\[ \int \frac {x^3}{\text {sech}^{\frac {3}{2}}(2 \log (c x))} \, dx \]
Optimal antiderivative \[ \frac {2}{7 \left (c^{4}+\frac {1}{x^{4}}\right ) \mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}+\frac {x^{4}}{7 \mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}-\frac {2 \left (c^{2}+\frac {1}{x^{2}}\right ) \sqrt {\frac {\cos \left (4 \,\mathrm {arccot}\left (c x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \,\mathrm {arccot}\left (c x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c^{4}+\frac {1}{x^{4}}}{\left (c^{2}+\frac {1}{x^{2}}\right )^{2}}}}{7 \cos \left (2 \,\mathrm {arccot}\left (c x \right )\right ) c \left (c^{4}+\frac {1}{x^{4}}\right )^{2} x^{3} \mathrm {sech}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}} \]
command
integrate(x^3/sech(2*log(c*x))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 \, \sqrt {2} \sqrt {c^{4}} c \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} {\rm ellipticF}\left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}, -1\right ) + \sqrt {2} {\left (c^{8} x^{8} + 4 \, c^{4} x^{4} + 3\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{28 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {x^{3}}{\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}, x\right ) \]