Integrand size = 27, antiderivative size = 467 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{693 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{693 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
-4/693*a^3*(4*c^4-33*c^3*d+177*c^2*d^2+561*c*d^3+315*d^4)*cos(f*x+e)*(c+d* sin(f*x+e))^(1/2)/d^2/f-4/693*a^3*(4*c^3-33*c^2*d+182*c*d^2+231*d^3)*cos(f *x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f-4/693*a^3*(4*c^2-33*c*d+189*d^2)*cos(f* x+e)*(c+d*sin(f*x+e))^(5/2)/d^2/f+8/99*a^3*(c-6*d)*cos(f*x+e)*(c+d*sin(f*x +e))^(7/2)/d^2/f-2/11*cos(f*x+e)*(a^3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^(7/ 2)/d/f-4/693*a^3*(c+3*d)*(4*c^4-45*c^3*d+309*c^2*d^2+525*c*d^3+231*d^4)*El lipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e) )^(1/2)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-4/693*a^3*(c^2-d^2)*(4*c^4-33 *c^3*d+177*c^2*d^2+561*c*d^3+315*d^4)*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x ,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/f/(c+d*sin(f* x+e))^(1/2)
Time = 6.32 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.81 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\frac {a^3 (1+\sin (e+f x))^3 \left (-32 \left (d^2 \left (c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^5-33 c^4 d+174 c^3 d^2+1452 c^2 d^3+1806 c d^4+693 d^5\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (2 \left (32 c^4-264 c^3 d-8994 c^2 d^2-13926 c d^3-5859 d^4\right ) \cos (e+f x)+d^2 \left (452 c^2+2508 c d+1701 d^2\right ) \cos (3 (e+f x))-63 d^4 \cos (5 (e+f x))-4 d \left (6 c^3+990 c^2 d+2401 c d^2+1155 d^3\right ) \sin (2 (e+f x))+14 d^3 (23 c+33 d) \sin (4 (e+f x))\right )\right )}{5544 d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(a^3*(1 + Sin[e + f*x])^3*(-32*(d^2*(c^4 + 858*c^3*d + 1668*c^2*d^2 + 1254 *c*d^3 + 315*d^4)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^5 - 33*c^4*d + 174*c^3*d^2 + 1452*c^2*d^3 + 1806*c*d^4 + 693*d^5)*((c + d)* EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*(c + d* Sin[e + f*x])*(2*(32*c^4 - 264*c^3*d - 8994*c^2*d^2 - 13926*c*d^3 - 5859*d ^4)*Cos[e + f*x] + d^2*(452*c^2 + 2508*c*d + 1701*d^2)*Cos[3*(e + f*x)] - 63*d^4*Cos[5*(e + f*x)] - 4*d*(6*c^3 + 990*c^2*d + 2401*c*d^2 + 1155*d^3)* Sin[2*(e + f*x)] + 14*d^3*(23*c + 33*d)*Sin[4*(e + f*x)])))/(5544*d^3*f*(C os[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])
Time = 2.79 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.04, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.926, Rules used = {3042, 3242, 3042, 3447, 3042, 3502, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int (\sin (e+f x) a+a) \left (a^2 (c+9 d)-2 a^2 (c-6 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (\sin (e+f x) a+a) \left (a^2 (c+9 d)-2 a^2 (c-6 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {2 \int (c+d \sin (e+f x))^{5/2} \left (-2 (c-6 d) \sin ^2(e+f x) a^3+(c+9 d) a^3+\left (a^3 (c+9 d)-2 a^3 (c-6 d)\right ) \sin (e+f x)\right )dx}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (c+d \sin (e+f x))^{5/2} \left (-2 (c-6 d) \sin (e+f x)^2 a^3+(c+9 d) a^3+\left (a^3 (c+9 d)-2 a^3 (c-6 d)\right ) \sin (e+f x)\right )dx}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \left (\frac {2 \int -\frac {1}{2} (c+d \sin (e+f x))^{5/2} \left (5 a^3 (c-33 d) d-a^3 \left (4 c^2-33 d c+189 d^2\right ) \sin (e+f x)\right )dx}{9 d}+\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\int (c+d \sin (e+f x))^{5/2} \left (5 a^3 (c-33 d) d-a^3 \left (4 c^2-33 d c+189 d^2\right ) \sin (e+f x)\right )dx}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\int (c+d \sin (e+f x))^{5/2} \left (5 a^3 (c-33 d) d-a^3 \left (4 c^2-33 d c+189 d^2\right ) \sin (e+f x)\right )dx}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {2}{7} \int \frac {5}{2} (c+d \sin (e+f x))^{3/2} \left (3 a^3 d \left (c^2-66 d c-63 d^2\right )-a^3 \left (4 c^3-33 d c^2+182 d^2 c+231 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 a^3 d \left (c^2-66 d c-63 d^2\right )-a^3 \left (4 c^3-33 d c^2+182 d^2 c+231 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 a^3 d \left (c^2-66 d c-63 d^2\right )-a^3 \left (4 c^3-33 d c^2+182 d^2 c+231 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (a^3 d \left (c^3-297 d c^2-497 d^2 c-231 d^3\right )-a^3 \left (4 c^4-33 d c^3+177 d^2 c^2+561 d^3 c+315 d^4\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (a^3 d \left (c^3-297 d c^2-497 d^2 c-231 d^3\right )-a^3 \left (4 c^4-33 d c^3+177 d^2 c^2+561 d^3 c+315 d^4\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (a^3 d \left (c^3-297 d c^2-497 d^2 c-231 d^3\right )-a^3 \left (4 c^4-33 d c^3+177 d^2 c^2+561 d^3 c+315 d^4\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int -\frac {d \left (c^4+858 d c^3+1668 d^2 c^2+1254 d^3 c+315 d^4\right ) a^3+(c+3 d) \left (4 c^4-45 d c^3+309 d^2 c^2+525 d^3 c+231 d^4\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^4+858 d c^3+1668 d^2 c^2+1254 d^3 c+315 d^4\right ) a^3+(c+3 d) \left (4 c^4-45 d c^3+309 d^2 c^2+525 d^3 c+231 d^4\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^4+858 d c^3+1668 d^2 c^2+1254 d^3 c+315 d^4\right ) a^3+(c+3 d) \left (4 c^4-45 d c^3+309 d^2 c^2+525 d^3 c+231 d^4\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}+\frac {5}{7} \left (\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {3}{5} \left (\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )\right )\right )}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(-2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(7/2))/(11* d*f) + (2*((4*a^3*(c - 6*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d* f) - ((2*a^3*(4*c^2 - 33*c*d + 189*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^ (5/2))/(7*f) + (5*((2*a^3*(4*c^3 - 33*c^2*d + 182*c*d^2 + 231*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (3*((2*a^3*(4*c^4 - 33*c^3*d + 1 77*c^2*d^2 + 561*c*d^3 + 315*d^4)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/( 3*f) + ((-2*a^3*(c + 3*d)*(4*c^4 - 45*c^3*d + 309*c^2*d^2 + 525*c*d^3 + 23 1*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x ]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a^3*(c^2 - d^2)*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*EllipticF[(e - Pi/2 + f*x)/ 2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[ e + f*x]]))/3))/5))/7)/(9*d)))/(11*d)
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[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1925\) vs. \(2(440)=880\).
Time = 47.17 (sec) , antiderivative size = 1926, normalized size of antiderivative = 4.12
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1926\) |
| parts | \(\text {Expression too large to display}\) | \(4388\) |
Input:
int((a+sin(f*x+e)*a)^3*(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
2/693*a^3*(-8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1 /2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/ 2),((c-d)/(c+d))^(1/2))*c^7+1386*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+si n(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin (f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^7-2016*((c+d*sin(f*x+e))/(c-d ))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)* EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^7+8*((c+d* sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e ))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/ 2))*c^6*d-72*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/ 2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2 ),((c-d)/(c+d))^(1/2))*c^5*d^2+2128*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1 +sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d* sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d^3+4176*((c+d*sin(f*x+e ))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d)) ^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d ^4-120*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d *(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c- d)/(c+d))^(1/2))*c^2*d^5-4104*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f *x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.75 \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
-2/2079*(2*(8*a^3*c^6 - 66*a^3*c^5*d + 345*a^3*c^4*d^2 + 330*a^3*c^3*d^3 - 1392*a^3*c^2*d^4 - 2376*a^3*c*d^5 - 945*a^3*d^6)*sqrt(1/2*I*d)*weierstras sPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*( 3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + 2*(8*a^3*c^6 - 66*a^3* c^5*d + 345*a^3*c^4*d^2 + 330*a^3*c^3*d^3 - 1392*a^3*c^2*d^4 - 2376*a^3*c* d^5 - 945*a^3*d^6)*sqrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2) /d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin( f*x + e) + 2*I*c)/d) + 6*(4*I*a^3*c^5*d - 33*I*a^3*c^4*d^2 + 174*I*a^3*c^3 *d^3 + 1452*I*a^3*c^2*d^4 + 1806*I*a^3*c*d^5 + 693*I*a^3*d^6)*sqrt(1/2*I*d )*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^ 3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^ 2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 6*(-4*I* a^3*c^5*d + 33*I*a^3*c^4*d^2 - 174*I*a^3*c^3*d^3 - 1452*I*a^3*c^2*d^4 - 18 06*I*a^3*c*d^5 - 693*I*a^3*d^6)*sqrt(-1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3* (4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) + 3*(63*a^3*d^6*cos(f*x + e)^5 - (113 *a^3*c^2*d^4 + 627*a^3*c*d^5 + 504*a^3*d^6)*cos(f*x + e)^3 - (4*a^3*c^4*d^ 2 - 33*a^3*c^3*d^3 - 1209*a^3*c^2*d^4 - 2211*a^3*c*d^5 - 1071*a^3*d^6)*cos (f*x + e) - (7*(23*a^3*c*d^5 + 33*a^3*d^6)*cos(f*x + e)^3 - 3*(a^3*c^3*...
\[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=a^{3} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{5}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**3*(c+d*sin(f*x+e))**(5/2),x)
Output:
a**3*(Integral(c**2*sqrt(c + d*sin(e + f*x)), x) + Integral(3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral(3*c**2*sqrt(c + d*sin(e + f* x))*sin(e + f*x)**2, x) + Integral(c**2*sqrt(c + d*sin(e + f*x))*sin(e + f *x)**3, x) + Integral(d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(3*d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3, x) + Integral(3* d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4, x) + Integral(d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**5, x) + Integral(2*c*d*sqrt(c + d*sin(e + f *x))*sin(e + f*x), x) + Integral(6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f* x)**2, x) + Integral(6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3, x) + Integral(2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4, x))
\[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(5/2), x)
\[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2),x)
Output:
int((a + a*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2), x)
\[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=a^{3} \left (\left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x \right ) c^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{5}d x \right ) d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) c d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) d^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) c^{2}+6 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) c d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) d^{2}+3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) c^{2}+6 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) c d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) d^{2}+3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) c d \right ) \] Input:
int((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x)
Output:
a**3*(int(sqrt(sin(e + f*x)*d + c),x)*c**2 + int(sqrt(sin(e + f*x)*d + c)* sin(e + f*x)**5,x)*d**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**4,x )*c*d + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**4,x)*d**2 + int(sqrt( sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*c**2 + 6*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*c*d + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3 ,x)*d**2 + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*c**2 + 6*int( sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*c*d + int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*d**2 + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x) ,x)*c**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x)*c*d)