\(\int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx\) [141]

Optimal result

Integrand size = 21, antiderivative size = 807 \[ \int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Output:

3*x^2/a/f^2/(a+I*a*sinh(f*x+e))^(1/2)+24*I*x*arctanh(exp(1/2*e+3/4*I*Pi+1/ 
2*f*x))*cosh(1/2*e+1/4*I*Pi+1/2*f*x)/a/f^3/(a+I*a*sinh(f*x+e))^(1/2)-I*x^3 
*arctanh(exp(1/2*e+3/4*I*Pi+1/2*f*x))*cosh(1/2*e+1/4*I*Pi+1/2*f*x)/a/f/(a+ 
I*a*sinh(f*x+e))^(1/2)-24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,exp(1/2 
*e+3/4*I*Pi+1/2*f*x))/a/f^4/(a+I*a*sinh(f*x+e))^(1/2)+3*I*x^2*cosh(1/2*e+1 
/4*I*Pi+1/2*f*x)*polylog(2,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^2/(a+I*a*sinh( 
f*x+e))^(1/2)+24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(2,-exp(1/2*e+3/4*I 
*Pi+1/2*f*x))/a/f^4/(a+I*a*sinh(f*x+e))^(1/2)-3*I*x^2*cosh(1/2*e+1/4*I*Pi+ 
1/2*f*x)*polylog(2,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^2/(a+I*a*sinh(f*x+e)) 
^(1/2)-12*I*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(3,exp(1/2*e+3/4*I*Pi+1/ 
2*f*x))/a/f^3/(a+I*a*sinh(f*x+e))^(1/2)+12*I*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x 
)*polylog(3,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^3/(a+I*a*sinh(f*x+e))^(1/2)+ 
24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(4,exp(1/2*e+3/4*I*Pi+1/2*f*x))/a 
/f^4/(a+I*a*sinh(f*x+e))^(1/2)-24*I*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*polylog(4 
,-exp(1/2*e+3/4*I*Pi+1/2*f*x))/a/f^4/(a+I*a*sinh(f*x+e))^(1/2)+1/2*x^3*tan 
h(1/2*e+1/4*I*Pi+1/2*f*x)/a/f/(a+I*a*sinh(f*x+e))^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.02 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.68 \[ \int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \left (f^2 x^2 (6+i f x) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) (-1)^{3/4} \left (-48 e \text {arctanh}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+2 e^3 \text {arctanh}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 e \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+e^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 f x \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+f^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 e \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-e^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-f^3 x^3 \log \left (1+(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-6 \left (-8+f^2 x^2\right ) \operatorname {PolyLog}\left (2,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+6 \left (-8+f^2 x^2\right ) \operatorname {PolyLog}\left (2,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \operatorname {PolyLog}\left (3,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 f x \operatorname {PolyLog}\left (3,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-48 \operatorname {PolyLog}\left (4,-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+48 \operatorname {PolyLog}\left (4,(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2+2 f^3 x^3 \sinh \left (\frac {1}{2} (e+f x)\right )\right )}{2 f^4 (a+i a \sinh (e+f x))^{3/2}} \] Input:

Integrate[x^3/(a + I*a*Sinh[e + f*x])^(3/2),x]
 

Output:

((Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])*(f^2*x^2*(6 + I*f*x)*(Cosh[(e + 
 f*x)/2] + I*Sinh[(e + f*x)/2]) + (1/2 - I/2)*(-1)^(3/4)*(-48*e*ArcTanh[(- 
1)^(3/4)*E^((e + f*x)/2)] + 2*e^3*ArcTanh[(-1)^(3/4)*E^((e + f*x)/2)] - 24 
*e*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] + e^3*Log[1 - (-1)^(3/4)*E^((e + f* 
x)/2)] - 24*f*x*Log[1 - (-1)^(3/4)*E^((e + f*x)/2)] + f^3*x^3*Log[1 - (-1) 
^(3/4)*E^((e + f*x)/2)] + 24*e*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - e^3*L 
og[1 + (-1)^(3/4)*E^((e + f*x)/2)] + 24*f*x*Log[1 + (-1)^(3/4)*E^((e + f*x 
)/2)] - f^3*x^3*Log[1 + (-1)^(3/4)*E^((e + f*x)/2)] - 6*(-8 + f^2*x^2)*Pol 
yLog[2, -((-1)^(3/4)*E^((e + f*x)/2))] + 6*(-8 + f^2*x^2)*PolyLog[2, (-1)^ 
(3/4)*E^((e + f*x)/2)] + 24*f*x*PolyLog[3, -((-1)^(3/4)*E^((e + f*x)/2))] 
- 24*f*x*PolyLog[3, (-1)^(3/4)*E^((e + f*x)/2)] - 48*PolyLog[4, -((-1)^(3/ 
4)*E^((e + f*x)/2))] + 48*PolyLog[4, (-1)^(3/4)*E^((e + f*x)/2)])*(Cosh[(e 
 + f*x)/2] + I*Sinh[(e + f*x)/2])^2 + 2*f^3*x^3*Sinh[(e + f*x)/2]))/(2*f^4 
*(a + I*a*Sinh[e + f*x])^(3/2))
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.59, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3800, 3042, 4674, 3042, 4670, 2715, 2838, 3011, 7163, 2720, 7143}

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.

\(\displaystyle \int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \int x^3 \text {sech}^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \int x^3 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 4674

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {12 \int x \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f^2}+\frac {1}{2} \int x^3 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {12 \int x \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{f^2}+\frac {1}{2} \int x^3 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {12 \left (\frac {2 i \int \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {2 i \int \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}+\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )}{f^2}+\frac {1}{2} \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {12 \left (\frac {4 i \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}-\frac {4 i \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}+\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )}{f^2}+\frac {1}{2} \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f^2}+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (-\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f^2}+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {2 \int \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {2 \int \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )dx}{f}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f^2}+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \int e^{\frac {1}{4} (i \pi -2 e)-\frac {f x}{2}} \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )de^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f^2}+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \left (\frac {1}{2} \left (\frac {4 i x^3 \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}-\frac {4 \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f}\right )}{f}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{f^2}\right )}{f^2}+\frac {6 x^2 \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{2 a \sqrt {a+i a \sinh (e+f x)}}\)

Input:

Int[x^3/(a + I*a*Sinh[e + f*x])^(3/2),x]
 

Output:

(Cosh[e/2 + (I/4)*Pi + (f*x)/2]*((-12*(((4*I)*x*ArcTanh[E^((2*e - I*Pi)/4 
+ (f*x)/2)])/f + ((4*I)*PolyLog[2, -E^((2*e - I*Pi)/4 + (f*x)/2)])/f^2 - ( 
(4*I)*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/f^2))/f^2 + (((4*I)*x^3*Ar 
cTanh[E^((2*e - I*Pi)/4 + (f*x)/2)])/f - ((6*I)*((-2*x^2*PolyLog[2, -E^((2 
*e - I*Pi)/4 + (f*x)/2)])/f + (4*((2*x*PolyLog[3, -E^((2*e - I*Pi)/4 + (f* 
x)/2)])/f - (4*PolyLog[4, -E^((2*e - I*Pi)/4 + (f*x)/2)])/f^2))/f))/f + (( 
6*I)*((-2*x^2*PolyLog[2, E^((2*e - I*Pi)/4 + (f*x)/2)])/f + (4*((2*x*PolyL 
og[3, E^((2*e - I*Pi)/4 + (f*x)/2)])/f - (4*PolyLog[4, E^((2*e - I*Pi)/4 + 
 (f*x)/2)])/f^2))/f))/f)/2 + (6*x^2*Sech[e/2 + (I/4)*Pi + (f*x)/2])/f^2 + 
(x^3*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/f))/(2 
*a*Sqrt[a + I*a*Sinh[e + f*x]])
 

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

Maple [F]

\[\int \frac {x^{3}}{\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)
 

Output:

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)
 

Fricas [F]

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

integrate(x^3/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

((a^2*f^2*e^(2*f*x + 2*e) - 2*I*a^2*f^2*e^(f*x + e) - a^2*f^2)*integral(1/ 
2*(-I*f^2*x^3 + 24*I*x)*sqrt(1/2*I*a*e^(-f*x - e))*e^(f*x + e)/(a^2*f^2*e^ 
(f*x + e) - I*a^2*f^2), x) + ((-I*f*x^3 - 6*I*x^2)*e^(2*f*x + 2*e) + (f*x^ 
3 - 6*x^2)*e^(f*x + e))*sqrt(1/2*I*a*e^(-f*x - e)))/(a^2*f^2*e^(2*f*x + 2* 
e) - 2*I*a^2*f^2*e^(f*x + e) - a^2*f^2)
 

Sympy [F]

\[ \int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {x^{3}}{\left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**3/(a+I*a*sinh(f*x+e))**(3/2),x)
 

Output:

Integral(x**3/(I*a*(sinh(e + f*x) - I))**(3/2), x)
 

Maxima [F]

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

integrate(x^3/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^3/(I*a*sinh(f*x + e) + a)^(3/2), x)
 

Giac [F]

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

integrate(x^3/(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

integrate(x^3/(I*a*sinh(f*x + e) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx=\int \frac {x^3}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:

int(x^3/(a + a*sinh(e + f*x)*1i)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int(x^3/(a + a*sinh(e + f*x)*1i)^(3/2), x)
 

Reduce [F]

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

int(x^3/(a+I*a*sinh(f*x+e))^(3/2),x)
 

Output:

(sqrt(a)*(int((sqrt(sinh(e + f*x)*i + 1)*x**3)/(sinh(e + f*x)**3*i + sinh( 
e + f*x)**2 + sinh(e + f*x)*i + 1),x) - int((sqrt(sinh(e + f*x)*i + 1)*sin 
h(e + f*x)*x**3)/(sinh(e + f*x)**3*i + sinh(e + f*x)**2 + sinh(e + f*x)*i 
+ 1),x)*i))/a**2