\[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b \sinh \left (x \right )}{5 \left (a^{2}-b^{2}\right ) \left (a +b \cosh \left (x \right )\right )^{\frac {5}{2}}}-\frac {16 a b \sinh \left (x \right )}{15 \left (a^{2}-b^{2}\right )^{2} \left (a +b \cosh \left (x \right )\right )^{\frac {3}{2}}}-\frac {2 b \left (23 a^{2}+9 b^{2}\right ) \sinh \left (x \right )}{15 \left (a^{2}-b^{2}\right )^{3} \sqrt {a +b \cosh \left (x \right )}}-\frac {2 i \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cosh \left (x \right )}}{15 \cosh \left (\frac {x}{2}\right ) \left (a^{2}-b^{2}\right )^{3} \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}+\frac {16 i a \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}{15 \cosh \left (\frac {x}{2}\right ) \left (a^{2}-b^{2}\right )^{2} \sqrt {a +b \cosh \left (x \right )}} \]
command
integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {b \cosh \left (x\right ) + a}}{b^{4} \cosh \left (x\right )^{4} + 4 \, a b^{3} \cosh \left (x\right )^{3} + 6 \, a^{2} b^{2} \cosh \left (x\right )^{2} + 4 \, a^{3} b \cosh \left (x\right ) + a^{4}}, x\right ) \]