\[ \int (a \cosh (x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (a \cosh \left (x \right )\right )^{\frac {3}{2}} \sinh \left (x \right )}{5}-\frac {6 i a^{2} \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \cosh \left (x \right )}}{5 \cosh \left (\frac {x}{2}\right ) \sqrt {\cosh \left (x \right )}} \]
command
integrate((a*cosh(x))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {12 \, {\left (\sqrt {2} a^{2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} a^{2} \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (a^{2} \cosh \left (x\right )^{4} + 4 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{2} \sinh \left (x\right )^{4} - 12 \, a^{2} \cosh \left (x\right )^{2} + 6 \, {\left (a^{2} \cosh \left (x\right )^{2} - 2 \, a^{2}\right )} \sinh \left (x\right )^{2} - a^{2} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} - 6 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}}{10 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \cosh \left (x\right )} a^{2} \cosh \left (x\right )^{2}, x\right ) \]