\[ \int (a \cosh (x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (a \cosh \left (x \right )\right )^{\frac {5}{2}} \sinh \left (x \right )}{7}-\frac {10 i a^{4} \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cosh }\left (x \right )\right )}{21 \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (x \right )}}+\frac {10 a^{3} \sinh \left (x \right ) \sqrt {a \cosh \left (x \right )}}{21} \]
command
integrate((a*cosh(x))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {40 \, {\left (\sqrt {2} a^{3} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} a^{3} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} a^{3} \sinh \left (x\right )^{3}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (3 \, a^{3} \cosh \left (x\right )^{6} + 18 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, a^{3} \sinh \left (x\right )^{6} + 23 \, a^{3} \cosh \left (x\right )^{4} - 23 \, a^{3} \cosh \left (x\right )^{2} + {\left (45 \, a^{3} \cosh \left (x\right )^{2} + 23 \, a^{3}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (15 \, a^{3} \cosh \left (x\right )^{3} + 23 \, a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 3 \, a^{3} + {\left (45 \, a^{3} \cosh \left (x\right )^{4} + 138 \, a^{3} \cosh \left (x\right )^{2} - 23 \, a^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (9 \, a^{3} \cosh \left (x\right )^{5} + 46 \, a^{3} \cosh \left (x\right )^{3} - 23 \, a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}}{84 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \cosh \left (x\right )} a^{3} \cosh \left (x\right )^{3}, x\right ) \]