\[ \int \frac {\cosh (x)}{a+b \cosh (x)+c \cosh ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {2 \arctanh \left (\frac {\sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \tanh \left (\frac {x}{2}\right )}{\sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}\right ) \left (1-\frac {b}{\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {b -2 c -\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c -\sqrt {-4 a c +b^{2}}}}+\frac {2 \arctanh \left (\frac {\sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \tanh \left (\frac {x}{2}\right )}{\sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}}\right ) \left (1+\frac {b}{\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {b -2 c +\sqrt {-4 a c +b^{2}}}\, \sqrt {b +2 c +\sqrt {-4 a c +b^{2}}}} \]
command
integrate(cosh(x)/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 0 \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________