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