\[ \int \frac {\sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\left (2 i c -\left (i b +\sqrt {4 a c -b^{2}}\right ) \tanh \left (\frac {x}{2}\right )\right ) \sqrt {2}}{2 \sqrt {b^{2}-2 \left (a -c \right ) c -i b \sqrt {4 a c -b^{2}}}}\right ) \sqrt {2}\, \left (i-\frac {b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {b^{2}-2 \left (a -c \right ) c -i b \sqrt {4 a c -b^{2}}}}+\frac {\arctan \left (\frac {\left (2 i c -i b \tanh \left (\frac {x}{2}\right )+\sqrt {4 a c -b^{2}}\, \tanh \left (\frac {x}{2}\right )\right ) \sqrt {2}}{2 \sqrt {b^{2}-2 \left (a -c \right ) c +i b \sqrt {4 a c -b^{2}}}}\right ) \sqrt {2}\, \left (i+\frac {b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {b^{2}-2 \left (a -c \right ) c +i b \sqrt {4 a c -b^{2}}}} \]
command
integrate(sinh(x)/(a+b*sinh(x)+c*sinh(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
\[ \text {Timed out} \]________________________________________________________________________________________