Integrand size = 27, antiderivative size = 378 \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {4 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{315 d^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
4/315*a^2*(5*c^3-45*c^2*d-141*c*d^2-75*d^3)*cos(f*x+e)*(c+d*sin(f*x+e))^(1 /2)/d/f+4/315*a^2*(5*c*(c-9*d)-56*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d /f+4/63*a^2*(c-9*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d/f-2/9*a^2*cos(f*x+ e)*(c+d*sin(f*x+e))^(7/2)/d/f+4/315*a^2*(5*c^4-45*c^3*d-381*c^2*d^2-435*c* d^3-168*d^4)*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^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4/315*a^2*(c^2 -d^2)*(5*c^3-45*c^2*d-141*c*d^2-75*d^3)*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^2/f/(c+d*sin( f*x+e))^(1/2)
Time = 4.76 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.85 \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {a^2 (1+\sin (e+f x))^2 \left (16 \left (-d^2 \left (235 c^3+405 c^2 d+309 c d^2+75 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d (c+d \sin (e+f x)) \left (2 \left (20 c^3+1080 c^2 d+1671 c d^2+690 d^3\right ) \cos (e+f x)+2 d \left (-5 d (19 c+18 d) \cos (3 (e+f x))+\left (150 c^2+540 c d+259 d^2-35 d^2 \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )\right )\right )}{1260 d^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(a^2*(1 + Sin[e + f*x])^2*(16*(-(d^2*(235*c^3 + 405*c^2*d + 309*c*d^2 + 75 *d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (5*c^4 - 45*c^3*d - 381*c^2*d^2 - 435*c*d^3 - 168*d^4)*((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*(20*c^3 + 1080*c^2*d + 1671*c*d^2 + 690*d^3)*Cos[e + f*x] + 2*d*(-5*d*(19*c + 18*d)* Cos[3*(e + f*x)] + (150*c^2 + 540*c*d + 259*d^2 - 35*d^2*Cos[2*(e + f*x)]) *Sin[2*(e + f*x)]))))/(1260*d^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4* Sqrt[c + d*Sin[e + f*x]])
Time = 1.94 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {3042, 3242, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int \left (8 a^2 d-a^2 (c-9 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \left (8 a^2 d-a^2 (c-9 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {2}{7} \int \frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (3 a^2 d (17 c+15 d)-a^2 \left (5 c (c-9 d)-56 d^2\right ) \sin (e+f x)\right )dx+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 a^2 d (17 c+15 d)-a^2 \left (5 c (c-9 d)-56 d^2\right ) \sin (e+f x)\right )dx+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 a^2 d (17 c+15 d)-a^2 \left (5 c (c-9 d)-56 d^2\right ) \sin (e+f x)\right )dx+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (8 a^2 d \left (10 c^2+15 d c+7 d^2\right )-a^2 \left (5 c^3-45 d c^2-141 d^2 c-75 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (8 a^2 d \left (10 c^2+15 d c+7 d^2\right )-a^2 \left (5 c^3-45 d c^2-141 d^2 c-75 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (8 a^2 d \left (10 c^2+15 d c+7 d^2\right )-a^2 \left (5 c^3-45 d c^2-141 d^2 c-75 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {a^2 d \left (235 c^3+405 d c^2+309 d^2 c+75 d^3\right )-a^2 \left (5 c^4-45 d c^3-381 d^2 c^2-435 d^3 c-168 d^4\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {a^2 d \left (235 c^3+405 d c^2+309 d^2 c+75 d^3\right )-a^2 \left (5 c^4-45 d c^3-381 d^2 c^2-435 d^3 c-168 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {a^2 d \left (235 c^3+405 d c^2+309 d^2 c+75 d^3\right )-a^2 \left (5 c^4-45 d c^3-381 d^2 c^2-435 d^3 c-168 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (\frac {1}{7} \left (\frac {3}{5} \left (\frac {2 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )\right )+\frac {2 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\right )}{9 d}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(-2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d*f) + (2*((2*a^2*(c - 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + ((2*a^2*(5*c*(c - 9 *d) - 56*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (3*((2*a^2* (5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f* x]])/(3*f) + ((-2*a^2*(5*c^4 - 45*c^3*d - 381*c^2*d^2 - 435*c*d^3 - 168*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^2*(c^2 - d^2)*(5*c^3 - 45* c^2*d - 141*c*d^2 - 75*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*S qrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/3))/5)/ 7))/(9*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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1613\) vs. \(2(355)=710\).
Time = 12.62 (sec) , antiderivative size = 1614, normalized size of antiderivative = 4.27
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1614\) |
| parts | \(\text {Expression too large to display}\) | \(3031\) |
Input:
int((a+sin(f*x+e)*a)^2*(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/315*a^2*(90*((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^5*d+772*((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*s in(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d^2+780*((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^3*d^3 -426*((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^2*d^4-870*((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*d^5+10*((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)*Ellipt icF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^5*d-570*((c+d*si n(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^2-1012*((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^3+84*((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...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.89 \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
2/945*(2*(10*a^2*c^5 - 90*a^2*c^4*d - 57*a^2*c^3*d^2 + 345*a^2*c^2*d^3 + 5 91*a^2*c*d^4 + 225*a^2*d^5)*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) + 2*(10*a^2*c^5 - 90*a^2*c^4*d - 57*a^2*c^3*d^ 2 + 345*a^2*c^2*d^3 + 591*a^2*c*d^4 + 225*a^2*d^5)*sqrt(-1/2*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) - 6*(-5*I*a^2*c^4*d + 45*I*a^2*c^3*d^2 + 381*I*a^2*c^2*d^3 + 435*I*a^2*c*d^4 + 168*I*a^2*d^5)*s qrt(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*(5*I*a^2*c^4*d - 45*I*a^2*c^3*d^2 - 381*I*a^2*c^2*d^3 - 435*I*a^2*c*d ^4 - 168*I*a^2*d^5)*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*s in(f*x + e) + 2*I*c)/d)) + 3*(5*(19*a^2*c*d^4 + 18*a^2*d^5)*cos(f*x + e)^3 - (5*a^2*c^3*d^2 + 270*a^2*c^2*d^3 + 489*a^2*c*d^4 + 240*a^2*d^5)*cos(f*x + e) + (35*a^2*d^5*cos(f*x + e)^3 - 3*(25*a^2*c^2*d^3 + 90*a^2*c*d^4 + 49 *a^2*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^3*f)
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=a^{2} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 2 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 2 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 4 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**2*(c+d*sin(f*x+e))**(5/2),x)
Output:
a**2*(Integral(c**2*sqrt(c + d*sin(e + f*x)), x) + Integral(2*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral(c**2*sqrt(c + d*sin(e + f*x) )*sin(e + f*x)**2, x) + Integral(d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x )**2, x) + Integral(2*d**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)**4, x) + Integral(2*c* d*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral(4*c*d*sqrt(c + d*si n(e + f*x))*sin(e + f*x)**2, x) + Integral(2*c*d*sqrt(c + d*sin(e + f*x))* sin(e + f*x)**3, x))
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(5/2), x)
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2),x)
Output:
int((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2), x)
\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=a^{2} \left (\left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x \right ) c^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) c d +2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) d^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) c^{2}+4 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) c d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) c d \right ) \] Input:
int((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x)
Output:
a**2*(int(sqrt(sin(e + f*x)*d + c),x)*c**2 + int(sqrt(sin(e + f*x)*d + c)* sin(e + f*x)**4,x)*d**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x )*c*d + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*d**2 + int(sqrt( sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*c**2 + 4*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 + 2*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)