\[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sinh \left (x \right )}{5 a \left (a \cosh \left (x \right )\right )^{\frac {5}{2}}}+\frac {6 \sinh \left (x \right )}{5 a^{3} \sqrt {a \cosh \left (x \right )}}+\frac {6 i \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \cosh \left (x \right )}}{5 \cosh \left (\frac {x}{2}\right ) a^{4} \sqrt {\cosh \left (x \right )}} \]
command
integrate(1/(a*cosh(x))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{4} + 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, {\left (3 \, \cosh \left (x\right )^{6} + 18 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, \sinh \left (x\right )^{6} + {\left (45 \, \cosh \left (x\right )^{2} + 8\right )} \sinh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{4} + 4 \, {\left (15 \, \cosh \left (x\right )^{3} + 8 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (45 \, \cosh \left (x\right )^{4} + 48 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (9 \, \cosh \left (x\right )^{5} + 16 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}\right )}}{5 \, {\left (a^{4} \cosh \left (x\right )^{6} + 6 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{5} + a^{4} \sinh \left (x\right )^{6} + 3 \, a^{4} \cosh \left (x\right )^{4} + 3 \, a^{4} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{4} + a^{4} + 4 \, {\left (5 \, a^{4} \cosh \left (x\right )^{3} + 3 \, a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{4} + 6 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a^{4} \cosh \left (x\right )^{5} + 2 \, a^{4} \cosh \left (x\right )^{3} + a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \cosh \left (x\right )}}{a^{4} \cosh \left (x\right )^{4}}, x\right ) \]