Integrand size = 27, antiderivative size = 445 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {12 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{77 d^2 f}-\frac {12 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{77 d^2 f}-\frac {12 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{77 d^2 f}+\frac {24 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{11 d^2 f}-\frac {2 \cos (e+f x) (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {12 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{77 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {12 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{77 d^3 f \sqrt {c+d \sin (e+f x)}} \]
-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*c os(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*(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*(c+3*d)*(4*c^4-45*c^3*d+309*c^2*d^2+525*c*d^3+231*d^4)*(s in(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(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)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/ 4*Pi+1/2*f*x)*EllipticF(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))/(c+d))^(1/2)/d^3/f/(c+d*sin(f*x+e))^(1/2)
Time = 0.90 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.77 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\frac {3 \left (-32 \left (d^2 \left (c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^5-33 c^4 d+174 c^3 d^2+1452 c^2 d^3+1806 c d^4+693 d^5\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (2 \left (32 c^4-264 c^3 d-8994 c^2 d^2-13926 c d^3-5859 d^4\right ) \cos (e+f x)+d^2 \left (452 c^2+2508 c d+1701 d^2\right ) \cos (3 (e+f x))-63 d^4 \cos (5 (e+f x))-4 d \left (6 c^3+990 c^2 d+2401 c d^2+1155 d^3\right ) \sin (2 (e+f x))+14 d^3 (23 c+33 d) \sin (4 (e+f x))\right )\right )}{616 d^3 f \sqrt {c+d \sin (e+f x)}} \]
(3*(-32*(d^2*(c^4 + 858*c^3*d + 1668*c^2*d^2 + 1254*c*d^3 + 315*d^4)*Ellip ticF[(-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)])))/(616*d^3*f*Sqrt[c + d*Sin[e + f*x]])
Time = 2.65 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.09, 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 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {2 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}+\frac {5}{7} \left (\frac {2 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {3}{5} \left (\frac {2 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )\right )\right )}{9 d}\right )}{11 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
(-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)
3.5.96.3.1 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[(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. \(2(501)=1002\).
Time = 43.33 (sec) , antiderivative size = 1926, normalized size of antiderivative = 4.33
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1926\) |
| parts | \(\text {Expression too large to display}\) | \(4388\) |
2/693*a^3*(4*c^5*d^2-1096*c^3*d^4-1584*c^2*d^5-33*c^4*d^3-630*c*d^6-120*(( c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+ e)+1)/(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)*(-(sin(f*x+e)-1)*d/(c+d ))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d) )^(1/2),((c-d)/(c+d))^(1/2))*c*d^6+66*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(si n(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d *sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^6*d-340*((c+d*sin(f*x+e)) /(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1 /2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^5*d^2- 2970*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(s in(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/ (c+d))^(1/2))*c^4*d^3-3264*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1) *d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e) )/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^4+1518*((c+d*sin(f*x+e))/(c-d))^ (1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*Elli pticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^5+3612*((c +d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e )+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^( 1/2))*c*d^6+8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.86 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (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}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (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}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (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}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-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}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (63 \, a^{3} d^{6} \cos \left (f x + e\right )^{5} - {\left (113 \, a^{3} c^{2} d^{4} + 627 \, a^{3} c d^{5} + 504 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - {\left (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}\right )} \cos \left (f x + e\right ) - {\left (7 \, {\left (23 \, a^{3} c d^{5} + 33 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c^{3} d^{3} + 165 \, a^{3} c^{2} d^{4} + 427 \, a^{3} c d^{5} + 231 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{2079 \, d^{4} f} \]
-2/2079*(sqrt(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(I*d)*weierstr assPInverse(-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) + sqrt(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 - 237 6*a^3*c*d^5 - 945*a^3*d^6)*sqrt(-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) + 3*sqrt(2)*(4*I*a^3*c^5*d - 33*I*a^3*c^4*d^2 + 1 74*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(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*sqrt(2)*(-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(-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, weierstrassPI nverse(-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 -...
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=a^{3} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{5}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]
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))
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]