\[ \int \frac {1}{\sinh ^{\frac {7}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx \]
Optimal antiderivative \[ -\frac {x \left (1-{\mathrm e}^{-2 a} \left (c \,x^{n}\right )^{-\frac {4}{n}}\right )}{6 \sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {7}{2}}}+\frac {x \left (1-{\mathrm e}^{-2 a} \left (c \,x^{n}\right )^{-\frac {4}{n}}\right ) {\mathrm e}^{-2 a} \left (c \,x^{n}\right )^{-\frac {4}{n}}}{15 \sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {7}{2}}} \]
command
integrate(1/sinh(a+2*log(c*x^n)/n)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {4 \, \sqrt {2} c^{\frac {7}{n}} {\left (\frac {5 \, e^{a}}{c^{\frac {4}{n}} \mathrm {sgn}\left (x\right )} - \frac {2 \, e^{\left (-a\right )}}{c^{\frac {8}{n}} x^{4} \mathrm {sgn}\left (x\right )}\right )} e^{\left (3 \, a\right )}}{15 \, {\left (c^{\frac {4}{n}} e^{\left (3 \, a\right )} - \frac {e^{a}}{x^{4}}\right )}^{\frac {5}{2}} x^{6}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________