\[ \int \frac {x^7}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx \]
Optimal antiderivative \[ \frac {4}{77 c^{4} \left (c^{4}-\frac {1}{x^{4}}\right ) \mathrm {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}-\frac {6 x^{4}}{77 \left (c^{4}-\frac {1}{x^{4}}\right ) \mathrm {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}+\frac {x^{8}}{11 \mathrm {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}-\frac {4 \EllipticF \left (\frac {1}{c x}, i\right )}{77 c^{11} \left (1-\frac {1}{c^{4} x^{4}}\right )^{\frac {3}{2}} x^{3} \mathrm {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}} \]
command
integrate(x^7/csch(2*log(c*x))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (7 \, c^{14} x^{12} - 20 \, c^{10} x^{8} + 17 \, c^{6} x^{4} - 4 \, c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 4 \, \sqrt {2} \sqrt {c^{4}} {\rm ellipticF}\left (\frac {1}{c x}, -1\right )}{308 \, c^{10}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {x^{7}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}, x\right ) \]