Integrand size = 27, antiderivative size = 336 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}+\frac {8 a^3 (c-d) (c+3 d) \cos (e+f x)}{15 d^2 (c+d)^2 f (c+d \sin (e+f x))^{3/2}}-\frac {4 a^3 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{15 d^2 (c+d)^3 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^3 \left (4 c^2+15 c d+27 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^3 (c+d)^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^3 \left (4 c^2+11 c d+15 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
2/5*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d)/f/(c+d*sin(f*x+e))^(5/2) +8/15*a^3*(c-d)*(c+3*d)*cos(f*x+e)/d^2/(c+d)^2/f/(c+d*sin(f*x+e))^(3/2)-4/ 15*a^3*(4*c^2+15*c*d+27*d^2)*cos(f*x+e)/d^2/(c+d)^3/f/(c+d*sin(f*x+e))^(1/ 2)+4/15*a^3*(4*c^2+15*c*d+27*d^2)*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/(c+d)^3/f/((c+d*sin(f*x+e) )/(c+d))^(1/2)+4/15*a^3*(4*c^2+11*c*d+15*d^2)*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/(c+d) ^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 6.73 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.89 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 a^3 (1+\sin (e+f x))^3 \left (-2 \left ((c-15 d) d^2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^2+15 c d+27 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) (c+d \sin (e+f x))^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d \cos (e+f x) \left (4 c^4+15 c^3 d+55 c^2 d^2+15 c d^3+3 d^4+d \left (9 c^3+45 c^2 d+115 c d^2+15 d^3\right ) \sin (e+f x)+2 d^2 \left (4 c^2+15 c d+27 d^2\right ) \sin ^2(e+f x)\right )\right )}{15 d^3 (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c+d \sin (e+f x))^{5/2}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]
Output:
(-2*a^3*(1 + Sin[e + f*x])^3*(-2*((c - 15*d)*d^2*EllipticF[(-2*e + Pi - 2* f*x)/4, (2*d)/(c + d)] + (4*c^2 + 15*c*d + 27*d^2)*((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)]))*(c + d*Sin[e + f*x])^2*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + d*Cos[e + f*x]*(4*c^4 + 15*c^3*d + 55*c^2*d^2 + 15*c*d^3 + 3*d^4 + d*(9*c^ 3 + 45*c^2*d + 115*c*d^2 + 15*d^3)*Sin[e + f*x] + 2*d^2*(4*c^2 + 15*c*d + 27*d^2)*Sin[e + f*x]^2)))/(15*d^3*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c + d*Sin[e + f*x])^(5/2))
Time = 2.12 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.15, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 3241, 3042, 3447, 3042, 3500, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c+d \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c+d \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3241 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a) (a (c-6 d)-a (2 c+3 d) \sin (e+f x))}{(c+d \sin (e+f x))^{5/2}}dx}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a) (a (c-6 d)-a (2 c+3 d) \sin (e+f x))}{(c+d \sin (e+f x))^{5/2}}dx}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \int \frac {-\left ((2 c+3 d) \sin ^2(e+f x) a^2\right )+(c-6 d) a^2+\left (a^2 (c-6 d)-a^2 (2 c+3 d)\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \int \frac {-\left ((2 c+3 d) \sin (e+f x)^2 a^2\right )+(c-6 d) a^2+\left (a^2 (c-6 d)-a^2 (2 c+3 d)\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {2 \int \frac {3 (c-d) d (c+9 d) a^2+(c-d) \left (4 c^2+11 d c+15 d^2\right ) \sin (e+f x) a^2}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {\int \frac {3 (c-d) d (c+9 d) a^2+(c-d) \left (4 c^2+11 d c+15 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {\int \frac {3 (c-d) d (c+9 d) a^2+(c-d) \left (4 c^2+11 d c+15 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {2 \int \frac {a^2 (c-15 d) d (c-d)^2+a^2 \left (4 c^2+15 d c+27 d^2\right ) \sin (e+f x) (c-d)^2}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\int \frac {a^2 (c-15 d) d (c-d)^2+a^2 \left (4 c^2+15 d c+27 d^2\right ) \sin (e+f x) (c-d)^2}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\int \frac {a^2 (c-15 d) d (c-d)^2+a^2 \left (4 c^2+15 d c+27 d^2\right ) \sin (e+f x) (c-d)^2}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {2 a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {2 a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {\frac {2 a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}}-\frac {2 a \left (-\frac {-\frac {2 a^2 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 a^2 (c-d)^2 \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 (c-d)^2 (c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}}{3 d \left (c^2-d^2\right )}-\frac {4 a^2 (c-d) (c+3 d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^{3/2}}\right )}{5 d (c+d)}\) |
Input:
Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]
Output:
(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(5*d*(c + d)*f*(c + d*Si n[e + f*x])^(5/2)) - (2*a*((-4*a^2*(c - d)*(c + 3*d)*Cos[e + f*x])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - ((-2*a^2*(c - d)*(4*c^2 + 15*c*d + 2 7*d^2)*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - ((2*a^2*(c - d )^2*(4*c^2 + 15*c*d + 27*d^2)*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 - d)^2*(c + d)*(4*c^2 + 11*c*d + 15*d^2)*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]]))/(c^2 - d^2))/(3*d*(c^2 - d^2))))/(5*d*(c + 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[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*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] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b *Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, 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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1582\) vs. \(2(317)=634\).
Time = 6.78 (sec) , antiderivative size = 1583, normalized size of antiderivative = 4.71
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1583\) |
| parts | \(\text {Expression too large to display}\) | \(4597\) |
Input:
int((a+sin(f*x+e)*a)^3/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*a^3*(2/d^3*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/( c-d))^(1/2),((c-d)/(c+d))^(1/2))-3/d^3*(c-d)*(2*d*cos(f*x+e)^2/(c^2-d^2)/( -(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e)) /(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f *x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d) )^(1/2),((c-d)/(c+d))^(1/2))))+3/d^3*(c^2-2*c*d+d^2)*(2/3/(c^2-d^2)/d*(-(- c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/ (c^2-d^2)^2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4 -6*c^2*d^2+3*d^4)*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f*x+ e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+ 8/3*c*d/(c^2-d^2)^2*(c/d-1)*((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)/(-(-c-d*sin(f*x+e))*cos(f* x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 1530, normalized size of antiderivative = 4.55 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")
Output:
-2/45*(2*(8*a^3*c^6 + 30*a^3*c^5*d + 75*a^3*c^4*d^2 + 135*a^3*c^3*d^3 + 15 3*a^3*c^2*d^4 + 135*a^3*c*d^5 - 3*(8*a^3*c^4*d^2 + 30*a^3*c^3*d^3 + 51*a^3 *c^2*d^4 + 45*a^3*c*d^5)*cos(f*x + e)^2 + (24*a^3*c^5*d + 90*a^3*c^4*d^2 + 161*a^3*c^3*d^3 + 165*a^3*c^2*d^4 + 51*a^3*c*d^5 + 45*a^3*d^6 - (8*a^3*c^ 3*d^3 + 30*a^3*c^2*d^4 + 51*a^3*c*d^5 + 45*a^3*d^6)*cos(f*x + e)^2)*sin(f* x + e))*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*(8*a^3*c^6 + 30*a^3*c^5*d + 75*a^3*c^4*d^2 + 135*a^3*c^3*d^3 + 153*a^3*c^2*d^4 + 135*a^3*c*d^5 - 3*(8*a^3*c^4*d^2 + 30*a^3*c^3*d^3 + 51* a^3*c^2*d^4 + 45*a^3*c*d^5)*cos(f*x + e)^2 + (24*a^3*c^5*d + 90*a^3*c^4*d^ 2 + 161*a^3*c^3*d^3 + 165*a^3*c^2*d^4 + 51*a^3*c*d^5 + 45*a^3*d^6 - (8*a^3 *c^3*d^3 + 30*a^3*c^2*d^4 + 51*a^3*c*d^5 + 45*a^3*d^6)*cos(f*x + e)^2)*sin (f*x + e))*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 + 15*I*a^3*c^4*d^2 + 39*I*a^3*c^3*d^3 + 45 *I*a^3*c^2*d^4 + 81*I*a^3*c*d^5 + 3*(-4*I*a^3*c^3*d^3 - 15*I*a^3*c^2*d^4 - 27*I*a^3*c*d^5)*cos(f*x + e)^2 + (12*I*a^3*c^4*d^2 + 45*I*a^3*c^3*d^3 + 8 5*I*a^3*c^2*d^4 + 15*I*a^3*c*d^5 + 27*I*a^3*d^6 + (-4*I*a^3*c^2*d^4 - 15*I *a^3*c*d^5 - 27*I*a^3*d^6)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(1/2*I*d)*wei erstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3,...
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(7/2), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(7/2), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2),x)
Output:
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=a^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x +\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right )+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right )\right ) \] Input:
int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x)
Output:
a**3*(int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)* *3*c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x) + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3)/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)**3*c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c* *3*d + c**4),x) + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)**3*c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x) + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)**3*c*d**3 + 6*sin(e + f*x)** 2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x))