31.13 Problem number 491

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

Optimal antiderivative \[ -\frac {\left (2 a +b \right ) \arctanh \! \left (\frac {\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{\sqrt {a -b}}\right )}{2 \left (a -b \right )^{\frac {5}{2}} f}+\frac {2 a +b}{2 \left (a -b \right )^{2} f \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}+\frac {\mathrm {sech}\! \left (f x +e \right )^{2}}{2 \left (a -b \right ) f \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}} \]

command

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

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {Exception raised: RuntimeError} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {output too large to display} \]