Integrand size = 48, antiderivative size = 435 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=-\frac {64 c^3 \left (B \left (45-8 m-4 m^2\right )-C \left (39-16 m+4 m^2\right )-A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) (9+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 \left (B \left (45-8 m-4 m^2\right )-C \left (39-16 m+4 m^2\right )-A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (7+2 m) (9+2 m) \left (15+16 m+4 m^2\right )}-\frac {2 c \left (B \left (45-8 m-4 m^2\right )-C \left (39-16 m+4 m^2\right )-A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac {2 (9 B+2 C+2 B m+4 C m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)} \]
-2*c*(B*(-4*m^2-8*m+45)-C*(4*m^2-16*m+39)-A*(4*m^2+32*m+63))*cos(f*x+e)*(a +a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2)/f/(8*m^3+84*m^2+286*m+315)-2*(2*B* m+4*C*m+9*B+2*C)*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)/f/(4 *m^2+32*m+63)+2*C*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(7/2)/c/f /(9+2*m)-64*c^3*(B*(-4*m^2-8*m+45)-C*(4*m^2-16*m+39)-A*(4*m^2+32*m+63))*co s(f*x+e)*(a+a*sin(f*x+e))^m/f/(32*m^5+400*m^4+1840*m^3+3800*m^2+3378*m+945 )/(c-c*sin(f*x+e))^(1/2)-16*c^2*(B*(-4*m^2-8*m+45)-C*(4*m^2-16*m+39)-A*(4* m^2+32*m+63))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2)/f/(16*m ^4+192*m^3+824*m^2+1488*m+945)
Result contains complex when optimal does not.
Time = 13.21 (sec) , antiderivative size = 1029, normalized size of antiderivative = 2.37 \[ \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=\frac {(a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{5/2} \left (\frac {\left (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\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 )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (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\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 )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (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\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 )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (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\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 )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (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\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 )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {\left (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\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 )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {(18 B-45 C+4 B m-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 )}{(7+2 m) (9+2 m)}+\frac {(18 B-45 C+4 B m-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 )}{(7+2 m) (9+2 m)}+\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 )}{9+2 m}+\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 )}{9+2 m}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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]
((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 - 346 5*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...
Time = 1.49 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.77, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3042, 3518, 27, 3042, 3452, 3042, 3219, 3042, 3219, 3042, 3217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 3518 |
| \(\displaystyle \frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac {2 \int -\frac {1}{2} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2} (a c (C (7-2 m)+A (2 m+9))+a c (2 m B+9 B+2 C+4 C m) \sin (e+f x))dx}{a c (2 m+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2} (a c (C (7-2 m)+A (2 m+9))+a c (2 m B+9 B+2 C+4 C m) \sin (e+f x))dx}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2} (a c (C (7-2 m)+A (2 m+9))+a c (2 m B+9 B+2 C+4 C m) \sin (e+f x))dx}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3452 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2}dx}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{5/2}dx}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \int (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{3/2}dx}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \left (\frac {4 c \int (\sin (e+f x) a+a)^m \sqrt {c-c \sin (e+f x)}dx}{2 m+3}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {-\frac {a c \left (-A \left (4 m^2+32 m+63\right )+B \left (-4 m^2-8 m+45\right )-C \left (4 m^2-16 m+39\right )\right ) \left (\frac {8 c \left (\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) (2 m+3) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)}\right )}{2 m+5}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5)}\right )}{2 m+7}-\frac {2 a c (2 B m+9 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)}}{a c (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}\) |
Int[(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2),x]
(2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f* (9 + 2*m)) + ((-2*a*c*(9*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[ e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)) - (a*c*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*((2*c*Cos[e + f* x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)) + (8*c *((8*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*(3 + 2*m)*Sqrt[ c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c *Sin[e + f*x]])/(f*(3 + 2*m))))/(5 + 2*m)))/(7 + 2*m))/(a*c*(9 + 2*m))
3.1.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] - Simp[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)) Int[( a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (b*B*d*(m + n + 2) - b*c*C*(2*m + 1))* Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]
Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (403) = 806\).
Time = 0.38 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.15 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)+C*sin( f*x+e)^2),x, algorithm="fricas")
2*((16*C*c^2*m^4 + 128*C*c^2*m^3 + 344*C*c^2*m^2 + 352*C*c^2*m + 105*C*c^2 )*cos(f*x + e)^5 + 128*(A + B + C)*c^2*m^2 + (16*(B - C)*c^2*m^4 + 16*(9*B - 14*C)*c^2*m^3 + 8*(52*B - 97*C)*c^2*m^2 + 4*(111*B - 226*C)*c^2*m + 15* (9*B - 19*C)*c^2)*cos(f*x + e)^4 + 256*(4*A + B - 2*C)*c^2*m - (16*(A - 2* B + 3*C)*c^2*m^4 + 16*(10*A - 23*B + 32*C)*c^2*m^3 + 8*(65*A - 169*B + 253 *C)*c^2*m^2 + 4*(150*A - 417*B + 656*C)*c^2*m + 3*(63*A - 180*B + 289*C)*c ^2)*cos(f*x + e)^3 + 96*(21*A - 15*B + 13*C)*c^2 + (16*(A - B + C)*c^2*m^4 + 32*(7*A - 5*B + 7*C)*c^2*m^3 + 8*(133*A - 97*B + 85*C)*c^2*m^2 + 8*(233 *A - 235*B + 233*C)*c^2*m + 3*(231*A - 255*B + 263*C)*c^2)*cos(f*x + e)^2 + 2*(16*(A - B + C)*c^2*m^4 + 192*(A - B + C)*c^2*m^3 + 8*(107*A - 99*B + 107*C)*c^2*m^2 + 16*(109*A - 89*B + 85*C)*c^2*m + 3*(483*A - 435*B + 419*C )*c^2)*cos(f*x + e) + (128*(A + B + C)*c^2*m^2 + (16*C*c^2*m^4 + 128*C*c^2 *m^3 + 344*C*c^2*m^2 + 352*C*c^2*m + 105*C*c^2)*cos(f*x + e)^4 + 256*(4*A + B - 2*C)*c^2*m - (16*(B - 2*C)*c^2*m^4 + 16*(9*B - 22*C)*c^2*m^3 + 32*(1 3*B - 35*C)*c^2*m^2 + 4*(111*B - 314*C)*c^2*m + 15*(9*B - 26*C)*c^2)*cos(f *x + e)^3 + 96*(21*A - 15*B + 13*C)*c^2 - (16*(A - B + C)*c^2*m^4 + 32*(5* A - 7*B + 5*C)*c^2*m^3 + 8*(65*A - 117*B + 113*C)*c^2*m^2 + 24*(25*A - 51* B + 57*C)*c^2*m + 9*(21*A - 45*B + 53*C)*c^2)*cos(f*x + e)^2 - 2*(16*(A - B + C)*c^2*m^4 + 192*(A - B + C)*c^2*m^3 + 8*(99*A - 107*B + 99*C)*c^2*m^2 + 16*(77*A - 97*B + 101*C)*c^2*m + 3*(147*A - 195*B + 211*C)*c^2)*cos(...
Timed out. \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1324 vs. \(2 (403) = 806\).
Time = 0.42 (sec) , antiderivative size = 1324, normalized size of antiderivative = 3.04 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)+C*sin( f*x+e)^2),x, algorithm="maxima")
-2*(((4*m^2 + 24*m + 43)*a^m*c^(5/2) - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*si n(f*x + e)/(cos(f*x + e) + 1) + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e) ^3/(cos(f*x + e) + 1)^3 - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)^4/ (cos(f*x + e) + 1)^4 + (4*m^2 + 24*m + 43)*a^m*c^(5/2)*sin(f*x + e)^5/(cos (f*x + e) + 1)^5)*A*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m*lo g(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((8*m^3 + 36*m^2 + 46*m + 15)* (sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2)) - 2*((4*m^2 + 40*m + 115) *a^m*c^(5/2) - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)/(cos(f* x + e) + 1) + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m*c^(5/2) *sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^ m*c^(5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 2*(4*m^3 + 40*m^2 + 115*m) *a^m*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + (4*m^2 + 40*m + 115)*a^ m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*B*e^(2*m*log(sin(f*x + e)/( cos(f*x + e) + 1) + 1) - m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/( (16*m^4 + 128*m^3 + 344*m^2 + 352*m + (16*m^4 + 128*m^3 + 344*m^2 + 352*m + 105)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 105)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2)) + 4*(2*(4*m^2 + 56*m + 219)*a^m*c^(5/2) - 4*(4...
\[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e)+C*sin( f*x+e)^2),x, algorithm="giac")
integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(-c*sin(f*x + e) + c)^(5 /2)*(a*sin(f*x + e) + a)^m, x)
Time = 23.24 (sec) , antiderivative size = 1253, normalized size of antiderivative = 2.88 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
int((a + a*sin(e + f*x))^m*(c - c*sin(e + f*x))^(5/2)*(A + B*sin(e + f*x) + C*sin(e + f*x)^2),x)
((c - c*sin(e + f*x))^(1/2)*((C*c^2*(a + a*sin(e + f*x))^m*(m*352i + m^2*3 44i + m^3*128i + m^4*16i + 105i))/(8*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x)) ^m*(18900*A - 14175*B + 12285*C + 15648*A*m - 4140*B*m + 648*C*m + 5280*A* m^2 + 896*A*m^3 + 64*A*m^4 - 832*B*m^2 - 208*B*m^3 - 16*B*m^4 + 1416*C*m^2 + 224*C*m^3 + 16*C*m^4))/(4*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^m*(A*18 900i - B*14175i + C*12285i + A*m*15648i - B*m*4140i + C*m*648i + A*m^2*528 0i + A*m^3*896i + A*m^4*64i - B*m^2*832i - B*m^3*208i - B*m^4*16i + C*m^2* 1416i + C*m^3*224i + C*m^4*16i))/(4*f*(m*3378i + m^2*3800i + m^3*1840i + m ^4*400i + m^5*32i + 945i)) + (C*c^2*exp(e*9i + f*x*9i)*(a + a*sin(e + f*x) )^m*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105))/(8*f*(m*3378i + m^2*3800i + m^3*1840i + m^4*400i + m^5*32i + 945i)) - (c^2*exp(e*7i + f*x*7i)*(a + a *sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(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))/(4*f*(m*3378i + m^2*3800i + m ^3*1840i + m^4*400i + m^5*32i + 945i)) - (c^2*exp(e*2i + f*x*2i)*(a + a*si n(e + f*x))^m*(8*m + 4*m^2 + 3)*(A*126i - B*315i + C*378i + A*m*64i - B*m* 124i + C*m*88i + A*m^2*8i - B*m^2*12i + C*m^2*8i))/(4*f*(m*3378i + m^2*380 0i + m^3*1840i + m^4*400i + m^5*32i + 945i)) + (c^2*exp(e*3i + f*x*3i)*(2* m + 1)*(a + a*sin(e + f*x))^m*(3150*A - 3465*B + 3150*C + 2356*A*m - 17...