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