\[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {14 i a \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{15 \cosh \left (\frac {x}{2}\right ) \cosh \left (x \right )^{\frac {3}{2}}}+\frac {14 a \sinh \left (x \right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{45}+\frac {2 a \left (\cosh ^{2}\left (x \right )\right ) \sinh \left (x \right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{9} \]
command
integrate((a*cosh(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {336 \, {\left (\sqrt {2} a \cosh \left (x\right )^{4} + 4 \, \sqrt {2} a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \sqrt {2} a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} a \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} a \sinh \left (x\right )^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (5 \, a \cosh \left (x\right )^{8} + 40 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + 5 \, a \sinh \left (x\right )^{8} + 38 \, a \cosh \left (x\right )^{6} + 2 \, {\left (70 \, a \cosh \left (x\right )^{2} + 19 \, a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (70 \, a \cosh \left (x\right )^{3} + 57 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{4} + 2 \, {\left (175 \, a \cosh \left (x\right )^{4} + 285 \, a \cosh \left (x\right )^{2} - 168 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (35 \, a \cosh \left (x\right )^{5} + 95 \, a \cosh \left (x\right )^{3} - 168 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 38 \, a \cosh \left (x\right )^{2} + 2 \, {\left (70 \, a \cosh \left (x\right )^{6} + 285 \, a \cosh \left (x\right )^{4} - 1008 \, a \cosh \left (x\right )^{2} - 19 \, a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (10 \, a \cosh \left (x\right )^{7} + 57 \, a \cosh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{3} - 19 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {a \cosh \left (x\right )}}{360 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \cosh \left (x\right )^{3}} a \cosh \left (x\right )^{3}, x\right ) \]