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