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