Integrand size = 27, antiderivative size = 310 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (3+3 \sin (e+f x))^3}-\frac {2 (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{45 f (3+3 \sin (e+f x))^2}-\frac {\left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 (c-d) f (27+27 \sin (e+f x))}-\frac {\left (4 c^2+5 c d-3 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{810 (c-d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) (4 c+5 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{810 f \sqrt {c+d \sin (e+f x)}} \]
-1/5*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^3-2/15*(c+ 2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/f/(a+a*sin(f*x+e))^2-1/30*(4*c^2+ 5*c*d-3*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(c-d)/f/(a^3+a^3*sin(f*x+e) )+1/30*(4*c^2+5*c*d-3*d^2)*(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)/a^3/(c-d)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/30*( c+d)*(4*c+5*d)*(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)/a^3/f/(c+d*sin(f*x+e))^(1/2)
Time = 4.29 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.41 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (\frac {2 \left (6 (c-d) \sin \left (\frac {1}{2} (e+f x)\right )+3 (-c+d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (c+2 d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 (c+2 d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+\frac {\left (4 c^2+5 c d-3 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}{c-d}\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {\left (4 c^2+5 c d-3 d^2\right ) (c+d \sin (e+f x))-d^2 (c+5 d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (-4 c^2-5 c d+3 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{c-d}\right )}{810 f (1+\sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*((2*(6*(c - d)*Sin[(e + f*x)/2] + 3*(-c + d)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c + 2*d)*Sin[(e + f *x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c + 2*d)*(Cos[(e + f*x )/2] + Sin[(e + f*x)/2])^3 + ((4*c^2 + 5*c*d - 3*d^2)*Sin[(e + f*x)/2]*(Co s[(e + f*x)/2] + Sin[(e + f*x)/2])^4)/(c - d))*(c + d*Sin[e + f*x]))/(Cos[ (e + f*x)/2] + Sin[(e + f*x)/2])^5 - ((4*c^2 + 5*c*d - 3*d^2)*(c + d*Sin[e + f*x]) - d^2*(c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*S qrt[(c + d*Sin[e + f*x])/(c + d)] + (-4*c^2 - 5*c*d + 3*d^2)*((c + d)*Elli pticE[(-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)])/(c - d)))/(810* f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])
Time = 1.96 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.13, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3244, 27, 3042, 3457, 25, 3042, 3457, 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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{3/2}}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{3/2}}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int -\frac {a \left (4 c^2+7 d c-d^2\right )+a d (3 c+7 d) \sin (e+f x)}{2 (\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a \left (4 c^2+7 d c-d^2\right )+a d (3 c+7 d) \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a \left (4 c^2+7 d c-d^2\right )+a d (3 c+7 d) \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {(c-d) \left (4 c^2+7 d c+d^2\right ) a^2+2 (c-d) d (c+2 d) \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(c-d) \left (4 c^2+7 d c+d^2\right ) a^2+2 (c-d) d (c+2 d) \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(c-d) \left (4 c^2+7 d c+d^2\right ) a^2+2 (c-d) d (c+2 d) \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {(c-d) d^2 (c+5 d) a^3+(c-d) d \left (4 c^2+5 d c-3 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {(c-d) d^2 (c+5 d) a^3+(c-d) d \left (4 c^2+5 d c-3 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {(c-d) d^2 (c+5 d) a^3+(c-d) d \left (4 c^2+5 d c-3 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {-\frac {a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (c-d)^2 (c+d) (4 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (c-d)^2 (c+d) (4 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d)^2 (c+d) (4 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d)^2 (c+d) (4 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d)^2 (c+d) (4 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d)^2 (c+d) (4 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d)^2 (c+d) (4 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {-\frac {a^2 \left (4 c^2+5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac {\frac {2 a^3 (c-d) \left (4 c^2+5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 (c-d)^2 (c+d) (4 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}}{3 a^2 (c-d)}-\frac {4 a (c+2 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\) |
-1/5*((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x ])^3) + ((-4*a*(c + 2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f*(a + a*Sin[e + f*x])^2) + (-((a^2*(4*c^2 + 5*c*d - 3*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) - ((2*a^3*(c - d)*(4*c^2 + 5*c *d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a^3*(c - d)^2*(c + d) *(4*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[ e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(2*a^2*(c - d)))/(3*a^2* (c - d)))/(10*a^2)
3.6.17.3.1 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_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*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] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (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_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1461\) vs. \(2(365)=730\).
Time = 4.43 (sec) , antiderivative size = 1462, normalized size of antiderivative = 4.72
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(d^2*(-(-d*sin(f*x+e)^2-c*sin( f*x+e)+d*sin(f*x+e)+c)/(c-d)/((sin(f*x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e) -c))^(1/2)-2*d/(2*c-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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c) *cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d) )^(1/2))-d/(c-d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/ (c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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)))) +2*d*(c-d)*(-1/3/(c-d)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e) +1)^2-1/3*(-d*sin(f*x+e)^2-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/(( sin(f*x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e)-c))^(1/2)+2*d^2/(3*c^2-6*c*d+3 *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 )*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2 )*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-1/3*d*(c-3 *d)/(c-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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*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))))+(c^2 -2*c*d+d^2)*(-1/5/(c-d)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 1589, normalized size of antiderivative = 5.13 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=\text {Too large to display} \]
1/180*((sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e)^3 + 3*s qrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8 *c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e) + (sqrt(2)*(8*c^3 + 10*c^ 2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c *d^2 - 15*d^3)*cos(f*x + e) - 4*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d ^3))*sin(f*x + e) - 4*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3))*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*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e)^3 + 3*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 10*c^2* d - 9*c*d^2 - 15*d^3)*cos(f*x + e) + (sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3) *cos(f*x + e) - 4*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3))*sin(f*x + e) - 4*sqrt(2)*(8*c^3 + 10*c^2*d - 9*c*d^2 - 15*d^3))*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) + 3*(sqrt(2)*(4*I*c^2 *d + 5*I*c*d^2 - 3*I*d^3)*cos(f*x + e)^3 + 3*sqrt(2)*(4*I*c^2*d + 5*I*c*d^ 2 - 3*I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^2*d - 5*I*c*d^2 + 3*I*d^3) *cos(f*x + e) + (sqrt(2)*(4*I*c^2*d + 5*I*c*d^2 - 3*I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^2*d - 5*I*c*d^2 + 3*I*d^3)*cos(f*x + e) + 4*sqrt(2)...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]