1,1,899,384,6.9809098,"\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","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2),x]","\frac{(a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{5/2} \left(\frac{\left(64 A m^4+16 C m^4+896 A m^3+224 C m^3+5280 A m^2+1416 C m^2+15648 A m+648 C m+18900 A+12285 C\right) \left(\left(\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)+\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(64 A m^4+16 C m^4+896 A m^3+224 C m^3+5280 A m^2+1416 C m^2+15648 A m+648 C m+18900 A+12285 C\right) \left(\left(\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)+\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(24 A m^3+8 C m^3+292 A m^2+100 C m^2+1178 A m+414 C m+1575 A+1575 C\right) \left(\left(\frac{1}{4}-\frac{i}{4}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{4}+\frac{i}{4}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(24 A m^3+8 C m^3+292 A m^2+100 C m^2+1178 A m+414 C m+1575 A+1575 C\right) \left(\left(\frac{1}{4}+\frac{i}{4}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{4}-\frac{i}{4}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(4 A m^2+4 C m^2+32 A m+44 C m+63 A+189 C\right) \left(\left(-\frac{1}{4}+\frac{i}{4}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{4}+\frac{i}{4}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7) (2 m+9)}+\frac{\left(4 A m^2+4 C m^2+32 A m+44 C m+63 A+189 C\right) \left(\left(-\frac{1}{4}-\frac{i}{4}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{4}-\frac{i}{4}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+15) \left(\left(\frac{3}{16}-\frac{3 i}{16}\right) C \sin \left(\frac{7}{2} (e+f x)\right)-\left(\frac{3}{16}+\frac{3 i}{16}\right) C \cos \left(\frac{7}{2} (e+f x)\right)\right)}{(2 m+7) (2 m+9)}+\frac{(2 m+15) \left(\left(\frac{3}{16}+\frac{3 i}{16}\right) C \sin \left(\frac{7}{2} (e+f x)\right)-\left(\frac{3}{16}-\frac{3 i}{16}\right) C \cos \left(\frac{7}{2} (e+f x)\right)\right)}{(2 m+7) (2 m+9)}+\frac{\left(\frac{1}{16}+\frac{i}{16}\right) C \cos \left(\frac{9}{2} (e+f x)\right)+\left(\frac{1}{16}-\frac{i}{16}\right) C \sin \left(\frac{9}{2} (e+f x)\right)}{2 m+9}+\frac{\left(\frac{1}{16}-\frac{i}{16}\right) C \cos \left(\frac{9}{2} (e+f x)\right)+\left(\frac{1}{16}+\frac{i}{16}\right) C \sin \left(\frac{9}{2} (e+f x)\right)}{2 m+9}\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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{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{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,"((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((18900*A + 12285*C + 15648*A*m + 648*C*m + 5280*A*m^2 + 1416*C*m^2 + 896*A*m^3 + 224*C*m^3 + 64*A*m^4 + 16*C*m^4)*((1/8 + I/8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((18900*A + 12285*C + 15648*A*m + 648*C*m + 5280*A*m^2 + 1416*C*m^2 + 896*A*m^3 + 224*C*m^3 + 64*A*m^4 + 16*C*m^4)*((1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((1575*A + 1575*C + 1178*A*m + 414*C*m + 292*A*m^2 + 100*C*m^2 + 24*A*m^3 + 8*C*m^3)*((1/4 - I/4)*Cos[(3*(e + f*x))/2] - (1/4 + I/4)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((1575*A + 1575*C + 1178*A*m + 414*C*m + 292*A*m^2 + 100*C*m^2 + 24*A*m^3 + 8*C*m^3)*((1/4 + I/4)*Cos[(3*(e + f*x))/2] - (1/4 - I/4)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((63*A + 189*C + 32*A*m + 44*C*m + 4*A*m^2 + 4*C*m^2)*((-1/4 + I/4)*Cos[(5*(e + f*x))/2] - (1/4 + I/4)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((63*A + 189*C + 32*A*m + 44*C*m + 4*A*m^2 + 4*C*m^2)*((-1/4 - I/4)*Cos[(5*(e + f*x))/2] - (1/4 - I/4)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((-3/16 - (3*I)/16)*C*Cos[(7*(e + f*x))/2] + (3/16 - (3*I)/16)*C*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((-3/16 + (3*I)/16)*C*Cos[(7*(e + f*x))/2] + (3/16 + (3*I)/16)*C*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((1/16 + I/16)*C*Cos[(9*(e + f*x))/2] + (1/16 - I/16)*C*Sin[(9*(e + f*x))/2])/(9 + 2*m) + ((1/16 - I/16)*C*Cos[(9*(e + f*x))/2] + (1/16 + I/16)*C*Sin[(9*(e + f*x))/2])/(9 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)","C",1
2,1,264,285,3.6288919,"\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","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2),x]","\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(-(2 m+1) \left(4 A \left(4 m^2+24 m+35\right)+C \left(12 m^2+80 m+253\right)\right) \sin (e+f x)+32 A m^3+272 A m^2+760 A m+700 A+8 C m^3 \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))-2 C \left(8 m^3+68 m^2+110 m+39\right) \cos (2 (e+f x))+46 C m \sin (3 (e+f x))+15 C \sin (3 (e+f x))+16 C m^3+136 C m^2+284 C m+494 C\right)}{2 f (2 m+1) (2 m+3) (2 m+5) (2 m+7) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(700*A + 494*C + 760*A*m + 284*C*m + 272*A*m^2 + 136*C*m^2 + 32*A*m^3 + 16*C*m^3 - 2*C*(39 + 110*m + 68*m^2 + 8*m^3)*Cos[2*(e + f*x)] - (1 + 2*m)*(4*A*(35 + 24*m + 4*m^2) + C*(253 + 80*m + 12*m^2))*Sin[e + f*x] + 15*C*Sin[3*(e + f*x)] + 46*C*m*Sin[3*(e + f*x)] + 36*C*m^2*Sin[3*(e + f*x)] + 8*C*m^3*Sin[3*(e + f*x)]))/(2*f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
3,1,160,180,0.8066332,"\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","Integrate[(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]]*(A + C*Sin[e + f*x]^2),x]","-\frac{\sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(-8 A m^2-32 A m-30 A+C \left(4 m^2+8 m+3\right) \cos (2 (e+f x))+8 C (2 m+1) \sin (e+f x)-4 C m^2-8 C m-19 C\right)}{f (2 m+1) (2 m+3) (2 m+5) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"-(((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(-30*A - 19*C - 32*A*m - 8*C*m - 8*A*m^2 - 4*C*m^2 + C*(3 + 8*m + 4*m^2)*Cos[2*(e + f*x)] + 8*C*(1 + 2*m)*Sin[e + f*x]))/(f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))","A",1
4,-1,0,123,180.0016009,"\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","Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/Sqrt[c - c*Sin[e + f*x]],x]","\text{\$Aborted}","\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,"$Aborted","F",-1
5,1,4061,202,7.1304943,"\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","Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\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,"(2^(-1/2 - 2*m)*C*(-(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Cos[(-e + Pi/2 - f*x)/2]]) + Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*m*(c - c*Sin[e + f*x])^(3/2)) - (C*(((-1/2*I)*(((-I)*2^(1 - 2*m)*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(1 + 2*m)*Hypergeometric2F1[1, 1/2 + m, 1/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(1 + 2*m) + (I*2^(1 - 2*m)*(1 + E^(I*(-e + Pi/2 - f*x)))^2*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(-1 + 2*m)*Hypergeometric2F1[1, 3/2 + m, 3/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(-1 + 2*m)))/Sqrt[2] - (Sqrt[2]*Cos[(-e + Pi/2 - f*x)/2]^(2 + 2*m)*Hypergeometric2F1[1/2, (2 + 2*m)/2, (4 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2])/((2 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2]))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(c - c*Sin[e + f*x])^(3/2)) + (C*(((I/2)*(((-I)*2^(1 - 2*m)*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(1 + 2*m)*Hypergeometric2F1[1, 1/2 + m, 1/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(1 + 2*m) + (I*2^(1 - 2*m)*(1 + E^(I*(-e + Pi/2 - f*x)))^2*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(-1 + 2*m)*Hypergeometric2F1[1, 3/2 + m, 3/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(-1 + 2*m)))/Sqrt[2] - (Sqrt[2]*Cos[(-e + Pi/2 - f*x)/2]^(2 + 2*m)*Hypergeometric2F1[1/2, (2 + 2*m)/2, (4 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2])/((2 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2]))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(c - c*Sin[e + f*x])^(3/2)) - ((A + C)*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/(8*Sqrt[2]*f*(c - c*Sin[e + f*x])^(3/2)*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^3*(-1/8*(m*Cos[(-e + Pi/2 - f*x)/4]*(Cos[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/4]*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/Sqrt[2] + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Tan[(-e + Pi/2 - f*x)/4])/2 + m*AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^3 + (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/2*(m*AppellF1[2, 1 - 2*m, 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[2, -2*m, 1 + 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/2) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m)*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((m*AppellF1[2, 1 - 2*m, 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2 + (m*AppellF1[2, -2*m, 1 + 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Csc[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sec[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(2^(2*m)*(1 + 2*m)) + (2^(1 - 2*m)*(-1/2*((1 + 2*m)*AppellF1[2 + 2*m, 2*m, 2, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2 + 2*m) - (m*(1 + 2*m)*AppellF1[2 + 2*m, 1 + 2*m, 1, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(2 + 2*m)))*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m) - (2^(2 - 2*m)*m*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(-1 + 2*m))/(1 + 2*m)))/(8*Sqrt[2])))","C",0
6,1,8316,207,7.0813991,"\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","Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c - c*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\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,"Result too large to show","C",0
7,1,190,167,0.6216619,"\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","Integrate[(A + C*Sin[e + f*x]^2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-\left((A-3 C) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+(A+C) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+A+C\right)}{2 f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{3/2}}","\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 - (A - 3*C)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (A + C)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(2*f*Sqrt[a*(1 + Sin[e + f*x])]*(c - c*Sin[e + f*x])^(3/2))","A",1
8,1,4861,257,15.3244312,"\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","Integrate[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2),x]","\text{Result too large to show}","\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,"(4*(16*C*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 128*C*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - A*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - C*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 80*C*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 64*C*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*(2*A*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*n) + C*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*n) + C*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[2*(-e + Pi/2 - f*x)]*Sin[(-e + Pi/2 - f*x)/2]^(2*n))*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*Tan[(-e + Pi/2 - f*x)/4])/(f*(1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((-8*m*(16*C*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 128*C*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - A*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - C*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 80*C*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 64*C*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m))/(1 + 2*n) - (2*(16*C*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 128*C*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - A*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - C*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 80*C*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 64*C*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n))/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (8*n*(16*C*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 128*C*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - A*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - C*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 80*C*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 64*C*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(-1 + 2*n)*Tan[(-e + Pi/2 - f*x)/4])/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (8*m*(16*C*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 128*C*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - A*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - C*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 80*C*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 64*C*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(1 + 2*n)*Tan[(-e + Pi/2 - f*x)/4])/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (8*(m + n)*(16*C*AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 128*C*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - A*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - C*AppellF1[1/2 + n, -2*m, 1 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 80*C*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 64*C*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]^2)/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (8*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]*(-(A*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 1 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(1 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 2 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n)))) - C*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 1 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(1 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 2 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n))) - 80*C*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 3 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(3 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 4 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n))) - 64*C*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 5 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(5 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 6 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n))) + 16*C*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 2*(1 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(1 + m + n)*AppellF1[3/2 + n, -2*m, 1 + 2*(1 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) + 128*C*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 2*(2 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(2 + m + n)*AppellF1[3/2 + n, -2*m, 1 + 2*(2 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n))))/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
9,1,1873,366,8.5462207,"\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","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2),x]","-\frac{\cos ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(-\frac{6 C (c+d) F_1\left(\frac{1}{2};-m-\frac{3}{2},-n;\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m+3}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m+\frac{3}{2}} \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^n \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2} (-2 m-4)+\frac{1}{2}}}{\left(4 d n F_1\left(\frac{3}{2};-m-\frac{3}{2},1-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+(c+d) (2 m+3) F_1\left(\frac{3}{2};-m-\frac{1}{2},-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3 (c+d) F_1\left(\frac{1}{2};-m-\frac{3}{2},-n;\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)}-4 C F_1\left(\frac{3}{2};\frac{1}{2} (-2 m-1),-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m+1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^3\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{\frac{1}{2} (-2 m-1)+\frac{1}{2} (2 m+1)} \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^n \left(\frac{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d}{c+d}\right)^{-n} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2} (-2 m-1)}+\frac{2}{5} C F_1\left(\frac{5}{2};\frac{1}{2} (1-2 m),-n;\frac{7}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m-1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^5\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{\frac{1}{2} (1-2 m)+\frac{1}{2} (2 m-1)} \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^n \left(\frac{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d}{c+d}\right)^{-n} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2} (1-2 m)}+\frac{12 A (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n;\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m-1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m-\frac{1}{2}} \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^n \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2}-m}}{3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n;\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)-\left(4 d n F_1\left(\frac{3}{2};\frac{1}{2}-m,1-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+(c+d) (2 m-1) F_1\left(\frac{3}{2};\frac{3}{2}-m,-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}+\frac{6 C (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n;\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m-1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m-\frac{1}{2}} \left(-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d\right)^n \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2}-m}}{3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-n;\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)-\left(4 d n F_1\left(\frac{3}{2};\frac{1}{2}-m,1-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+(c+d) (2 m-1) F_1\left(\frac{3}{2};\frac{3}{2}-m,-n;\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}\right) (\sin (e+f x) a+a)^m}{2 f}","\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,"-1/2*(((2*C*AppellF1[5/2, (1 - 2*m)/2, -n, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(5*((c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d))^n) - (4*C*AppellF1[3/2, (-1 - 2*m)/2, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/((c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d))^n - (6*C*(c + d)*AppellF1[1/2, -3/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-4 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(3/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(-3*(c + d)*AppellF1[1/2, -3/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (4*d*n*AppellF1[3/2, -3/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(3 + 2*m)*AppellF1[3/2, -1/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (12*A*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(3*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (4*d*n*AppellF1[3/2, 1/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (6*C*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(3*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (4*d*n*AppellF1[3/2, 1/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2))*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m))","B",0
10,1,7530,392,59.053772,"\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","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-2 - m)*(A + C*Sin[e + f*x]^2),x]","\text{Result too large to show}","-\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)-\left(c^2 (2 C m+C)\right)\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,"Result too large to show","B",0
11,1,4492,385,18.1579421,"\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","Integrate[(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2),x]","\text{Result too large to show}","\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,"(((4*A*d*AppellF1[3/2, (1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) + (C*d*AppellF1[3/2, (1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*c*C*AppellF1[5/2, (1 - 2*m)/2, -1/2, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(5*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) + (C*d*AppellF1[7/2, (1 - 2*m)/2, -1/2, 9/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^7*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(7*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) + (4*c*C*AppellF1[3/2, (-1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (3*C*d*AppellF1[5/2, (-1 - 2*m)/2, -1/2, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (5*C*d*AppellF1[3/2, (-3 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-3 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-3 - 2*m)/2 + (3 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (3*C*d*(c + d)*AppellF1[1/2, -5/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(5 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-6 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(5/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -5/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -5/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(5 + 2*m)*AppellF1[3/2, -3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (6*c*C*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-4 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(3/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -3/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(3 + 2*m)*AppellF1[3/2, -1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (12*A*d*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-2 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(1 + 2*m)*AppellF1[3/2, 1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (9*C*d*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-2 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(1 + 2*m)*AppellF1[3/2, 1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (12*A*c*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (6*c*C*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2))*(a + a*Sin[e + f*x])^m)/(2*f*Cos[(-e + Pi/2 - f*x)/2]^(2*m))","B",0
12,1,1874,375,7.7693688,"\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","Integrate[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]]*(A + C*Sin[e + f*x]^2),x]","\frac{\cos ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\frac{6 C (c+d) F_1\left(\frac{1}{2};-m-\frac{3}{2},-\frac{1}{2};\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m+3}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m+\frac{3}{2}} \sqrt{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2} (-2 m-4)+\frac{1}{2}}}{\left(2 d F_1\left(\frac{3}{2};-m-\frac{3}{2},\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+(c+d) (2 m+3) F_1\left(\frac{3}{2};-m-\frac{1}{2},-\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-3 (c+d) F_1\left(\frac{1}{2};-m-\frac{3}{2},-\frac{1}{2};\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)}+\frac{4 C F_1\left(\frac{3}{2};\frac{1}{2} (-2 m-1),-\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m+1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^3\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{\frac{1}{2} (-2 m-1)+\frac{1}{2} (2 m+1)} \sqrt{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2} (-2 m-1)}}{\sqrt{\frac{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d}{c+d}}}-\frac{2 C F_1\left(\frac{5}{2};\frac{1}{2} (1-2 m),-\frac{1}{2};\frac{7}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m-1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^5\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{\frac{1}{2} (1-2 m)+\frac{1}{2} (2 m-1)} \sqrt{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2} (1-2 m)}}{5 \sqrt{\frac{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d}{c+d}}}-\frac{12 A (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m-1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m-\frac{1}{2}} \sqrt{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2}-m}}{3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)-\left(2 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+(c+d) (2 m-1) F_1\left(\frac{3}{2};\frac{3}{2}-m,-\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}-\frac{6 C (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^{2 m-1}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{m-\frac{1}{2}} \sqrt{-2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+c+d} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)^{\frac{1}{2}-m}}{3 (c+d) F_1\left(\frac{1}{2};\frac{1}{2}-m,-\frac{1}{2};\frac{3}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)-\left(2 d F_1\left(\frac{3}{2};\frac{1}{2}-m,\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+(c+d) (2 m-1) F_1\left(\frac{3}{2};\frac{3}{2}-m,-\frac{1}{2};\frac{5}{2};\sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{2 d \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}\right) (\sin (e+f x) a+a)^m}{2 f}","\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*AppellF1[5/2, (1 - 2*m)/2, -1/2, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(5*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) + (4*C*AppellF1[3/2, (-1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (6*C*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-4 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(3/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -3/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(3 + 2*m)*AppellF1[3/2, -1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (12*A*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (6*C*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2))*(a + a*Sin[e + f*x])^m)/(2*f*Cos[(-e + Pi/2 - f*x)/2]^(2*m))","B",0
13,1,9629,365,31.2351197,"\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","Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/Sqrt[c + d*Sin[e + f*x]],x]","\text{Result too large to show}","\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,"Result too large to show","B",0
14,1,19634,413,32.1113707,"\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","Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(3/2),x]","\text{Result too large to show}","\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,"Result too large to show","B",0
15,1,25065,424,32.4558181,"\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","Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\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,"Result too large to show","B",0
16,1,196,174,0.7331117,"\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","Integrate[(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{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left((-A+B+3 C) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+(A-B+C) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+A+B+C\right)}{2 f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{3/2}}","\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 + (-A + B + 3*C)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (A - B + C)*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(2*f*Sqrt[a*(1 + Sin[e + f*x])]*(c - c*Sin[e + f*x])^(3/2))","A",1
17,1,6226,269,16.5039422,"\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","Integrate[(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]","\text{Result too large to show}","\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,"Result too large to show","C",0
18,1,1029,435,7.2347325,"\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","Integrate[(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{(a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{5/2} \left(\frac{\left(64 A m^4-16 B m^4+16 C m^4+896 A m^3-208 B m^3+224 C m^3+5280 A m^2-832 B m^2+1416 C m^2+15648 A m-4140 B m+648 C m+18900 A-14175 B+12285 C\right) \left(\left(\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)+\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(64 A m^4-16 B m^4+16 C m^4+896 A m^3-208 B m^3+224 C m^3+5280 A m^2-832 B m^2+1416 C m^2+15648 A m-4140 B m+648 C m+18900 A-14175 B+12285 C\right) \left(\left(\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)+\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+1) (2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(48 A m^3-24 B m^3+16 C m^3+584 A m^2-316 B m^2+200 C m^2+2356 A m-1706 B m+828 C m+3150 A-3465 B+3150 C\right) \left(\left(\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(48 A m^3-24 B m^3+16 C m^3+584 A m^2-316 B m^2+200 C m^2+2356 A m-1706 B m+828 C m+3150 A-3465 B+3150 C\right) \left(\left(\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{3}{2} (e+f x)\right)-\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(8 A m^2-12 B m^2+8 C m^2+64 A m-124 B m+88 C m+126 A-315 B+378 C\right) \left(\left(-\frac{1}{8}+\frac{i}{8}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{8}+\frac{i}{8}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7) (2 m+9)}+\frac{\left(8 A m^2-12 B m^2+8 C m^2+64 A m-124 B m+88 C m+126 A-315 B+378 C\right) \left(\left(-\frac{1}{8}-\frac{i}{8}\right) \cos \left(\frac{5}{2} (e+f x)\right)-\left(\frac{1}{8}-\frac{i}{8}\right) \sin \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(4 m B+18 B-45 C-6 C m) \left(\left(\frac{1}{16}-\frac{i}{16}\right) \cos \left(\frac{7}{2} (e+f x)\right)-\left(\frac{1}{16}+\frac{i}{16}\right) \sin \left(\frac{7}{2} (e+f x)\right)\right)}{(2 m+7) (2 m+9)}+\frac{(4 m B+18 B-45 C-6 C m) \left(\left(\frac{1}{16}+\frac{i}{16}\right) \cos \left(\frac{7}{2} (e+f x)\right)-\left(\frac{1}{16}-\frac{i}{16}\right) \sin \left(\frac{7}{2} (e+f x)\right)\right)}{(2 m+7) (2 m+9)}+\frac{\left(\frac{1}{16}+\frac{i}{16}\right) C \cos \left(\frac{9}{2} (e+f x)\right)+\left(\frac{1}{16}-\frac{i}{16}\right) C \sin \left(\frac{9}{2} (e+f x)\right)}{2 m+9}+\frac{\left(\frac{1}{16}-\frac{i}{16}\right) C \cos \left(\frac{9}{2} (e+f x)\right)+\left(\frac{1}{16}+\frac{i}{16}\right) C \sin \left(\frac{9}{2} (e+f x)\right)}{2 m+9}\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\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{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{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,"((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((18900*A - 14175*B + 12285*C + 15648*A*m - 4140*B*m + 648*C*m + 5280*A*m^2 - 832*B*m^2 + 1416*C*m^2 + 896*A*m^3 - 208*B*m^3 + 224*C*m^3 + 64*A*m^4 - 16*B*m^4 + 16*C*m^4)*((1/8 + I/8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((18900*A - 14175*B + 12285*C + 15648*A*m - 4140*B*m + 648*C*m + 5280*A*m^2 - 832*B*m^2 + 1416*C*m^2 + 896*A*m^3 - 208*B*m^3 + 224*C*m^3 + 64*A*m^4 - 16*B*m^4 + 16*C*m^4)*((1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((3150*A - 3465*B + 3150*C + 2356*A*m - 1706*B*m + 828*C*m + 584*A*m^2 - 316*B*m^2 + 200*C*m^2 + 48*A*m^3 - 24*B*m^3 + 16*C*m^3)*((1/8 - I/8)*Cos[(3*(e + f*x))/2] - (1/8 + I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((3150*A - 3465*B + 3150*C + 2356*A*m - 1706*B*m + 828*C*m + 584*A*m^2 - 316*B*m^2 + 200*C*m^2 + 48*A*m^3 - 24*B*m^3 + 16*C*m^3)*((1/8 + I/8)*Cos[(3*(e + f*x))/2] - (1/8 - I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((126*A - 315*B + 378*C + 64*A*m - 124*B*m + 88*C*m + 8*A*m^2 - 12*B*m^2 + 8*C*m^2)*((-1/8 + I/8)*Cos[(5*(e + f*x))/2] - (1/8 + I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((126*A - 315*B + 378*C + 64*A*m - 124*B*m + 88*C*m + 8*A*m^2 - 12*B*m^2 + 8*C*m^2)*((-1/8 - I/8)*Cos[(5*(e + f*x))/2] - (1/8 - I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((18*B - 45*C + 4*B*m - 6*C*m)*((1/16 - I/16)*Cos[(7*(e + f*x))/2] - (1/16 + I/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((18*B - 45*C + 4*B*m - 6*C*m)*((1/16 + I/16)*Cos[(7*(e + f*x))/2] - (1/16 - I/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((1/16 + I/16)*C*Cos[(9*(e + f*x))/2] + (1/16 - I/16)*C*Sin[(9*(e + f*x))/2])/(9 + 2*m) + ((1/16 - I/16)*C*Cos[(9*(e + f*x))/2] + (1/16 + I/16)*C*Sin[(9*(e + f*x))/2])/(9 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)","C",1
19,1,306,322,5.111873,"\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","Integrate[(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{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(-(2 m+1) \left(4 A \left(4 m^2+24 m+35\right)-4 B \left(4 m^2+32 m+63\right)+C \left(12 m^2+80 m+253\right)\right) \sin (e+f x)+32 A m^3+272 A m^2+760 A m+700 A+2 \left(4 m^2+8 m+3\right) (B (2 m+7)-C (2 m+13)) \cos (2 (e+f x))-16 B m^3-120 B m^2-380 B m-546 B+8 C m^3 \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))+46 C m \sin (3 (e+f x))+15 C \sin (3 (e+f x))+16 C m^3+136 C m^2+284 C m+494 C\right)}{2 f (2 m+1) (2 m+3) (2 m+5) (2 m+7) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\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,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(700*A - 546*B + 494*C + 760*A*m - 380*B*m + 284*C*m + 272*A*m^2 - 120*B*m^2 + 136*C*m^2 + 32*A*m^3 - 16*B*m^3 + 16*C*m^3 + 2*(3 + 8*m + 4*m^2)*(B*(7 + 2*m) - C*(13 + 2*m))*Cos[2*(e + f*x)] - (1 + 2*m)*(4*A*(35 + 24*m + 4*m^2) - 4*B*(63 + 32*m + 4*m^2) + C*(253 + 80*m + 12*m^2))*Sin[e + f*x] + 15*C*Sin[3*(e + f*x)] + 46*C*m*Sin[3*(e + f*x)] + 36*C*m^2*Sin[3*(e + f*x)] + 8*C*m^3*Sin[3*(e + f*x)]))/(2*f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
20,1,177,197,1.0629197,"\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","Integrate[(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{\sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (a (\sin (e+f x)+1))^m \left(8 A m^2+32 A m+30 A+2 (2 m+1) (2 B m+5 B-4 C) \sin (e+f x)-8 B m-20 B-C \left(4 m^2+8 m+3\right) \cos (2 (e+f x))+4 C m^2+8 C m+19 C\right)}{f (2 m+1) (2 m+3) (2 m+5) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(30*A - 20*B + 19*C + 32*A*m - 8*B*m + 8*C*m + 8*A*m^2 + 4*C*m^2 - C*(3 + 8*m + 4*m^2)*Cos[2*(e + f*x)] + 2*(1 + 2*m)*(5*B - 4*C + 2*B*m)*Sin[e + f*x]))/(f*(1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
21,-1,0,170,180.0044958,"\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","Integrate[((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]","\text{\$Aborted}","\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,"$Aborted","F",-1
22,1,4066,216,7.1712935,"\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","Integrate[((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]","\text{Result too large to show}","\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,"(2^(-3/2 - 2*m)*(B + 2*C)*(-(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Cos[(-e + Pi/2 - f*x)/2]]) + Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*m*(c - c*Sin[e + f*x])^(3/2)) - (C*(((-1/2*I)*(((-I)*2^(1 - 2*m)*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(1 + 2*m)*Hypergeometric2F1[1, 1/2 + m, 1/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(1 + 2*m) + (I*2^(1 - 2*m)*(1 + E^(I*(-e + Pi/2 - f*x)))^2*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(-1 + 2*m)*Hypergeometric2F1[1, 3/2 + m, 3/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(-1 + 2*m)))/Sqrt[2] - (Sqrt[2]*Cos[(-e + Pi/2 - f*x)/2]^(2 + 2*m)*Hypergeometric2F1[1/2, (2 + 2*m)/2, (4 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2])/((2 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2]))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(c - c*Sin[e + f*x])^(3/2)) + (C*(((I/2)*(((-I)*2^(1 - 2*m)*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(1 + 2*m)*Hypergeometric2F1[1, 1/2 + m, 1/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(1 + 2*m) + (I*2^(1 - 2*m)*(1 + E^(I*(-e + Pi/2 - f*x)))^2*((1 + E^(I*(-e + Pi/2 - f*x)))/E^((I/2)*(-e + Pi/2 - f*x)))^(-1 + 2*m)*Hypergeometric2F1[1, 3/2 + m, 3/2 - m, -E^(I*(-e + Pi/2 - f*x))])/(-1 + 2*m)))/Sqrt[2] - (Sqrt[2]*Cos[(-e + Pi/2 - f*x)/2]^(2 + 2*m)*Hypergeometric2F1[1/2, (2 + 2*m)/2, (4 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*Sin[(-e + Pi/2 - f*x)/2])/((2 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2]))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(c - c*Sin[e + f*x])^(3/2)) - ((A + B + C)*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/(8*Sqrt[2]*f*(c - c*Sin[e + f*x])^(3/2)*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^3*(-1/8*(m*Cos[(-e + Pi/2 - f*x)/4]*(Cos[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/4]*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/Sqrt[2] + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Tan[(-e + Pi/2 - f*x)/4])/2 + m*AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^3 + (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/2*(m*AppellF1[2, 1 - 2*m, 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[2, -2*m, 1 + 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/2) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m)*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((m*AppellF1[2, 1 - 2*m, 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2 + (m*AppellF1[2, -2*m, 1 + 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Csc[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sec[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(2^(2*m)*(1 + 2*m)) + (2^(1 - 2*m)*(-1/2*((1 + 2*m)*AppellF1[2 + 2*m, 2*m, 2, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2 + 2*m) - (m*(1 + 2*m)*AppellF1[2 + 2*m, 1 + 2*m, 1, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(2 + 2*m)))*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m) - (2^(2 - 2*m)*m*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(-1 + 2*m))/(1 + 2*m)))/(8*Sqrt[2])))","C",0
23,1,8321,230,7.3155482,"\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","Integrate[((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]","\text{Result too large to show}","-\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,"Result too large to show","C",0
24,1,1087,232,17.3829865,"\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","Integrate[(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{2^{-m-5} (2 m-3) \cot ^3\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-2)} (\sin (e+f x) a+a)^m \left(2 A+C+C \cos \left(2 \left(-e-f x+\frac{\pi }{2}\right)\right)+2 B \sin (e+f x)\right) (c-c \sin (e+f x))^{-m-2} \left(\frac{64 C F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \tan ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)}{1-2 m}-\frac{\left((A+B+C) (2 m-1) \, _2F_1\left(\frac{3}{2}-m,-2 m;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+(3 A-5 B-13 C) (2 m-3) \, _2F_1\left(\frac{1}{2}-m,-2 m;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right) \tan ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)}{4 m^2-8 m+3}-\frac{(3 A-5 B-13 C) \, _2F_1\left(-m-\frac{1}{2},-2 m;\frac{1}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)}{2 m+1}-\frac{(A+B+C) \, _2F_1\left(-m-\frac{3}{2},-2 m;-m-\frac{1}{2};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)}{2 m+3}\right)}{f \left(-3 A \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}+3 C \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}+2 A m \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}-2 C m \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}-3 B \sin (e+f x) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}-6 C \sin (e+f x) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}+2 B m \sin (e+f x) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}+4 C m \sin (e+f x) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}+256 C m F_1\left(\frac{3}{2}-m;1-2 m,1;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^6\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+128 C F_1\left(\frac{3}{2}-m;-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^6\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+64 C (2 m-3) F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)}","\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^(-5 - m)*(-3 + 2*m)*Cot[(-e + Pi/2 - f*x)/4]^3*(a + a*Sin[e + f*x])^m*(2*A + C + C*Cos[2*(-e + Pi/2 - f*x)] + 2*B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-2 - m)*(-(((A + B + C)*Hypergeometric2F1[-3/2 - m, -2*m, -1/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2])/(3 + 2*m)) - ((3*A - 5*B - 13*C)*Hypergeometric2F1[-1/2 - m, -2*m, 1/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2)/(1 + 2*m) + (64*C*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^4)/(1 - 2*m) - (Tan[(-e + Pi/2 - f*x)/4]^4*((3*A - 5*B - 13*C)*(-3 + 2*m)*Hypergeometric2F1[1/2 - m, -2*m, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2] + (A + B + C)*(-1 + 2*m)*Hypergeometric2F1[3/2 - m, -2*m, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2))/(3 - 8*m + 4*m^2)))/(f*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-2 - m))*(64*C*(-3 + 2*m)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2*Sin[(-e + Pi/2 - f*x)/4]^4 + 256*C*m*AppellF1[3/2 - m, 1 - 2*m, 1, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^6 + 128*C*AppellF1[3/2 - m, -2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^6 - 3*A*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + 3*C*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + 2*A*m*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) - 2*C*m*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) - 3*B*Sin[e + f*x]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) - 6*C*Sin[e + f*x]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + 2*B*m*Sin[e + f*x]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + 4*C*m*Sin[e + f*x]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
25,1,2572,383,8.7671571,"\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","Integrate[(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]","\text{Result too large to show}","\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,"-1/2*(((-4*B*AppellF1[3/2, (1 - 2*m)/2, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(3*((c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d))^n) + (2*C*AppellF1[5/2, (1 - 2*m)/2, -n, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(5*((c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d))^n) - (4*C*AppellF1[3/2, (-1 - 2*m)/2, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/((c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d))^n - (6*C*(c + d)*AppellF1[1/2, -3/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-4 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(3/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(-3*(c + d)*AppellF1[1/2, -3/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (4*d*n*AppellF1[3/2, -3/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(3 + 2*m)*AppellF1[3/2, -1/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (12*B*(c + d)*AppellF1[1/2, -1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-2 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(-3*(c + d)*AppellF1[1/2, -1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (4*d*n*AppellF1[3/2, -1/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(1 + 2*m)*AppellF1[3/2, 1/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (12*A*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(3*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (4*d*n*AppellF1[3/2, 1/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (6*C*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)^n)/(3*(c + d)*AppellF1[1/2, 1/2 - m, -n, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (4*d*n*AppellF1[3/2, 1/2 - m, 1 - n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -n, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2))*(a + a*Sin[e + f*x])^m)/(f*Cos[(-e + Pi/2 - f*x)/2]^(2*m))","B",0
26,1,5175,410,40.0775013,"\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","Integrate[(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]","\text{Result too large to show}","-\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))-\left(c^2 (2 C m+C)\right)\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,"Result too large to show","B",0
27,1,6591,406,9.6280512,"\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","Integrate[(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]","\text{Result too large to show}","\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,"Result too large to show","B",0
28,1,2574,396,8.5178532,"\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","Integrate[(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]","\text{Result too large to show}","\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,"(((4*B*AppellF1[3/2, (1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) - (2*C*AppellF1[5/2, (1 - 2*m)/2, -1/2, 7/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^5*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((1 - 2*m)/2 + (-1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(5*Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]) + (4*C*AppellF1[3/2, (-1 - 2*m)/2, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]^3*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^((-1 - 2*m)/2 + (1 + 2*m)/2)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/Sqrt[(c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (6*C*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(3 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-4 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(3/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -3/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -3/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(3 + 2*m)*AppellF1[3/2, -1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) + (12*B*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 + (-2 - 2*m)/2)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(-3*(c + d)*AppellF1[1/2, -1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (2*d*AppellF1[3/2, -1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(1 + 2*m)*AppellF1[3/2, 1/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (12*A*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2) - (6*C*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/2]^2)^(1/2 - m)*Sin[(-e + Pi/2 - f*x)/2]*(1 - Sin[(-e + Pi/2 - f*x)/2]^2)^(-1/2 + m)*Sqrt[c + d - 2*d*Sin[(-e + Pi/2 - f*x)/2]^2])/(3*(c + d)*AppellF1[1/2, 1/2 - m, -1/2, 3/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - (2*d*AppellF1[3/2, 1/2 - m, 1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)] + (c + d)*(-1 + 2*m)*AppellF1[3/2, 3/2 - m, -1/2, 5/2, Sin[(-e + Pi/2 - f*x)/2]^2, (2*d*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Sin[(-e + Pi/2 - f*x)/2]^2))*(a + a*Sin[e + f*x])^m)/(2*f*Cos[(-e + Pi/2 - f*x)/2]^(2*m))","B",0
29,1,9760,389,32.5079205,"\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","Integrate[((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]","\text{Result too large to show}","\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,"Result too large to show","B",0
30,1,31369,433,33.0511209,"\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","Integrate[((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]","\text{Result too large to show}","-\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,"Result too large to show","B",0
31,1,20654,451,33.4758427,"\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","Integrate[((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]","\text{Result too large to show}","\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,"Result too large to show","B",0
32,1,92,81,0.2140161,"\int (a+b \sin (c+d x)) \left(A+B \sin (c+d x)+C \sin ^2(c+d x)\right) \, dx","Integrate[(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x] + C*Sin[c + d*x]^2),x]","\frac{-3 \cos (c+d x) (4 a B+4 A b+3 b C)+12 a A d x-3 a C \sin (2 (c+d x))+6 a c C+6 a C d x-3 b B \sin (2 (c+d x))+6 b B c+6 b B d x+b C \cos (3 (c+d x))}{12 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,"(6*b*B*c + 6*a*c*C + 12*a*A*d*x + 6*b*B*d*x + 6*a*C*d*x - 3*(4*A*b + 4*a*B + 3*b*C)*Cos[c + d*x] + b*C*Cos[3*(c + d*x)] - 3*b*B*Sin[2*(c + d*x)] - 3*a*C*Sin[2*(c + d*x)])/(12*d)","A",1
33,1,97,117,0.8752463,"\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","Integrate[((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}{4} (-2 e-2 f x+\pi )\right|2\right) (3 a B+3 A b+b C)+6 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right) (a (C-A)+b B)+\frac{2 \cos (e+f x) (3 a A+b C \sin (e+f x))}{\sqrt{\sin (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,"-1/3*(6*(b*B + a*(-A + C))*EllipticE[(-2*e + Pi - 2*f*x)/4, 2] + 2*(3*A*b + 3*a*B + b*C)*EllipticF[(-2*e + Pi - 2*f*x)/4, 2] + (2*Cos[e + f*x]*(3*a*A + b*C*Sin[e + f*x]))/Sqrt[Sin[e + f*x]])/f","A",1
34,0,0,48,46.5225718,"\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","Integrate[(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,"Integrate[(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",-1