Integrand size = 27, antiderivative size = 499 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{810 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{5 (c-d) f (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac {2 (c-5 d) \cos (e+f x)}{45 (c-d)^2 f (3+3 \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac {\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{810 (c-d)^5 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{810 (c-d)^5 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{810 (c-d)^4 (c+d) f \sqrt {c+d \sin (e+f x)}} \]
-1/30*d*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*cos(f*x+e)/a^3/(c-d)^4/(c+d)/f/ (c+d*sin(f*x+e))^(3/2)-1/5*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^3/(c+d*sin( f*x+e))^(3/2)-2/15*(c-5*d)*cos(f*x+e)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2/(c+d* sin(f*x+e))^(3/2)-1/30*(4*c^2-27*c*d+119*d^2)*cos(f*x+e)/(c-d)^3/f/(a^3+a^ 3*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2)-1/30*d*(4*c^4-27*c^3*d+111*c^2*d^2+57 9*c*d^3+357*d^4)*cos(f*x+e)/a^3/(c-d)^5/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)+1 /30*(4*c^4-27*c^3*d+111*c^2*d^2+579*c*d^3+357*d^4)*(sin(1/2*e+1/4*Pi+1/2*f *x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x) ,2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/a^3/(c-d)^5/(c+d)^2/f/((c +d*sin(f*x+e))/(c+d))^(1/2)-1/30*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*(sin(1 /2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2* e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/ a^3/(c-d)^4/(c+d)/f/(c+d*sin(f*x+e))^(1/2)
Time = 11.93 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, 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^4-27 c^3 d+111 c^2 d^2+449 c d^3+267 d^4}{15 (c-d)^5 (c+d)^2}+\frac {2 \sin \left (\frac {1}{2} (e+f x)\right )}{5 (c-d)^3 \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)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {2 (c-8 d)}{15 (c-d)^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {4 \left (c \sin \left (\frac {1}{2} (e+f x)\right )-8 d \sin \left (\frac {1}{2} (e+f x)\right )\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 )^3}+\frac {4 c^2 \sin \left (\frac {1}{2} (e+f x)\right )-35 c d \sin \left (\frac {1}{2} (e+f x)\right )+177 d^2 \sin \left (\frac {1}{2} (e+f x)\right )}{15 (c-d)^5 \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)}{3 (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac {2 \left (13 c d^4 \cos (e+f x)+9 d^5 \cos (e+f x)\right )}{3 (c-d)^5 (c+d)^2 (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^3 d-387 c^2 d^2-471 c d^3-165 d^4\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^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\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^4+27 c^3 d-111 c^2 d^2-579 c d^3-357 d^4\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)^5 (c+d)^2 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^4 - 27*c^3*d + 111*c^2*d^2 + 449*c*d^3 + 267*d^4)/((c - d)^5*(c + d)^2 ) + (2*Sin[(e + f*x)/2])/(5*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2] )^5) - 1/(5*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - (2*(c - 8 *d))/(15*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (4*(c*Sin[(e + f*x)/2] - 8*d*Sin[(e + f*x)/2]))/(15*(c - d)^4*(Cos[(e + f*x)/2] + Sin[ (e + f*x)/2])^3) + (4*c^2*Sin[(e + f*x)/2] - 35*c*d*Sin[(e + f*x)/2] + 177 *d^2*Sin[(e + f*x)/2])/(15*(c - d)^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) ) - (2*d^4*Cos[e + f*x])/(3*(c - d)^4*(c + d)*(c + d*Sin[e + f*x])^2) - (2 *(13*c*d^4*Cos[e + f*x] + 9*d^5*Cos[e + f*x]))/(3*(c - d)^5*(c + d)^2*(c + d*Sin[e + f*x]))))/(f*(3 + 3*Sin[e + f*x])^3) + (d*(Cos[(e + f*x)/2] + Si n[(e + f*x)/2])^6*((-2*(-(c^3*d) - 387*c^2*d^2 - 471*c*d^3 - 165*d^4)*Elli pticF[(-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*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c* d^3 + 357*d^4)*Cos[e + f*x]^2*Sqrt[c + d*Sin[e + f*x]])/(d*(1 - Sin[e + f* x]^2)) - ((-4*c^4 + 27*c^3*d - 111*c^2*d^2 - 579*c*d^3 - 357*d^4)*((2*(c + d)*EllipticE[(-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)^5*(c + d)^2*f*(3 + 3*Sin[e + f*x])^3)
Time = 2.99 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.13, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3245, 27, 3042, 3457, 25, 3042, 3457, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle -\frac {\int -\frac {a (4 c-13 d)+7 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^{5/2}}dx}{5 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (4 c-13 d)+7 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^{5/2}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (4 c-13 d)+7 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 (c+d \sin (e+f x))^{5/2}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (4 c^2-17 d c+69 d^2\right ) a^2+10 (c-5 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{5/2}}dx}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-17 d c+69 d^2\right ) a^2+10 (c-5 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{5/2}}dx}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-17 d c+69 d^2\right ) a^2+10 (c-5 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{5/2}}dx}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (5 (c-33 d) d^2 a^3+d \left (4 c^2-27 d c+119 d^2\right ) \sin (e+f x) a^3\right )}{2 (c+d \sin (e+f x))^{5/2}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {5 (c-33 d) d^2 a^3+d \left (4 c^2-27 d c+119 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^{5/2}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {5 (c-33 d) d^2 a^3+d \left (4 c^2-27 d c+119 d^2\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^{5/2}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {2 \int -\frac {3 d^2 \left (c^2-138 d c-119 d^2\right ) a^3+d \left (4 c^3-27 d c^2+114 d^2 c+165 d^3\right ) \sin (e+f x) a^3}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {3 d^2 \left (c^2-138 d c-119 d^2\right ) a^3+d \left (4 c^3-27 d c^2+114 d^2 c+165 d^3\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {3 d^2 \left (c^2-138 d c-119 d^2\right ) a^3+d \left (4 c^3-27 d c^2+114 d^2 c+165 d^3\right ) \sin (e+f x) a^3}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {2 \int \frac {d^2 \left (c^3+387 d c^2+471 d^2 c+165 d^3\right ) a^3+d \left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\int \frac {d^2 \left (c^3+387 d c^2+471 d^2 c+165 d^3\right ) a^3+d \left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\int \frac {d^2 \left (c^3+387 d c^2+471 d^2 c+165 d^3\right ) a^3+d \left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\frac {a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{c^2-d^2}-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {-\frac {2 a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 a^3 \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}}{3 a^2 (c-d)}-\frac {4 a (c-5 d) \cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}}{10 a^2 (c-d)}-\frac {\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}}\) |
-1/5*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^( 3/2)) + ((-4*a*(c - 5*d)*Cos[e + f*x])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2 *(c + d*Sin[e + f*x])^(3/2)) + (-((a^2*(4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))) + (3*(( -2*a^3*d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((-2*a^3*d*(4*c^4 - 27*c^3*d + 111*c^ 2*d^2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - ((2*a^3*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4) *EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(f *Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a^3*(c^2 - d^2)*(4*c^3 - 27*c^2* d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt [(c + d*Sin[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(c^2 - d^2)) /(3*(c^2 - d^2))))/(2*a^2*(c - d)))/(3*a^2*(c - d)))/(10*a^2*(c - d))
3.6.21.3.1 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. \(2310\) vs. \(2(552)=1104\).
Time = 8.16 (sec) , antiderivative size = 2311, normalized size of antiderivative = 4.63
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^3*(1/(c-d)^2*(-1/5/(c-d)*(-(-d*s in(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)*c os(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))))-2/(c-d)^3*d*(-1/3/(c-d)*(-(-d*sin(f*x+e)-c)*co s(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)/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.37 (sec) , antiderivative size = 4788, normalized size of antiderivative = 9.60 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
1/180*((sqrt(2)*(8*c^5*d^2 - 54*c^4*d^3 + 219*c^3*d^4 - 3*c^2*d^5 - 699*c* d^6 - 495*d^7)*cos(f*x + e)^5 + sqrt(2)*(16*c^6*d - 84*c^5*d^2 + 276*c^4*d ^3 + 651*c^3*d^4 - 1407*c^2*d^5 - 3087*c*d^6 - 1485*d^7)*cos(f*x + e)^4 - sqrt(2)*(8*c^7 - 22*c^6*d + 27*c^5*d^2 + 711*c^4*d^3 - 54*c^3*d^4 - 3300*c ^2*d^5 - 4077*c*d^6 - 1485*d^7)*cos(f*x + e)^3 - sqrt(2)*(24*c^7 - 82*c^6* d + 173*c^5*d^2 + 1803*c^4*d^3 - 594*c^3*d^4 - 8496*c^2*d^5 - 9843*c*d^6 - 3465*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 38 1*c^4*d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7)*cos(f*x + e ) + (sqrt(2)*(8*c^5*d^2 - 54*c^4*d^3 + 219*c^3*d^4 - 3*c^2*d^5 - 699*c*d^6 - 495*d^7)*cos(f*x + e)^4 - 2*sqrt(2)*(8*c^6*d - 46*c^5*d^2 + 165*c^4*d^3 + 216*c^3*d^4 - 702*c^2*d^5 - 1194*c*d^6 - 495*d^7)*cos(f*x + e)^3 - sqrt (2)*(8*c^7 - 6*c^6*d - 65*c^5*d^2 + 1041*c^4*d^3 + 378*c^3*d^4 - 4704*c^2* d^5 - 6465*c*d^6 - 2475*d^7)*cos(f*x + e)^2 + 2*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4*d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 49 5*d^7)*cos(f*x + e) + 4*sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4* d^3 - 486*c^3*d^4 - 1896*c^2*d^5 - 1689*c*d^6 - 495*d^7))*sin(f*x + e) + 4 *sqrt(2)*(8*c^7 - 38*c^6*d + 119*c^5*d^2 + 381*c^4*d^3 - 486*c^3*d^4 - 189 6*c^2*d^5 - 1689*c*d^6 - 495*d^7))*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^5*d^2 - 54*c^4*d^3 + 219...
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\int \frac {1}{c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )} + 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{5}{\left (e + f x \right )} + 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )} + 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a^{3}} \]
Integral(1/(c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2 + 3*c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + c**2*sqrt(c + d*sin(e + f*x)) + 2*c*d*sqrt(c + d*sin(e + f*x))*si n(e + f*x)**4 + 6*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 6*c*d*sqr t(c + d*sin(e + f*x))*sin(e + f*x)**2 + 2*c*d*sqrt(c + d*sin(e + f*x))*sin (e + f*x) + d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**5 + 3*d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**4 + 3*d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2), x)/a**3
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Hanged} \]