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