\(\int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx\) [524]

Optimal result

Integrand size = 27, antiderivative size = 322 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\left (4 c^2+15 c d+27 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{30 a^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \left (4 c^2+11 c d+15 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{30 a^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

-2/15*(c-d)*(c+3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/f/(a+a*sin(f*x+e)) 
^2-1/30*(4*c^2+15*c*d+27*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a^3+a^3 
*sin(f*x+e))-1/5*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e) 
)^3+1/30*(4*c^2+15*c*d+27*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/f/((c+d*sin(f*x+e))/(c+d))^(1 
/2)+1/30*(c+d)*(4*c^2+11*c*d+15*d^2)*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/f/(c+d*sin(f*x 
+e))^(1/2)
 

Mathematica [A] (verified)

Time = 4.89 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\left (\left (4 c^2+15 c d+27 d^2\right ) (c+d \sin (e+f x))\right )-\frac {\left (-2 d (35 c+57 d) \cos \left (\frac {1}{2} (e+f x)\right )+\left (20 c^2+74 c d+90 d^2\right ) \cos \left (\frac {3}{2} (e+f x)\right )+2 \left (-3 \left (6 c^2+11 c d+29 d^2\right )+2 \left (2 c^2+7 c d-9 d^2\right ) \cos (e+f x)+\left (4 c^2+15 c d+27 d^2\right ) \cos (2 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+(c-15 d) d^2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (4 c^2+15 c d+27 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{30 a^3 f (1+\sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \] Input:

Integrate[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^3,x]
 

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 + 15*c*d + 27*d^2)*(c + 
 d*Sin[e + f*x])) - ((-2*d*(35*c + 57*d)*Cos[(e + f*x)/2] + (20*c^2 + 74*c 
*d + 90*d^2)*Cos[(3*(e + f*x))/2] + 2*(-3*(6*c^2 + 11*c*d + 29*d^2) + 2*(2 
*c^2 + 7*c*d - 9*d^2)*Cos[e + f*x] + (4*c^2 + 15*c*d + 27*d^2)*Cos[2*(e + 
f*x)])*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(2*(Cos[(e + f*x)/2] + Sin[ 
(e + f*x)/2])^5) + (c - 15*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 + 15*c*d + 27*d^2)*((c 
 + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e 
+ Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(30* 
a^3*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 3244, 27, 3042, 3456, 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.

\(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a \sin (e+f x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3244

\(\displaystyle -\frac {\int -\frac {\sqrt {c+d \sin (e+f x)} \left (a \left (4 c^2+9 d c-3 d^2\right )+a d (c+9 d) \sin (e+f x)\right )}{2 (\sin (e+f x) a+a)^2}dx}{5 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (a \left (4 c^2+9 d c-3 d^2\right )+a d (c+9 d) \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (a \left (4 c^2+9 d c-3 d^2\right )+a d (c+9 d) \sin (e+f x)\right )}{(\sin (e+f x) a+a)^2}dx}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\int \frac {\left (4 c^3+13 d c^2+19 d^2 c-6 d^3\right ) a^2+d \left (2 c^2+7 d c+21 d^2\right ) \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (4 c^3+13 d c^2+19 d^2 c-6 d^3\right ) a^2+d \left (2 c^2+7 d c+21 d^2\right ) \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {-\frac {\int \frac {(c-15 d) (c-d) d^2 a^3+(c-d) d \left (4 c^2+15 d c+27 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {(c-15 d) (c-d) d^2 a^3+(c-d) d \left (4 c^2+15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int \frac {(c-15 d) (c-d) d^2 a^3+(c-d) d \left (4 c^2+15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {-\frac {a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {-\frac {\frac {a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {-\frac {a^2 \left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac {\frac {2 a^3 (c-d) \left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}}{3 a^2}-\frac {4 a (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}}{10 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}\)

Input:

Int[(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^3,x]
 

Output:

-1/5*((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f 
*x])^3) + ((-4*a*(c - d)*(c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/ 
(3*f*(a + a*Sin[e + f*x])^2) + (-((a^2*(4*c^2 + 15*c*d + 27*d^2)*Cos[e + f 
*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) - ((2*a^3*(c - d)* 
(4*c^2 + 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqr 
t[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 + 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c 
+ d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(2 
*a^2*(c - d)))/(3*a^2))/(10*a^2)
 

Defintions of rubi rules used

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 3244

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

rule 3456

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[(A*b - a*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*(a*d*n - b*c*(m + 1)) - B*(a*c*m + 
 b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B}, 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] && (In 
tegerQ[2*n] || 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. \(1607\) vs. \(2(303)=606\).

Time = 125.07 (sec) , antiderivative size = 1608, normalized size of antiderivative = 4.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1608\)

Input:

int((c+d*sin(f*x+e))^(5/2)/(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*(2*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)*EllipticF(((c+d*sin(f*x+e))/( 
c-d))^(1/2),((c-d)/(c+d))^(1/2))+(c^3-3*c^2*d+3*c*d^2-d^3)*(-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-s 
in(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))-1/30*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))*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))))+3*d*(c^2-2*c*d+d^2)*(-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* 
sin(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...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 1409, normalized size of antiderivative = 4.38 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
 

Output:

1/90*(((8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^3 - 32*c^3 - 12 
0*c^2*d - 204*c*d^2 - 180*d^3 + 3*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*c 
os(f*x + e)^2 - 2*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e) - (3 
2*c^3 + 120*c^2*d + 204*c*d^2 + 180*d^3 - (8*c^3 + 30*c^2*d + 51*c*d^2 + 4 
5*d^3)*cos(f*x + e)^2 + 2*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*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 + 30*c^2*d + 51*c*d^2 + 45*d^3)*cos(f*x + e)^ 
3 - 32*c^3 - 120*c^2*d - 204*c*d^2 - 180*d^3 + 3*(8*c^3 + 30*c^2*d + 51*c* 
d^2 + 45*d^3)*cos(f*x + e)^2 - 2*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45*d^3)*co 
s(f*x + e) - (32*c^3 + 120*c^2*d + 204*c*d^2 + 180*d^3 - (8*c^3 + 30*c^2*d 
 + 51*c*d^2 + 45*d^3)*cos(f*x + e)^2 + 2*(8*c^3 + 30*c^2*d + 51*c*d^2 + 45 
*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) - 3*((-4*I*c^2*d - 15*I*c*d^2 - 27*I*d 
^3)*cos(f*x + e)^3 + 16*I*c^2*d + 60*I*c*d^2 + 108*I*d^3 + 3*(-4*I*c^2*d - 
 15*I*c*d^2 - 27*I*d^3)*cos(f*x + e)^2 + 2*(4*I*c^2*d + 15*I*c*d^2 + 27*I* 
d^3)*cos(f*x + e) + (16*I*c^2*d + 60*I*c*d^2 + 108*I*d^3 + (-4*I*c^2*d - 1 
5*I*c*d^2 - 27*I*d^3)*cos(f*x + e)^2 + 2*(4*I*c^2*d + 15*I*c*d^2 + 27*I*d^ 
3)*cos(f*x + e))*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate((c+d*sin(f*x+e))**(5/2)/(a+a*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
 

Output:

integrate((d*sin(f*x + e) + c)^(5/2)/(a*sin(f*x + e) + a)^3, x)
 

Giac [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((d*sin(f*x + e) + c)^(5/2)/(a*sin(f*x + e) + a)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \] Input:

int((c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x))^3,x)
 

Output:

int((c + d*sin(e + f*x))^(5/2)/(a + a*sin(e + f*x))^3, x)
 

Reduce [F]

\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) c^{2}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) d^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right ) c d}{a^{3}} \] Input:

int((c+d*sin(f*x+e))^(5/2)/(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)*c**2 + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2)/(s 
in(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)*d**2 + 2*int(( 
sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)**3 + 3*sin(e + f*x)** 
2 + 3*sin(e + f*x) + 1),x)*c*d)/a**3