\(\int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx\) [131]

Optimal result

Integrand size = 21, antiderivative size = 506 \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=-\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {9536 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3} \] Output:

-256/15*a^2*x*(a+I*a*sinh(d*x+c))^(1/2)/d^2-128/45*a^2*x*cosh(1/2*c+1/4*I* 
Pi+1/2*d*x)^2*(a+I*a*sinh(d*x+c))^(1/2)/d^2-32/25*a^2*x*cosh(1/2*c+1/4*I*P 
i+1/2*d*x)^4*(a+I*a*sinh(d*x+c))^(1/2)/d^2+32/15*a^2*x^2*cosh(1/2*c+1/4*I* 
Pi+1/2*d*x)*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d+8/5*a 
^2*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^3*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a* 
sinh(d*x+c))^(1/2)/d+9536/225*a^2*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4 
*I*Pi+1/2*d*x)/d^3+64/15*a^2*x^2*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4* 
I*Pi+1/2*d*x)/d+2432/675*a^2*sinh(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*sinh(d* 
x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d^3+64/125*a^2*sinh(1/2*c+1/4*I*P 
i+1/2*d*x)^4*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d^3
 

Mathematica [A] (verified)

Time = 9.67 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.59 \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a+i a \sinh (c+d x)} \left (33750 i \left (8+4 i d x+d^2 x^2\right ) \cosh \left (\frac {1}{2} (c+d x)\right )+625 \left (8 i+12 d x+9 i d^2 x^2\right ) \cosh \left (\frac {3}{2} (c+d x)\right )-216 i \cosh \left (\frac {5}{2} (c+d x)\right )+540 d x \cosh \left (\frac {5}{2} (c+d x)\right )-675 i d^2 x^2 \cosh \left (\frac {5}{2} (c+d x)\right )+270000 \sinh \left (\frac {1}{2} (c+d x)\right )-135000 i d x \sinh \left (\frac {1}{2} (c+d x)\right )+33750 d^2 x^2 \sinh \left (\frac {1}{2} (c+d x)\right )-5000 \sinh \left (\frac {3}{2} (c+d x)\right )-7500 i d x \sinh \left (\frac {3}{2} (c+d x)\right )-5625 d^2 x^2 \sinh \left (\frac {3}{2} (c+d x)\right )-216 \sinh \left (\frac {5}{2} (c+d x)\right )+540 i d x \sinh \left (\frac {5}{2} (c+d x)\right )-675 d^2 x^2 \sinh \left (\frac {5}{2} (c+d x)\right )\right )}{6750 d^3 \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:

Integrate[x^2*(a + I*a*Sinh[c + d*x])^(5/2),x]
 

Output:

(a^2*Sqrt[a + I*a*Sinh[c + d*x]]*((33750*I)*(8 + (4*I)*d*x + d^2*x^2)*Cosh 
[(c + d*x)/2] + 625*(8*I + 12*d*x + (9*I)*d^2*x^2)*Cosh[(3*(c + d*x))/2] - 
 (216*I)*Cosh[(5*(c + d*x))/2] + 540*d*x*Cosh[(5*(c + d*x))/2] - (675*I)*d 
^2*x^2*Cosh[(5*(c + d*x))/2] + 270000*Sinh[(c + d*x)/2] - (135000*I)*d*x*S 
inh[(c + d*x)/2] + 33750*d^2*x^2*Sinh[(c + d*x)/2] - 5000*Sinh[(3*(c + d*x 
))/2] - (7500*I)*d*x*Sinh[(3*(c + d*x))/2] - 5625*d^2*x^2*Sinh[(3*(c + d*x 
))/2] - 216*Sinh[(5*(c + d*x))/2] + (540*I)*d*x*Sinh[(5*(c + d*x))/2] - 67 
5*d^2*x^2*Sinh[(5*(c + d*x))/2]))/(6750*d^3*(Cosh[(c + d*x)/2] + I*Sinh[(c 
 + d*x)/2]))
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.91, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3800, 3042, 3792, 3042, 3113, 2009, 3792, 3042, 3113, 2009, 3777, 26, 3042, 26, 3777, 3042, 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.

\(\displaystyle \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 (a+a \sin (i c+i d x))^{5/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \int x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^5dx\)

\(\Big \downarrow \) 3792

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {8 \int \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{25 d^2}+\frac {4}{5} \int x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {8 \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^5dx}{25 d^2}+\frac {4}{5} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3113

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {16 i \int \left (\sinh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )+2 \sinh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )+1\right )d\left (-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}+\frac {4}{5} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3792

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{9 d^2}+\frac {2}{3} \int x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{9 d^2}+\frac {2}{3} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3113

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {16 i \int \left (\sinh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )+1\right )d\left (-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}+\frac {2}{3} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \int -i x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \int x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \int -i x \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {4 i \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i \int \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{d}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx}{d}\right )}{d}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {16 i \left (-\frac {1}{5} i \sinh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-\frac {2}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{25 d^3}-\frac {8 x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{9 d^3}+\frac {2}{3} \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )}{d}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {8 x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

Input:

Int[x^2*(a + I*a*Sinh[c + d*x])^(5/2),x]
 

Output:

4*a^2*Sech[c/2 + (I/4)*Pi + (d*x)/2]*((-8*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2] 
^5)/(25*d^2) + (2*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sinh[c/2 + (I/4)*Pi 
 + (d*x)/2])/(5*d) + (((16*I)/25)*((-I)*Sinh[c/2 + (I/4)*Pi + (d*x)/2] - ( 
(2*I)/3)*Sinh[c/2 + (I/4)*Pi + (d*x)/2]^3 - (I/5)*Sinh[c/2 + (I/4)*Pi + (d 
*x)/2]^5))/d^3 + (4*((-8*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3)/(9*d^2) + (2* 
x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^2*Sinh[c/2 + (I/4)*Pi + (d*x)/2])/(3*d) 
 + (((16*I)/9)*((-I)*Sinh[c/2 + (I/4)*Pi + (d*x)/2] - (I/3)*Sinh[c/2 + (I/ 
4)*Pi + (d*x)/2]^3))/d^3 + (2*((2*x^2*Sinh[c/2 + (I/4)*Pi + (d*x)/2])/d + 
((4*I)*(((2*I)*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2])/d - ((4*I)*Sinh[c/2 + (I/ 
4)*Pi + (d*x)/2])/d^2))/d))/3))/5)*Sqrt[a + I*a*Sinh[c + d*x]]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

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 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])
 

Maple [F]

\[\int x^{2} \left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int(x^2*(a+I*a*sinh(d*x+c))^(5/2),x)
 

Output:

int(x^2*(a+I*a*sinh(d*x+c))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2*(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+I*a*sinh(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((I*a*sinh(d*x + c) + a)^(5/2)*x^2, x)
                                                                                    
                                                                                    
 

Giac [F]

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

integrate(x^2*(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((I*a*sinh(d*x + c) + a)^(5/2)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\int x^2\,{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \] Input:

int(x^2*(a + a*sinh(c + d*x)*1i)^(5/2),x)
 

Output:

int(x^2*(a + a*sinh(c + d*x)*1i)^(5/2), x)
 

Reduce [F]

\[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right )^{2} x^{2}d x \right )+\int \sqrt {\sinh \left (d x +c \right ) i +1}\, x^{2}d x +2 \left (\int \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right ) x^{2}d x \right ) i \right ) \] Input:

int(x^2*(a+I*a*sinh(d*x+c))^(5/2),x)
 

Output:

sqrt(a)*a**2*( - int(sqrt(sinh(c + d*x)*i + 1)*sinh(c + d*x)**2*x**2,x) + 
int(sqrt(sinh(c + d*x)*i + 1)*x**2,x) + 2*int(sqrt(sinh(c + d*x)*i + 1)*si 
nh(c + d*x)*x**2,x)*i)