\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {14 i \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right )}{15 \cosh \left (\frac {x}{2}\right ) a \cosh \left (x \right )^{\frac {3}{2}} \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {14 \sinh \left (x \right )}{45 a \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {2 \left (\cosh ^{2}\left (x \right )\right ) \sinh \left (x \right )}{9 a \sqrt {a \mathrm {sech}\left (x \right )^{3}}} \]
command
integrate(1/(a*sech(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {672 \, \sqrt {2} {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - \sqrt {2} {\left (5 \, \cosh \left (x\right )^{10} + 50 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + 5 \, \sinh \left (x\right )^{10} + {\left (225 \, \cosh \left (x\right )^{2} + 43\right )} \sinh \left (x\right )^{8} + 43 \, \cosh \left (x\right )^{8} + 8 \, {\left (75 \, \cosh \left (x\right )^{3} + 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 2 \, {\left (525 \, \cosh \left (x\right )^{4} + 602 \, \cosh \left (x\right )^{2} - 149\right )} \sinh \left (x\right )^{6} - 298 \, \cosh \left (x\right )^{6} + 4 \, {\left (315 \, \cosh \left (x\right )^{5} + 602 \, \cosh \left (x\right )^{3} - 447 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (525 \, \cosh \left (x\right )^{6} + 1505 \, \cosh \left (x\right )^{4} - 2235 \, \cosh \left (x\right )^{2} - 187\right )} \sinh \left (x\right )^{4} - 374 \, \cosh \left (x\right )^{4} + 8 \, {\left (75 \, \cosh \left (x\right )^{7} + 301 \, \cosh \left (x\right )^{5} - 745 \, \cosh \left (x\right )^{3} - 187 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (225 \, \cosh \left (x\right )^{8} + 1204 \, \cosh \left (x\right )^{6} - 4470 \, \cosh \left (x\right )^{4} - 2244 \, \cosh \left (x\right )^{2} - 43\right )} \sinh \left (x\right )^{2} - 43 \, \cosh \left (x\right )^{2} + 2 \, {\left (25 \, \cosh \left (x\right )^{9} + 172 \, \cosh \left (x\right )^{7} - 894 \, \cosh \left (x\right )^{5} - 748 \, \cosh \left (x\right )^{3} - 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{720 \, {\left (a^{2} \cosh \left (x\right )^{5} + 5 \, a^{2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, a^{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, a^{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + a^{2} \sinh \left (x\right )^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \operatorname {sech}\left (x\right )^{3}}}{a^{2} \operatorname {sech}\left (x\right )^{6}}, x\right ) \]