Integrand size = 27, antiderivative size = 371 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {12 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d^2 f}-\frac {12 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}+\frac {24 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d^2 f}-\frac {2 \cos (e+f x) (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {12 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {12 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^3 f \sqrt {c+d \sin (e+f x)}} \]
-4/315*a^3*(4*c^2-27*c*d+119*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f+ 8/63*a^3*(c-5*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d^2/f-2/9*cos(f*x+e)*(a ^3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2)/d/f-4/315*a^3*(4*c^3-27*c^2*d+11 4*c*d^2+165*d^3)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-4/315*a^3*(4*c^4- 27*c^3*d+111*c^2*d^2+579*c*d^3+357*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)*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) +4/315*a^3*(c^2-d^2)*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*(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*s in(f*x+e))^(1/2)
Time = 0.96 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.76 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\frac {3 \left (-16 \left (d^2 \left (c^3+387 c^2 d+471 c d^2+165 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (\left (32 c^3-216 c^2 d-3828 c d^2-2910 d^3\right ) \cos (e+f x)+2 d \left (5 d (10 c+27 d) \cos (3 (e+f x))-\left (6 c^2+432 c d+511 d^2-35 d^2 \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )\right )\right )}{140 d^3 f \sqrt {c+d \sin (e+f x)}} \]
(3*(-16*(d^2*(c^3 + 387*c^2*d + 471*c*d^2 + 165*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*Ell ipticF[(-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])*((32*c^3 - 216*c^2*d - 3828*c*d^2 - 2910* d^3)*Cos[e + f*x] + 2*d*(5*d*(10*c + 27*d)*Cos[3*(e + f*x)] - (6*c^2 + 432 *c*d + 511*d^2 - 35*d^2*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))))/(140*d^3*f* Sqrt[c + d*Sin[e + f*x]])
Time = 2.16 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.09, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {3042, 3242, 3042, 3447, 3042, 3502, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int (\sin (e+f x) a+a) \left (a^2 (c+7 d)-2 a^2 (c-5 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}dx}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (\sin (e+f x) a+a) \left (a^2 (c+7 d)-2 a^2 (c-5 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}dx}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {2 \int (c+d \sin (e+f x))^{3/2} \left (-2 (c-5 d) \sin ^2(e+f x) a^3+(c+7 d) a^3+\left (a^3 (c+7 d)-2 a^3 (c-5 d)\right ) \sin (e+f x)\right )dx}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (c+d \sin (e+f x))^{3/2} \left (-2 (c-5 d) \sin (e+f x)^2 a^3+(c+7 d) a^3+\left (a^3 (c+7 d)-2 a^3 (c-5 d)\right ) \sin (e+f x)\right )dx}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \left (\frac {2 \int -\frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (3 a^3 (c-33 d) d-a^3 \left (4 c^2-27 d c+119 d^2\right ) \sin (e+f x)\right )dx}{7 d}+\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\int (c+d \sin (e+f x))^{3/2} \left (3 a^3 (c-33 d) d-a^3 \left (4 c^2-27 d c+119 d^2\right ) \sin (e+f x)\right )dx}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\int (c+d \sin (e+f x))^{3/2} \left (3 a^3 (c-33 d) d-a^3 \left (4 c^2-27 d c+119 d^2\right ) \sin (e+f x)\right )dx}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {2}{5} \int \frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (a^3 d \left (c^2-138 d c-119 d^2\right )-a^3 \left (4 c^3-27 d c^2+114 d^2 c+165 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (a^3 d \left (c^2-138 d c-119 d^2\right )-a^3 \left (4 c^3-27 d c^2+114 d^2 c+165 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (a^3 d \left (c^2-138 d c-119 d^2\right )-a^3 \left (4 c^3-27 d c^2+114 d^2 c+165 d^3\right ) \sin (e+f x)\right )dx+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {2}{3} \int -\frac {d \left (c^3+387 d c^2+471 d^2 c+165 d^3\right ) a^3+\left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^3+387 d c^2+471 d^2 c+165 d^3\right ) a^3+\left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^3+387 d c^2+471 d^2 c+165 d^3\right ) a^3+\left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-5 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {2 a^3 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}+\frac {3}{5} \left (\frac {2 a^3 \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )\right )}{7 d}\right )}{9 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{5/2}}{9 d f}\) |
(-2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d *f) + (2*((4*a^3*(c - 5*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f ) - ((2*a^3*(4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^( 3/2))/(5*f) + (3*((2*a^3*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((-2*a^3*(4*c^4 - 27*c^3*d + 111*c^ 2*d^2 + 579*c*d^3 + 357*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^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*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*d)))/(9*d)
3.5.97.3.1 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. \(1612\) vs. \(2(428)=856\).
Time = 7.22 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.35
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1613\) |
| parts | \(\text {Expression too large to display}\) | \(3589\) |
2/315*a^3*(-1044*((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))*d^6-8*((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^6+714*((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))*d^6+4*c^4*d^2-46 6*c^2*d^4+85*c*d^5*sin(f*x+e)^5+53*c^2*d^4*sin(f*x+e)^4-c^3*d^3*sin(f*x+e) ^3-4*c^4*d^2*sin(f*x+e)^2+35*d^6*sin(f*x+e)^6-330*c*d^5-492*((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^2*d ^4+1158*((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^5+8*((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^5*d-60*((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^4*d^2+1048*((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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.94 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{5} - 54 \, a^{3} c^{4} d + 219 \, a^{3} c^{3} d^{2} - 3 \, a^{3} c^{2} d^{3} - 699 \, a^{3} c d^{4} - 495 \, a^{3} d^{5}\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^{5} - 54 \, a^{3} c^{4} d + 219 \, a^{3} c^{3} d^{2} - 3 \, a^{3} c^{2} d^{3} - 699 \, a^{3} c d^{4} - 495 \, a^{3} d^{5}\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^{4} d - 27 i \, a^{3} c^{3} d^{2} + 111 i \, a^{3} c^{2} d^{3} + 579 i \, a^{3} c d^{4} + 357 i \, a^{3} d^{5}\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^{4} d + 27 i \, a^{3} c^{3} d^{2} - 111 i \, a^{3} c^{2} d^{3} - 579 i \, a^{3} c d^{4} - 357 i \, a^{3} d^{5}\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 (5 \, {\left (10 \, a^{3} c d^{4} + 27 \, a^{3} d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (4 \, a^{3} c^{3} d^{2} - 27 \, a^{3} c^{2} d^{3} - 516 \, a^{3} c d^{4} - 465 \, a^{3} d^{5}\right )} \cos \left (f x + e\right ) + {\left (35 \, a^{3} d^{5} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c^{2} d^{3} + 72 \, a^{3} c d^{4} + 91 \, a^{3} d^{5}\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 )}}{945 \, d^{4} f} \]
-2/945*(sqrt(2)*(8*a^3*c^5 - 54*a^3*c^4*d + 219*a^3*c^3*d^2 - 3*a^3*c^2*d^ 3 - 699*a^3*c*d^4 - 495*a^3*d^5)*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) + sqrt(2)*(8*a^3*c^5 - 54*a^3*c^4*d + 219*a^3 *c^3*d^2 - 3*a^3*c^2*d^3 - 699*a^3*c*d^4 - 495*a^3*d^5)*sqrt(-I*d)*weierst rassPInverse(-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^4*d - 27*I*a^3*c^3*d^2 + 111*I*a^3*c^2*d^3 + 579*I*a^3*c*d^4 + 357*I*a^ 3*d^5)*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^4*d + 27*I*a^3*c^3*d^2 - 111*I*a^3*c^2*d^3 - 579*I*a^3*c*d^4 - 357*I*a^3*d^5)*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*(5*(10*a^3*c*d^4 + 27*a^3*d^5)*cos(f *x + e)^3 + (4*a^3*c^3*d^2 - 27*a^3*c^2*d^3 - 516*a^3*c*d^4 - 465*a^3*d^5) *cos(f*x + e) + (35*a^3*d^5*cos(f*x + e)^3 - 3*(a^3*c^2*d^3 + 72*a^3*c*d^4 + 91*a^3*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4* f)
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=a^{3} \left (\int c \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 3 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]
a**3*(Integral(c*sqrt(c + d*sin(e + f*x)), x) + Integral(3*c*sqrt(c + d*si n(e + f*x))*sin(e + f*x), x) + Integral(3*c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral(3*d*sqrt( c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(3*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3, x) + Integral(d*sqrt(c + d*sin(e + f*x))*sin(e + f* x)**4, x))
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]