\[ \int \frac {1}{\sinh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx \]
Optimal antiderivative \[ -\frac {x \left (1-{\mathrm e}^{-2 a} \left (c \,x^{n}\right )^{-\frac {4}{n}}\right )}{2 \sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {3}{2}}} \]
command
integrate(1/sinh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2}}{\sqrt {c^{\frac {4}{n}} e^{\left (3 \, a\right )} - \frac {e^{a}}{x^{4}}} c^{\left (\frac {1}{n}\right )} x^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________