\(\int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx\) [526]

Optimal result

Integrand size = 27, antiderivative size = 334 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}-\frac {(2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a (c-d) f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 (c-d)^2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (4 c^2-5 c d-3 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{30 a^3 (c-d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(4 c-5 d) (c+d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{30 a^3 (c-d) f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/5*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^3-1/15*(2*c-d)*c 
os(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/(c-d)/f/(a+a*sin(f*x+e))^2-1/30*(4*c^2- 
5*c*d-3*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(c-d)^2/f/(a^3+a^3*sin(f*x+ 
e))+1/30*(4*c^2-5*c*d-3*d^2)*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)^2/f/((c+d*sin(f*x+e))/(c+ 
d))^(1/2)+1/30*(4*c-5*d)*(c+d)*InverseJacobiAM(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)/f/(c+d*sin(f*x 
+e))^(1/2)
 

Mathematica [A] (verified)

Time = 4.97 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\left (\left (4 c^2-5 c d-3 d^2\right ) (c+d \sin (e+f x))\right )+\frac {2 \left (6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (c-d) (2 c-d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-(c-d) (2 c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+\left (4 c^2-5 c d-3 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+(c-5 d) d^2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (4 c^2-5 c d-3 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{30 a^3 (c-d)^2 f (1+\sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \] Input:

Integrate[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x])^3,x]
 

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 - 5*c*d - 3*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]) + 2*(c - d)*(2*c - d)*Sin[(e + f*x)/2]*(Cos[( 
e + f*x)/2] + Sin[(e + f*x)/2])^2 - (c - d)*(2*c - d)*(Cos[(e + f*x)/2] + 
Sin[(e + f*x)/2])^3 + (4*c^2 - 5*c*d - 3*d^2)*Sin[(e + f*x)/2]*(Cos[(e + f 
*x)/2] + Sin[(e + f*x)/2])^4)*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Si 
n[(e + f*x)/2])^5 + (c - 5*d)*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/( 
c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (4*c^2 - 5*c*d - 3*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)]))/(30*a^ 
3*(c - d)^2*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.95 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3243, 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.

\(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a \sin (e+f x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a \sin (e+f x)+a)^3}dx\)

\(\Big \downarrow \) 3243

\(\displaystyle \frac {\int \frac {a (4 c+d)+3 a d \sin (e+f x)}{2 (\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{5 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (4 c+d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (4 c+d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {\left (4 c^2-3 d c-4 d^2\right ) a^2+(2 c-d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-3 d c-4 d^2\right ) a^2+(2 c-d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-3 d c-4 d^2\right ) a^2+(2 c-d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {-\frac {\int \frac {(c-5 d) d^2 a^3+d \left (4 c^2-5 d c-3 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {(c-5 d) d^2 a^3+d \left (4 c^2-5 d c-3 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int \frac {(c-5 d) d^2 a^3+d \left (4 c^2-5 d c-3 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {-\frac {a^3 \left (4 c^2-5 c d-3 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {a^3 \left (4 c^2-5 c d-3 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {-\frac {\frac {a^3 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {a^3 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {-\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}-\frac {\frac {2 a^3 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}}{3 a^2 (c-d)}-\frac {2 a (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3}\)

Input:

Int[Sqrt[c + d*Sin[e + f*x]]/(a + a*Sin[e + f*x])^3,x]
 

Output:

-1/5*(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x])^3) + 
((-2*a*(2*c - 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 - 5*c*d - 3*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 - 5*c 
*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e 
+ f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a^3*(4*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)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(2*a^2*(c - d)))/(3*a^2*(c - d 
)))/(10*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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]
 

rule 3134

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]
 

rule 3140

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]
 

rule 3142

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]
 

rule 3231

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]
 

rule 3243

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m* 
((c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a*d*n - b*c 
*(m + 1) - b*d*(m + n + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e 
, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && 
LtQ[m, -1] && LtQ[0, n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c 
, 0]))
 

rule 3457

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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1051\) vs. \(2(315)=630\).

Time = 1.61 (sec) , antiderivative size = 1052, normalized size of antiderivative = 3.15

method	result	size
default	\(\text {Expression too large to display}\)	\(1052\)

Input:

int((c+d*sin(f*x+e))^(1/2)/(a+sin(f*x+e)*a)^3,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/a^3*(d*(-1/3/(c-d)*(-(-c-d*sin(f*x 
+e))*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^2-1/3*(-sin(f*x+e)^2*d-c*sin(f*x+e 
)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/((1+sin(f*x+e))*(-1+sin(f*x+e))*(-c-d*si 
n(f*x+e)))^(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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-( 
-c-d*sin(f*x+e))*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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2) 
/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x 
+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^ 
(1/2),((c-d)/(c+d))^(1/2))))+(c-d)*(-1/5/(c-d)*(-(-c-d*sin(f*x+e))*cos(f*x 
+e)^2)^(1/2)/(1+sin(f*x+e))^3-2/15*(c-3*d)/(c-d)^2*(-(-c-d*sin(f*x+e))*cos 
(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^2-1/30*(-sin(f*x+e)^2*d-c*sin(f*x+e)+d*sin 
(f*x+e)+c)/(c-d)^3*(4*c^2-15*c*d+27*d^2)/((1+sin(f*x+e))*(-1+sin(f*x+e))*( 
-c-d*sin(f*x+e)))^(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)* 
(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*El 
lipticF(((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)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1554, normalized size of antiderivative = 4.65 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
 

Output:

1/90*(((8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f*x + e)^3 - 32*c^3 + 40* 
c^2*d + 36*c*d^2 - 60*d^3 + 3*(8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f* 
x + e)^2 - 2*(8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f*x + e) - (32*c^3 
- 40*c^2*d - 36*c*d^2 + 60*d^3 - (8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos 
(f*x + e)^2 + 2*(8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f*x + e))*sin(f* 
x + e))*sqrt(1/2*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) + ((8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f*x + e)^3 - 32*c^3 + 
 40*c^2*d + 36*c*d^2 - 60*d^3 + 3*(8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*co 
s(f*x + e)^2 - 2*(8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f*x + e) - (32* 
c^3 - 40*c^2*d - 36*c*d^2 + 60*d^3 - (8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3) 
*cos(f*x + e)^2 + 2*(8*c^3 - 10*c^2*d - 9*c*d^2 + 15*d^3)*cos(f*x + e))*si 
n(f*x + e))*sqrt(-1/2*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*((-4*I*c^2*d + 5*I*c*d^2 + 3*I*d^3)*cos(f*x + e)^3 + 16* 
I*c^2*d - 20*I*c*d^2 - 12*I*d^3 + 3*(-4*I*c^2*d + 5*I*c*d^2 + 3*I*d^3)*cos 
(f*x + e)^2 + 2*(4*I*c^2*d - 5*I*c*d^2 - 3*I*d^3)*cos(f*x + e) + (16*I*c^2 
*d - 20*I*c*d^2 - 12*I*d^3 + (-4*I*c^2*d + 5*I*c*d^2 + 3*I*d^3)*cos(f*x + 
e)^2 + 2*(4*I*c^2*d - 5*I*c*d^2 - 3*I*d^3)*cos(f*x + e))*sin(f*x + e))*sqr 
t(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9...
 

Sympy [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\frac {\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sin ^{3}{\left (e + f x \right )} + 3 \sin ^{2}{\left (e + f x \right )} + 3 \sin {\left (e + f x \right )} + 1}\, dx}{a^{3}} \] Input:

integrate((c+d*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))**3,x)
 

Output:

Integral(sqrt(c + d*sin(e + f*x))/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3 
*sin(e + f*x) + 1), x)/a**3
 

Maxima [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a)^3, x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:

int((c + d*sin(e + f*x))^(1/2)/(a + a*sin(e + f*x))^3,x)
 

Output:

int((c + d*sin(e + f*x))^(1/2)/(a + a*sin(e + f*x))^3, x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx=\frac {\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x}{a^{3}} \] Input:

int((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^3,x)
 

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin( 
e + f*x) + 1),x)/a**3