+B \left(-4 m^2-8 m+21\right)-C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}-\frac{2 c \left(-A \left(4 m^2+24 m+35\right)+B \left(-4 m^2-8 m+21\right)-C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}-\frac{2 (2 B m+7 B+4 C m+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) (2 m+7)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}","-\frac{8 c^2 \left(-A \left(4 m^2+24 m+35\right)+B \left(-4 m^2-8 m+21\right)-C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left(4 m^2+8 m+3\right) \sqrt{c-c \sin (e+f x)}}-\frac{2 c \left(-A \left(4 m^2+24 m+35\right)+B \left(-4 m^2-8 m+21\right)-C \left(4 m^2-8 m+19\right)\right) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}-\frac{2 (2 B m+7 B+4 C m+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) (2 m+7)}+\frac{2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)}",1,"(-8*c^2*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) - (2*c*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) - (2*(7*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f*(7 + 2*m))","A",4,4,48,0.08333,1,"{3039, 2973, 2740, 2738}"
20,1,197,0,0.6285871,"\int (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]]*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\frac{2 c (A (2 m+5)-B (2 m+5)-6 C m+C) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+5) \sqrt{c-c \sin (e+f x)}}+\frac{2 c (2 B m+5 B+4 C m+2 C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) (2 m+5) \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}{c f (2 m+5)}","\frac{2 c (A (2 m+5)-B (2 m+5)-6 C m+C) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+5) \sqrt{c-c \sin (e+f x)}}+\frac{2 c (2 B m+5 B+4 C m+2 C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) (2 m+5) \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}{c f (2 m+5)}",1,"(2*c*(C - 6*C*m + A*(5 + 2*m) - B*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(5*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(c*f*(5 + 2*m))","A",4,3,48,0.06250,1,"{3039, 2971, 2738}"
21,1,170,0,0.4682791,"\int \frac{(a+a \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{(A+B+C) \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{2 B \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+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}","\frac{(A+B+C) \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{2 B \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+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(-2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + ((A + B + C)*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]]) - (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",5,5,48,0.1042,1,"{3037, 2973, 2745, 2667, 68}"
22,1,216,0,0.6779424,"\int \frac{(a+a \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{(A (1-2 m)-B (2 m+3)-C (2 m+7)) \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)}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}","\frac{(A (1-2 m)-B (2 m+3)-C (2 m+7)) \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)}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}}",1,"((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) + ((A + B + 2*A*m + 2*B*m + C*(9 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + ((A*(1 - 2*m) - B*(3 + 2*m) - C*(7 + 2*m))*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)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",5,5,48,0.1042,1,"{3035, 2973, 2745, 2667, 68}"
23,1,230,0,0.6971499,"\int \frac{(a+a \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{\left(-A \left(4 m^2-8 m+3\right)+B \left(-4 m^2-8 m+5\right)-C \left(4 m^2+24 m+19\right)\right) \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)}{32 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A (5-2 m)-B (2 m+3)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{16 c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}","-\frac{\left(-A \left(4 m^2-8 m+3\right)+B \left(-4 m^2-8 m+5\right)-C \left(4 m^2+24 m+19\right)\right) \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)}{32 c^2 f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A (5-2 m)-B (2 m+3)-C (2 m+11)) \cos (e+f x) (a \sin (e+f x)+a)^m}{16 c f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{8 a f (c-c \sin (e+f x))^{5/2}}",1,"((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(8*a*f*(c - c*Sin[e + f*x])^(5/2)) + ((A*(5 - 2*m) - B*(3 + 2*m) - C*(11 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(16*c*f*(c - c*Sin[e + f*x])^(3/2)) - ((B*(5 - 8*m - 4*m^2) - A*(3 - 8*m + 4*m^2) - C*(19 + 24*m + 4*m^2))*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)/(32*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",5,5,48,0.1042,1,"{3035, 2972, 2745, 2667, 68}"
24,1,232,0,0.7197571,"\int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{2 a f (2 m+3)}+\frac{(A-B+C) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{2 c f (2 m+1)}-\frac{C 2^{-m-\frac{1}{2}} \cos ^3(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-2} \, _2F_1\left(\frac{1}{2} (2 m+3),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}","\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{2 a f (2 m+3)}+\frac{(A-B+C) \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{2 c f (2 m+1)}-\frac{C 2^{-m-\frac{1}{2}} \cos ^3(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-2} \, _2F_1\left(\frac{1}{2} (2 m+3),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}",1,"-((2^(-1/2 - m)*C*Cos[e + f*x]^3*Hypergeometric2F1[(3 + 2*m)/2, (3 + 2*m)/2, (5 + 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])^(-2 - m))/(f*(3 + 2*m))) + ((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(2*a*f*(3 + 2*m)) + ((A - B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(2*c*f*(1 + 2*m))","A",6,6,50,0.1200,1,"{3035, 2972, 2745, 2689, 70, 69}"
25,1,381,0,0.9035906,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (d (A (m+n+2)-B (m+n+2)+C (-m+n+1))+c (2 C m+C)) (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)}{d f (2 m+1) (m+n+2) \sqrt{1-\sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (B d (m+n+2)-c C (m+1)+C d 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{3}{2};\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (m+n+2) \sqrt{1-\sin (e+f x)}}-\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{n+1}}{d f (m+n+2)}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (d (A (m+n+2)-B (m+n+2)+C (-m+n+1))+c (2 C m+C)) (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)}{d f (2 m+1) (m+n+2) \sqrt{1-\sin (e+f x)}}-\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c C (m+1)-d (B (m+n+2)+C 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{3}{2};\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (m+n+2) \sqrt{1-\sin (e+f x)}}-\frac{C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{n+1}}{d f (m+n+2)}",1,"-((C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(2 + m + n))) + (Sqrt[2]*(c*(C + 2*C*m) + d*(C*(1 - m + n) + A*(2 + m + n) - B*(2 + m + n)))*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)/(d*f*(1 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n) + (Sqrt[2]*(C*d*m - c*C*(1 + m) + B*d*(2 + m + n))*AppellF1[3/2 + m, 1/2, -n, 5/2 + 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])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*d*f*(3 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n)","A",10,6,45,0.1333,1,"{3045, 2987, 2788, 140, 139, 138}"
26,1,410,0,1.0598844,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-2 - m)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","-\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left(c d (A m+A+B m+C m+C)-d^2 (A m+B (m+1)-C (m+1))+c^2 (-(2 C m+C))\right) \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)}{d f (m+1) (c-d) (c+d)^2}+\frac{\cos (e+f x) \left(A d^2-B c d+c^2 C\right) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left(c^2-d^2\right)}+\frac{\sqrt{2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^m (c+d \sin (e+f x))^{-m} F_1\left(m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}}","-\frac{a 2^{m+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left(c d (A m+A+B m+C m+C)-d^2 (A m+B (m+1)-C (m+1))+c^2 (-(2 C m+C))\right) \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)}{d f (m+1) (c-d) (c+d)^2}+\frac{\cos (e+f x) \left(A d^2-B c d+c^2 C\right) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left(c^2-d^2\right)}+\frac{\sqrt{2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left(\frac{c+d \sin (e+f x)}{c-d}\right)^m (c+d \sin (e+f x))^{-m} F_1\left(m+\frac{3}{2};\frac{1}{2},m+1;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (c-d) \sqrt{1-\sin (e+f x)}}",1,"((c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/(d*(c^2 - d^2)*f*(1 + m)) - (2^(1/2 + m)*a*(c*d*(A + C + A*m + B*m + C*m) - c^2*(C + 2*C*m) - d^2*(A*m + B*(1 + m) - C*(1 + m)))*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)*d*(c + d)^2*f*(1 + m)*(c + d*Sin[e + f*x])^m) + (Sqrt[2]*C*AppellF1[3/2 + m, 1/2, 1 + m, 5/2 + 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])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/(a*(c - d)*d*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*(c + d*Sin[e + f*x])^m)","A",8,7,49,0.1429,1,"{3043, 2987, 2788, 132, 140, 139, 138}"
27,1,403,0,1.0343896,"\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\frac{\sqrt{2} (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+7)+2 B m+7 B+2 C m-5 C)) \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)}{d f (2 m+1) (2 m+7) \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) (B d (2 m+7)-2 c C (m+1)+2 C d m) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+7) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)}","\frac{\sqrt{2} (c-d) \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+7)+2 B m+7 B+2 C m-5 C)) \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)}{d f (2 m+1) (2 m+7) \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) (2 c C (m+1)-d (B (2 m+7)+2 C m)) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+7) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)}",1,"(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2))/(d*f*(7 + 2*m)) + (Sqrt[2]*(c - d)*(2*c*(C + 2*C*m) - d*(7*B - 5*C + 2*B*m + 2*C*m - A*(7 + 2*m)))*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]])/(d*f*(1 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (Sqrt[2]*(c - d)*(2*C*d*m - 2*c*C*(1 + m) + B*d*(7 + 2*m))*AppellF1[3/2 + m, 1/2, -3/2, 5/2 + 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])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",10,6,47,0.1277,1,"{3045, 2987, 2788, 140, 139, 138}"
28,1,393,0,0.9234722,"\int (a+a \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]]*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+5)+2 B m+5 B+2 C m-3 C)) \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)}{d f (2 m+1) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}+\frac{\sqrt{2} \cos (e+f x) (B d (2 m+5)-2 c C (m+1)+2 C d m) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+5)+2 B m+5 B+2 C m-3 C)) \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)}{d f (2 m+1) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{\sqrt{2} \cos (e+f x) (2 c C (m+1)-d (B (2 m+5)+2 C m)) (a \sin (e+f x)+a)^{m+1} \sqrt{c+d \sin (e+f x)} F_1\left(m+\frac{3}{2};\frac{1}{2},-\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) (2 m+5) \sqrt{1-\sin (e+f x)} \sqrt{\frac{c+d \sin (e+f x)}{c-d}}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}",1,"(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2))/(d*f*(5 + 2*m)) + (Sqrt[2]*(2*c*(C + 2*C*m) - d*(5*B - 3*C + 2*B*m + 2*C*m - A*(5 + 2*m)))*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]])/(d*f*(1 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (Sqrt[2]*(2*C*d*m - 2*c*C*(1 + m) + B*d*(5 + 2*m))*AppellF1[3/2 + m, 1/2, -1/2, 5/2 + 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])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])","A",10,6,47,0.1277,1,"{3045, 2987, 2788, 140, 139, 138}"
29,1,386,0,0.9282851,"\int \frac{(a+a \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{\sqrt{c+d \sin (e+f x)}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/Sqrt[c + d*Sin[e + f*x]],x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+3)+2 B m+3 B+2 C m-C)) \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)}{d f (2 m+1) (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{\sqrt{2} \cos (e+f x) (B d (2 m+3)-2 c C (m+1)+2 C d m) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)}}{d f (2 m+3)}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m (2 c (2 C m+C)-d (-A (2 m+3)+2 B m+3 B+2 C m-C)) \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)}{d f (2 m+1) (2 m+3) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{2} \cos (e+f x) (2 c C (m+1)-d (B (2 m+3)+2 C m)) (a \sin (e+f x)+a)^{m+1} \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3)^2 \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt{c+d \sin (e+f x)}}{d f (2 m+3)}",1,"(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(3 + 2*m)) + (Sqrt[2]*(2*c*(C + 2*C*m) - d*(3*B - C + 2*B*m + 2*C*m - A*(3 + 2*m)))*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)])/(d*f*(1 + 2*m)*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(2*C*d*m - 2*c*C*(1 + m) + B*d*(3 + 2*m))*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + 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])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*f*(3 + 2*m)^2*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",10,6,47,0.1277,1,"{3045, 2987, 2788, 140, 139, 138}"
30,1,433,0,1.0899923,"\int \frac{(a+a \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{(c+d \sin (e+f x))^{3/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \left(-c d (A+4 B m+B+C)+d^2 (4 A m+A+B-C)+2 c^2 (2 C m+C)\right) \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)}{d f (2 m+1) \left(c^2-d^2\right) \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+1} \left(d (2 m+1) (B c-A d)-2 c^2 C (m+1)+C d^2\right) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) \left(c^2-d^2\right) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \cos (e+f x) \left(A d^2-B c d+c^2 C\right) (a \sin (e+f x)+a)^m}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}","-\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \left(-c d (A+4 B m+B+C)+d^2 (4 A m+A+B-C)+2 c^2 (2 C m+C)\right) \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)}{d f (2 m+1) \left(c^2-d^2\right) \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+1} \left(d (2 m+1) (B c-A d)+C \left(d^2-2 c^2 (m+1)\right)\right) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{1}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a d f (2 m+3) \left(c^2-d^2\right) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \cos (e+f x) \left(A d^2-B c d+c^2 C\right) (a \sin (e+f x)+a)^m}{d f \left(c^2-d^2\right) \sqrt{c+d \sin (e+f x)}}",1,"(2*(c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[2]*(d^2*(A + B - C + 4*A*m) - c*d*(A + B + C + 4*B*m) + 2*c^2*(C + 2*C*m))*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)])/(d*(c^2 - d^2)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[2]*(C*d^2 - 2*c^2*C*(1 + m) + d*(B*c - A*d)*(1 + 2*m))*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + 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])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*(c^2 - d^2)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",10,6,47,0.1277,1,"{3043, 2987, 2788, 140, 139, 138}"
31,1,451,0,1.1628659,"\int \frac{(a+a \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{(c+d \sin (e+f x))^{5/2}} \, dx","Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(5/2),x]","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \left(c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right) \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)}{3 d f (2 m+1) (c-d)^2 (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+1} \left(-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \cos (e+f x) \left(A d^2-B c d+c^2 C\right) (a \sin (e+f x)+a)^m}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}","\frac{\sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^m \left(c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right) \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)}{3 d f (2 m+1) (c-d)^2 (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+1} \left(-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right) \sqrt{\frac{c+d \sin (e+f x)}{c-d}} F_1\left(m+\frac{3}{2};\frac{1}{2},\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{3 a d f (2 m+3) (c-d)^2 (c+d) \sqrt{1-\sin (e+f x)} \sqrt{c+d \sin (e+f x)}}+\frac{2 \cos (e+f x) \left(A d^2-B c d+c^2 C\right) (a \sin (e+f x)+a)^m}{3 d f \left(c^2-d^2\right) (c+d \sin (e+f x))^{3/2}}",1,"(2*(c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (Sqrt[2]*(d^2*(A - 3*B + 3*C - 4*A*m) + c*d*(3*A - B + 3*C + 4*B*m) - 2*c^2*(C + 2*C*m))*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)])/(3*(c - d)^2*d*(c + d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(B*c*d*(1 - 2*m) + 2*c^2*C*(1 + m) - d^2*(A + 3*C - 2*A*m))*AppellF1[3/2 + m, 1/2, 3/2, 5/2 + 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])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(3*a*(c - d)^2*d*(c + d)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])","A",10,6,47,0.1277,1,"{3043, 2987, 2788, 140, 139, 138}"
32,1,113,0,0.1022173,"\int (a+b \sin (c+d x)) \left(A+B \sin (c+d x)+C \sin ^2(c+d x)\right) \, dx","Int[(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x] + C*Sin[c + d*x]^2),x]","-\frac{\cos (c+d x) \left(a (3 b B-a C)+b^2 (3 A+2 C)\right)}{3 b d}+\frac{1}{2} x (a (2 A+C)+b B)-\frac{(3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}-\frac{C \cos (c+d x) (a+b \sin (c+d x))^2}{3 b d}","-\frac{\cos (c+d x) (a B+A b+b C)}{d}+\frac{1}{2} x (a (2 A+C)+b B)-\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b C \cos ^3(c+d x)}{3 d}",1,"((b*B + a*(2*A + C))*x)/2 - ((b^2*(3*A + 2*C) + a*(3*b*B - a*C))*Cos[c + d*x])/(3*b*d) - ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (C*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(3*b*d)","A",2,2,31,0.06452,1,"{3023, 2734}"
33,1,117,0,0.2192007,"\int \frac{(a+b \sin (e+f x)) \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right)}{\sin ^{\frac{3}{2}}(e+f x)} \, dx","Int[((a + b*Sin[e + f*x])*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/Sin[e + f*x]^(3/2),x]","\frac{2 F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) (3 a B+3 A b+b C)}{3 f}+\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) (b B-a (A-C))}{f}-\frac{2 a A \cos (e+f x)}{f \sqrt{\sin (e+f x)}}-\frac{2 b C \sqrt{\sin (e+f x)} \cos (e+f x)}{3 f}","\frac{2 F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) (3 a B+3 A b+b C)}{3 f}+\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) (b B-a (A-C))}{f}-\frac{2 a A \cos (e+f x)}{f \sqrt{\sin (e+f x)}}-\frac{2 b C \sqrt{\sin (e+f x)} \cos (e+f x)}{3 f}",1,"(2*(b*B - a*(A - C))*EllipticE[(e - Pi/2 + f*x)/2, 2])/f + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(e - Pi/2 + f*x)/2, 2])/(3*f) - (2*a*A*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]]) - (2*b*C*Cos[e + f*x]*Sqrt[Sin[e + f*x]])/(3*f)","A",5,5,41,0.1220,1,"{3031, 3023, 2748, 2641, 2639}"
34,0,0,0,0.118066,"\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]","\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) \, dx","\text{Int}\left((a+b \sin (e+f x))^m \left(A+B \sin (e+f x)+C \sin ^2(e+f x)\right) (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*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x]","A",0,0,0,0,-1,"{}"