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