Integrand size = 27, antiderivative size = 331 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 (c-d) f (3+3 \sin (e+f x))^3}-\frac {2 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{45 (c-d)^2 f (3+3 \sin (e+f x))^2}-\frac {\left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 (c-d)^3 f (27+27 \sin (e+f x))}-\frac {\left (4 c^2-15 c d+27 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{810 (c-d)^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 c^2-11 c d+15 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{810 (c-d)^2 f \sqrt {c+d \sin (e+f x)}} \]
-1/5*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(c-d)/f/(a+a*sin(f*x+e))^3-2/15*(c- 3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2-1/30 *(4*c^2-15*c*d+27*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(c-d)^3/f/(a^3+a^ 3*sin(f*x+e))+1/30*(4*c^2-15*c*d+27*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)^3/f/((c+d*sin(f*x+e))/(c+ d))^(1/2)-1/30*(4*c^2-11*c*d+15*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/s in(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/(c-d)^2/f/(c+d*sin(f*x+e))^ (1/2)
Time = 8.96 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\left (\left (4 c^2-15 c d+27 d^2\right ) (c+d \sin (e+f x))\right )+\frac {2 \left (6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (c-3 d) (c-d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 (c-3 d) (c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+\left (4 c^2-15 c d+27 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+d^2 (c+15 d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (4 c^2-15 c d+27 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{810 (c-d)^3 f (1+\sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 - 15*c*d + 27*d^2)*(c + d*Sin[e + f*x])) + (2*(6*(c - d)^2*Sin[(e + f*x)/2] - 3*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - 3*d)*(c - d)*Sin[(e + f*x)/2]*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c - 3*d)*(c - d)*(Cos[(e + f*x)/2 ] + Sin[(e + f*x)/2])^3 + (4*c^2 - 15*c*d + 27*d^2)*Sin[(e + f*x)/2]*(Cos[ (e + f*x)/2] + Sin[(e + f*x)/2])^4)*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2 ] + Sin[(e + f*x)/2])^5 + d^2*(c + 15*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(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)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)] ))/(810*(c - d)^3*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])
Time = 1.89 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3245, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {a (4 c-9 d)+3 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{5 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (4 c-9 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (4 c-9 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (4 c^2-13 d c+21 d^2\right ) a^2+2 (c-3 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-13 d c+21 d^2\right ) a^2+2 (c-3 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-13 d c+21 d^2\right ) a^2+2 (c-3 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {d^2 (c+15 d) a^3+d \left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {d^2 (c+15 d) a^3+d \left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {d^2 (c+15 d) a^3+d \left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {-\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\) |
-1/5*(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x ])^3) + ((-4*a*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(c - d) *f*(a + a*Sin[e + f*x])^2) + (-((a^2*(4*c^2 - 15*c*d + 27*d^2)*Cos[e + f*x ]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x]))) - ((2*a^3*(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]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a^3*(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)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(2*a^2*( c - d)))/(3*a^2*(c - d)))/(10*a^2*(c - d))
3.6.19.3.1 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^2*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*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + 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] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[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])
Time = 2.93 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.79
| method | result | size |
| default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{5 \left (c -d \right ) \left (\sin \left (f x +e \right )+1\right )^{3}}-\frac {2 \left (c -3 d \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{15 \left (c -d \right )^{2} \left (\sin \left (f x +e \right )+1\right )^{2}}-\frac {\left (-d \left (\sin ^{2}\left (f x +e \right )\right )-c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+c \right ) \left (4 c^{2}-15 c d +27 d^{2}\right )}{30 \left (c -d \right )^{3} \sqrt {\left (\sin \left (f x +e \right )+1\right ) \left (\sin \left (f x +e \right )-1\right ) \left (-d \sin \left (f x +e \right )-c \right )}}+\frac {2 \left (-c \,d^{2}-15 d^{3}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (60 c^{3}-180 c^{2} d +180 c \,d^{2}-60 d^{3}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {d \left (4 c^{2}-15 c d +27 d^{2}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{30 \left (c -d \right )^{3} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{a^{3} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(593\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(-1/5/(c-d)*(-(-d*sin(f*x+e)-c )*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1)^3-2/15*(c-3*d)/(c-d)^2*(-(-d*sin(f*x+ e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1)^2-1/30*(-d*sin(f*x+e)^2-c*sin(f*x +e)+d*sin(f*x+e)+c)/(c-d)^3*(4*c^2-15*c*d+27*d^2)/((sin(f*x+e)+1)*(sin(f*x +e)-1)*(-d*sin(f*x+e)-c))^(1/2)+2*(-c*d^2-15*d^3)/(60*c^3-180*c^2*d+180*c* d^2-60*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)*(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))-1/30 *d*(4*c^2-15*c*d+27*d^2)/(c-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)*(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))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 1744, normalized size of antiderivative = 5.27 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\text {Too large to display} \]
1/180*((sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^3 + 3* sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)* (8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e) + (sqrt(2)*(8*c^3 - 30 *c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e) - 4*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3))*sin(f*x + e) - 4*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*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 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^3 + 3*sqrt(2) *(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e) + (sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d ^2 - 45*d^3)*cos(f*x + e) - 4*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^ 3))*sin(f*x + e) - 4*sqrt(2)*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*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) + 3*(sq rt(2)*(4*I*c^2*d - 15*I*c*d^2 + 27*I*d^3)*cos(f*x + e)^3 + 3*sqrt(2)*(4*I* c^2*d - 15*I*c*d^2 + 27*I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^2*d + 15 *I*c*d^2 - 27*I*d^3)*cos(f*x + e) + (sqrt(2)*(4*I*c^2*d - 15*I*c*d^2 + 27* I*d^3)*cos(f*x + e)^2 + 2*sqrt(2)*(-4*I*c^2*d + 15*I*c*d^2 - 27*I*d^3)*...
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\int \frac {1}{\sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx}{a^{3}} \]
Integral(1/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2 + 3*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + sqrt(c + d*sin(e + f*x))), x)/a**3
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]