Integrand size = 27, antiderivative size = 407 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{810 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{5 (c-d) f (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-4 d) \cos (e+f x)}{45 (c-d)^2 f (3+3 \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f (27+27 \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{810 (c-d)^4 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 c^2-21 c d+65 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{810 (c-d)^3 f \sqrt {c+d \sin (e+f x)}} \]
-1/30*d*(4*c^3-21*c^2*d+62*c*d^2+147*d^3)*cos(f*x+e)/a^3/(c-d)^4/(c+d)/f/( c+d*sin(f*x+e))^(1/2)-1/5*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^3/(c+d*sin(f *x+e))^(1/2)-2/15*(c-4*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2/(c+d*s in(f*x+e))^(1/2)-1/30*(4*c^2-21*c*d+65*d^2)*cos(f*x+e)/(c-d)^3/f/(a^3+a^3* sin(f*x+e))/(c+d*sin(f*x+e))^(1/2)+1/30*(4*c^3-21*c^2*d+62*c*d^2+147*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)*EllipticE(co s(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)^4/(c+d)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/30*(4*c^2-21*c*d+65*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)*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)^3/f/(c+d*sin(f*x+e))^(1/2)
Time = 11.70 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)} \left (-\frac {4 c^3-21 c^2 d+62 c d^2+117 d^3}{15 (c-d)^4 (c+d)}+\frac {2 \sin \left (\frac {1}{2} (e+f x)\right )}{5 (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {1}{5 (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {-2 c+11 d}{15 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 \left (2 c \sin \left (\frac {1}{2} (e+f x)\right )-11 d \sin \left (\frac {1}{2} (e+f x)\right )\right )}{15 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {4 c^2 \sin \left (\frac {1}{2} (e+f x)\right )-25 c d \sin \left (\frac {1}{2} (e+f x)\right )+87 d^2 \sin \left (\frac {1}{2} (e+f x)\right )}{15 (c-d)^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {2 d^4 \cos (e+f x)}{(c-d)^4 (c+d) (c+d \sin (e+f x))}\right )}{f (3+3 \sin (e+f x))^3}+\frac {d \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 (-c^2 d-126 c d^2-65 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}}}{\sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos ^2(e+f x) \sqrt {c+d \sin (e+f x)}}{d \left (1-\sin ^2(e+f x)\right )}-\frac {\left (-4 c^3+21 c^2 d-62 c d^2-147 d^3\right ) \left (\frac {2 (c+d) E\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}}}{\sqrt {c+d \sin (e+f x)}}-\frac {2 c \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}}}{\sqrt {c+d \sin (e+f x)}}\right )}{d}\right )}{60 (c-d)^4 (c+d) f (3+3 \sin (e+f x))^3} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]]*(-1/15*( 4*c^3 - 21*c^2*d + 62*c*d^2 + 117*d^3)/((c - d)^4*(c + d)) + (2*Sin[(e + f *x)/2])/(5*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5) - 1/(5*(c - d)^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (-2*c + 11*d)/(15*(c - d)^ 3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (2*(2*c*Sin[(e + f*x)/2] - 11 *d*Sin[(e + f*x)/2]))/(15*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^ 3) + (4*c^2*Sin[(e + f*x)/2] - 25*c*d*Sin[(e + f*x)/2] + 87*d^2*Sin[(e + f *x)/2])/(15*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - (2*d^4*Cos[ e + f*x])/((c - d)^4*(c + d)*(c + d*Sin[e + f*x]))))/(f*(3 + 3*Sin[e + f*x ])^3) + (d*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*((-2*(-(c^2*d) - 126*c* d^2 - 65*d^3)*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Si n[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] + (2*(4*c^3 - 21*c^2*d + 62 *c*d^2 + 147*d^3)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Sin[e + f*x]^2)) - ((-4*c^3 + 21*c^2*d - 62*c*d^2 - 147*d^3)*((2*(c + d)*Elliptic E[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/ Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]]))/d))/(60 *(c - d)^4*(c + d)*f*(3 + 3*Sin[e + f*x])^3)
Time = 2.41 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.13, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3245, 27, 3042, 3457, 25, 3042, 3457, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {a (4 c-11 d)+5 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^{3/2}}dx}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (4 c-11 d)+5 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^{3/2}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (4 c-11 d)+5 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^{3/2}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (4 c^2-15 d c+41 d^2\right ) a^2+6 (c-4 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{3/2}}dx}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-15 d c+41 d^2\right ) a^2+6 (c-4 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{3/2}}dx}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-15 d c+41 d^2\right ) a^2+6 (c-4 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{3/2}}dx}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 (c-49 d) d^2 a^3+d \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x) a^3}{2 (c+d \sin (e+f x))^{3/2}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 (c-49 d) d^2 a^3+d \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^{3/2}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 (c-49 d) d^2 a^3+d \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^{3/2}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \int \frac {d^2 \left (c^2+126 d c+65 d^2\right ) a^3+d \left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {d^2 \left (c^2+126 d c+65 d^2\right ) a^3+d \left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {d^2 \left (c^2+126 d c+65 d^2\right ) a^3+d \left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{c^2-d^2}-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}}{3 a^2 (c-d)}-\frac {4 a (c-4 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}}\) |
-1/5*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x ]]) + ((-4*a*(c - 4*d)*Cos[e + f*x])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*S qrt[c + d*Sin[e + f*x]]) + (-((a^2*(4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x]) /((c - d)*f*(a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])) + ((-2*a^3*d*( 4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - ((2*a^3*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Elli pticE[(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^2 - d^2)*(4*c^2 - 21*c*d + 65* 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]]))/(c^2 - d^2))/(2*a^2*(c - d)))/(3 *a^2*(c - d)))/(10*a^2*(c - d))
3.6.20.3.1 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[(-(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*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])
Leaf count of result is larger than twice the leaf count of optimal. \(1850\) vs. \(2(461)=922\).
Time = 6.20 (sec) , antiderivative size = 1851, normalized size of antiderivative = 4.55
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(1/(c-d)*(-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))))+d^2/(c-d)^3*(-(-d*sin(f*x+e)^2-c*sin(f*x+e)+d*si n(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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.23 (sec) , antiderivative size = 3067, normalized size of antiderivative = 7.54 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \]
1/180*((sqrt(2)*(8*c^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)* cos(f*x + e)^4 - sqrt(2)*(8*c^5 - 26*c^4*d + 37*c^3*d^2 + 158*c^2*d^3 - 36 3*c*d^4 - 390*d^5)*cos(f*x + e)^3 - sqrt(2)*(24*c^5 - 86*c^4*d + 153*c^3*d ^2 + 353*c^2*d^3 - 1005*c*d^4 - 975*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e) - (sqrt(2)*(8*c^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)*cos(f *x + e)^3 + sqrt(2)*(8*c^5 - 18*c^4*d - 5*c^3*d^2 + 279*c^2*d^3 - 447*c*d^ 4 - 585*d^5)*cos(f*x + e)^2 - 2*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 3 7*c^2*d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e) - 4*sqrt(2)*(8*c^5 - 34*c^4* d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5))*sin(f*x + e) + 4*sqrt( 2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5))*sqr t(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^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)*cos(f*x + e) ^4 - sqrt(2)*(8*c^5 - 26*c^4*d + 37*c^3*d^2 + 158*c^2*d^3 - 363*c*d^4 - 39 0*d^5)*cos(f*x + e)^3 - sqrt(2)*(24*c^5 - 86*c^4*d + 153*c^3*d^2 + 353*c^2 *d^3 - 1005*c*d^4 - 975*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^5 - 34*c^4*d + 79*c^3*d^2 + 37*c^2*d^3 - 279*c*d^4 - 195*d^5)*cos(f*x + e) - (sqrt(2)*( 8*c^4*d - 42*c^3*d^2 + 121*c^2*d^3 - 84*c*d^4 - 195*d^5)*cos(f*x + e)^3 + sqrt(2)*(8*c^5 - 18*c^4*d - 5*c^3*d^2 + 279*c^2*d^3 - 447*c*d^4 - 585*d...
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\int \frac {1}{c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 3 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 3 c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c \sqrt {c + d \sin {\left (e + f x \right )}} + d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )} + 3 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 3 d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx}{a^{3}} \]
Integral(1/(c*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*c*sqrt(c + d*si n(e + f*x))*sin(e + f*x)**2 + 3*c*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + c*sqrt(c + d*sin(e + f*x)) + d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4 + 3*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*d*sqrt(c + d*sin(e + f*x) )*sin(e + f*x)**2 + d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)), x)/a**3
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Hanged} \]