3.2.25 \(\int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx\) [125]

3.2.25.1 Optimal result

Integrand size = 21, antiderivative size = 377 \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {1280 a \sqrt {a+i a \sinh (e+f x)}}{9 f^4}-\frac {16 a x^2 \sqrt {a+i a \sinh (e+f x)}}{f^2}-\frac {64 a \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{27 f^4}-\frac {8 a x^2 \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{3 f^2}+\frac {32 a x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{9 f^3}+\frac {4 a x^3 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{3 f}+\frac {640 a x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{9 f^3}+\frac {8 a x^3 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 f} \]

output

-1280/9*a*(a+I*a*sinh(f*x+e))^(1/2)/f^4-16*a*x^2*(a+I*a*sinh(f*x+e))^(1/2) 
/f^2-64/27*a*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^2*(a+I*a*sinh(f*x+e))^(1/2)/f^4- 
8/3*a*x^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^2*(a+I*a*sinh(f*x+e))^(1/2)/f^2+32/ 
9*a*x*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*sinh(1/2*e+1/4*I*Pi+1/2*f*x)*(a+I*a*sin 
h(f*x+e))^(1/2)/f^3+4/3*a*x^3*cosh(1/2*e+1/4*I*Pi+1/2*f*x)*sinh(1/2*e+1/4* 
I*Pi+1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)/f+640/9*a*x*(a+I*a*sinh(f*x+e))^(1 
/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f^3+8/3*a*x^3*(a+I*a*sinh(f*x+e))^(1/2)*t 
anh(1/2*e+1/4*I*Pi+1/2*f*x)/f
 

3.2.25.2 Mathematica [A] (verified)

Time = 7.66 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.71 \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {a (-i+\sinh (e+f x)) \sqrt {a+i a \sinh (e+f x)} \left (81 \left (48 i+24 f x+6 i f^2 x^2+f^3 x^3\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (-16 i+24 f x-18 i f^2 x^2+9 f^3 x^3\right ) \cosh \left (\frac {3}{2} (e+f x)\right )-3888 \sinh \left (\frac {1}{2} (e+f x)\right )-1944 i f x \sinh \left (\frac {1}{2} (e+f x)\right )-486 f^2 x^2 \sinh \left (\frac {1}{2} (e+f x)\right )-81 i f^3 x^3 \sinh \left (\frac {1}{2} (e+f x)\right )-16 \sinh \left (\frac {3}{2} (e+f x)\right )+24 i f x \sinh \left (\frac {3}{2} (e+f x)\right )-18 f^2 x^2 \sinh \left (\frac {3}{2} (e+f x)\right )+9 i f^3 x^3 \sinh \left (\frac {3}{2} (e+f x)\right )\right )}{27 f^4 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3} \]

input

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

output

-1/27*(a*(-I + Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]]*(81*(48*I + 24*f 
*x + (6*I)*f^2*x^2 + f^3*x^3)*Cosh[(e + f*x)/2] + (-16*I + 24*f*x - (18*I) 
*f^2*x^2 + 9*f^3*x^3)*Cosh[(3*(e + f*x))/2] - 3888*Sinh[(e + f*x)/2] - (19 
44*I)*f*x*Sinh[(e + f*x)/2] - 486*f^2*x^2*Sinh[(e + f*x)/2] - (81*I)*f^3*x 
^3*Sinh[(e + f*x)/2] - 16*Sinh[(3*(e + f*x))/2] + (24*I)*f*x*Sinh[(3*(e + 
f*x))/2] - 18*f^2*x^2*Sinh[(3*(e + f*x))/2] + (9*I)*f^3*x^3*Sinh[(3*(e + f 
*x))/2]))/(f^4*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3)
 

3.2.25.3 Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.06, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.095, Rules used = {3042, 3800, 3042, 3792, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 3791, 3042, 3777, 26, 3042, 26, 3118}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{3 f^2}+\frac {2}{3} \int x^3 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \int x^3 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i \int -i x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 \int x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 \int -i x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \int x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \int x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int -i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int -i \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \int x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int -i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 \int -i \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (-\frac {4 x^2 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}\right )}{f}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {8 \left (-\frac {4 \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )+\frac {2 x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )}{3 f^2}+\frac {2 x^3 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

input

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

output

2*a*Sech[e/2 + (I/4)*Pi + (f*x)/2]*((-4*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2] 
^3)/(3*f^2) + (2*x^3*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sinh[e/2 + (I/4)*Pi 
+ (f*x)/2])/(3*f) + (8*((-4*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^3)/(9*f^2) + (2 
*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/(3*f) 
+ (2*((-4*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f^2 + (2*x*Sinh[e/2 + (I/4)*Pi + 
 (f*x)/2])/f))/3))/(3*f^2) + (2*((2*x^3*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f 
+ ((6*I)*(((2*I)*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f - ((4*I)*((-4*Cosh[ 
e/2 + (I/4)*Pi + (f*x)/2])/f^2 + (2*x*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f))/ 
f))/f))/3)*Sqrt[a + I*a*Sinh[e + f*x]]
 

3.2.25.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3792

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

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])
 

3.2.25.4 Maple [F]

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

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)
 

3.2.25.5 Fricas [F(-2)]

Exception generated. \[ \int x^3 (a+i a \sinh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

3.2.25.6 Sympy [F]

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

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)
 

3.2.25.7 Maxima [F]

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

input

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

output

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

3.2.25.8 Giac [F]

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

input

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

output

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

3.2.25.9 Mupad [F(-1)]

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

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)