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