\[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 i \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right )}{3 \cosh \left (\frac {x}{2}\right ) \cosh \left (x \right )^{\frac {3}{2}} \sqrt {a \mathrm {sech}\left (x \right )^{3}}}+\frac {2 \tanh \left (x \right )}{3 \sqrt {a \mathrm {sech}\left (x \right )^{3}}} \]
command
integrate(1/(a*sech(x)^3)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 \, \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{6 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \operatorname {sech}\left (x\right )^{3}}}{a \operatorname {sech}\left (x\right )^{3}}, x\right ) \]