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