Integrand size = 27, antiderivative size = 321 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {d (3 c+5 d) \cos (e+f x)}{9 (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{9 (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{9 (c-d)^3 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c+5 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{9 (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}} \]
-1/3*d*(3*c+5*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(3/2)-cos(f *x+e)/(c-d)/f/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2)-1/3*d*(3*c^2+20*c*d+ 9*d^2)*cos(f*x+e)/a/(c-d)^3/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)+1/3*(3*c^2+20 *c*d+9*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/(c-d)^3/(c+d)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/3*(3*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(c os(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/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(1/2)
Time = 3.04 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\frac {\left (3 c^2+20 c d+9 d^2\right ) (c+d \sin (e+f x))+d \left (15 c^2+12 c d+5 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (3 c^2+20 c d+9 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c+d)^2}+2 (c+d \sin (e+f x)) \left (\frac {3 \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}-\frac {3 c^2+13 c d+6 d^2+\frac {d^2 \cos (e+f x) \left (8 c^2+3 c d-d^2+d (7 c+3 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^2}}{(c+d)^2}\right )\right )}{9 (c-d)^3 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(((3*c^2 + 20*c*d + 9*d^2)*(c + d *Sin[e + f*x]) + d*(15*c^2 + 12*c*d + 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x) /4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (3*c^2 + 20*c*d + 9*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*Ellipt icF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(c + d)^2 + 2*(c + d*Sin[e + f*x])*((3*Sin[(e + f*x)/2])/(Cos[(e + f* x)/2] + Sin[(e + f*x)/2]) - (3*c^2 + 13*c*d + 6*d^2 + (d^2*Cos[e + f*x]*(8 *c^2 + 3*c*d - d^2 + d*(7*c + 3*d)*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2)/ (c + d)^2)))/(9*(c - d)^3*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])
Time = 1.62 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3247, 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.
\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a) (c+d \sin (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a) (c+d \sin (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 3247 |
\(\displaystyle \frac {d \int -\frac {5 a-3 a \sin (e+f x)}{2 (c+d \sin (e+f x))^{5/2}}dx}{a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {d \int \frac {5 a-3 a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d \int \frac {5 a-3 a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle -\frac {d \left (\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int -\frac {3 a (5 c+3 d)-a (3 c+5 d) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {d \left (\frac {\int \frac {3 a (5 c+3 d)-a (3 c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d \left (\frac {\int \frac {3 a (5 c+3 d)-a (3 c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3233 |
\(\displaystyle -\frac {d \left (\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {2 \int -\frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {d \left (\frac {\frac {\int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d \left (\frac {\frac {\int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {d \left (\frac {\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle -\frac {d \left (\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}+\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\) |
-(Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) ) - (d*((2*a*(3*c + 5*d)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x ])^(3/2)) + ((2*a*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/((c^2 - d^2)*f*Sq rt[c + d*Sin[e + f*x]]) + ((2*a*(3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi /2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin [e + f*x])/(c + d)]) - (2*a*(3*c + 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d *Sin[e + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^2))))/(2*a^2*(c - d))
3.6.9.3.1 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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{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[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1290\) vs. \(2(377)=754\).
Time = 3.34 (sec) , antiderivative size = 1291, normalized size of antiderivative = 4.02
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a*(1/(c-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 ))))-d/(c-d)^2*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^ 2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f* x+e))/(c+d))^(1/2)*(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))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+ e))/(c+d))^(1/2)*(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/(c-d)*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin (f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f* x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 2198, normalized size of antiderivative = 6.85 \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
1/18*((sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^3 + sqrt(2)*(12*c^4*d - 4*c^3*d^2 - 41*c^2*d^3 - 48*c*d^4 - 15*d^5)*cos(f*x + e)^2 - sqrt(2)*(6*c^5 - 5*c^4*d - 12*c^3*d^2 - 20*c^2*d^3 - 18*c*d^4 - 1 5*d^5)*cos(f*x + e) + (sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5) *cos(f*x + e)^2 - 2*sqrt(2)*(6*c^4*d - 5*c^3*d^2 - 18*c^2*d^3 - 15*c*d^4)* cos(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48*c*d ^4 - 15*d^5))*sin(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^ 2*d^3 - 48*c*d^4 - 15*d^5))*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*s in(f*x + e) - 2*I*c)/d) + (sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15* d^5)*cos(f*x + e)^3 + sqrt(2)*(12*c^4*d - 4*c^3*d^2 - 41*c^2*d^3 - 48*c*d^ 4 - 15*d^5)*cos(f*x + e)^2 - sqrt(2)*(6*c^5 - 5*c^4*d - 12*c^3*d^2 - 20*c^ 2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e) + (sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^2 - 2*sqrt(2)*(6*c^4*d - 5*c^3*d^2 - 18* c^2*d^3 - 15*c*d^4)*cos(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48*c*d^4 - 15*d^5))*sin(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48*c*d^4 - 15*d^5))*sqrt(-I*d)*weierstrassPInv erse(-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)*(3*I*c^2*d^3 + 20*I*c*d^4 + 9*I*d^5)*cos(f*x + e)^3 + sqrt(2)*(6*I*c^3*d^2 + 43*I*c^2*...
\[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\int \frac {1}{c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a} \]
Integral(1/(c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + c**2*sqrt(c + d*s in(e + f*x)) + 2*c*d*sqrt(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)**3 + d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2), x)/a
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Hanged} \]