\[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b \sinh \left (x \right )}{3 \left (a^{2}-b^{2}\right ) \left (a +b \cosh \left (x \right )\right )^{\frac {3}{2}}}-\frac {8 a b \sinh \left (x \right )}{3 \left (a^{2}-b^{2}\right )^{2} \sqrt {a +b \cosh \left (x \right )}}-\frac {8 i a \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cosh \left (x \right )}}{3 \cosh \left (\frac {x}{2}\right ) \left (a^{2}-b^{2}\right )^{2} \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}+\frac {2 i \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}{3 \cosh \left (\frac {x}{2}\right ) \left (a^{2}-b^{2}\right ) \sqrt {a +b \cosh \left (x \right )}} \]
command
integrate(1/(a+b*cosh(x))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (\sqrt {2} {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \cosh \left (x\right )^{4} + \sqrt {2} {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \sinh \left (x\right )^{4} + 4 \, \sqrt {2} {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (x\right )^{3} + 4 \, {\left (\sqrt {2} {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a^{3} b + 3 \, a b^{3}\right )}\right )} \sinh \left (x\right )^{3} + 2 \, \sqrt {2} {\left (2 \, a^{4} + 7 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \sqrt {2} {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \cosh \left (x\right )^{2} + 6 \, \sqrt {2} {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, a^{4} + 7 \, a^{2} b^{2} + 3 \, b^{4}\right )}\right )} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (x\right ) + 4 \, {\left (\sqrt {2} {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (2 \, a^{4} + 7 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a^{3} b + 3 \, a b^{3}\right )}\right )} \sinh \left (x\right ) + \sqrt {2} {\left (a^{2} b^{2} + 3 \, b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 12 \, {\left (\sqrt {2} a b^{3} \cosh \left (x\right )^{4} + \sqrt {2} a b^{3} \sinh \left (x\right )^{4} + 4 \, \sqrt {2} a^{2} b^{2} \cosh \left (x\right )^{3} + 4 \, \sqrt {2} a^{2} b^{2} \cosh \left (x\right ) + \sqrt {2} a b^{3} + 4 \, {\left (\sqrt {2} a b^{3} \cosh \left (x\right ) + \sqrt {2} a^{2} b^{2}\right )} \sinh \left (x\right )^{3} + 2 \, \sqrt {2} {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \sqrt {2} a b^{3} \cosh \left (x\right )^{2} + 6 \, \sqrt {2} a^{2} b^{2} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, a^{3} b + a b^{3}\right )}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} a b^{3} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} a^{2} b^{2} \cosh \left (x\right )^{2} + \sqrt {2} a^{2} b^{2} + \sqrt {2} {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 6 \, {\left (4 \, a b^{3} \cosh \left (x\right )^{4} + 4 \, a b^{3} \sinh \left (x\right )^{4} + {\left (13 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{3} + {\left (16 \, a b^{3} \cosh \left (x\right ) + 13 \, a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{3} + 4 \, {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (x\right )^{2} + {\left (24 \, a b^{3} \cosh \left (x\right )^{2} + 8 \, a^{3} b + 4 \, a b^{3} + 3 \, {\left (13 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (16 \, a b^{3} \cosh \left (x\right )^{3} + 3 \, a^{2} b^{2} + b^{4} + 3 \, {\left (13 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + 8 \, {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}\right )}}{9 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7} + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{4} + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right )^{3} + 4 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6} + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (2 \, a^{6} b - 3 \, a^{4} b^{3} + b^{7}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (2 \, a^{6} b - 3 \, a^{4} b^{3} + b^{7} + 3 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{2} + 6 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right ) + 4 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6} + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right )^{2} + {\left (2 \, a^{6} b - 3 \, a^{4} b^{3} + b^{7}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cosh \left (x\right ) + a}}{b^{3} \cosh \left (x\right )^{3} + 3 \, a b^{2} \cosh \left (x\right )^{2} + 3 \, a^{2} b \cosh \left (x\right ) + a^{3}}, x\right ) \]