\[ \int \frac {1}{a+b \sinh ^{-1}(c x)} \, dx \]
Optimal antiderivative \[ \frac {\hyperbolicCosineIntegral \! \left (\frac {a +b \arcsinh \left (c x \right )}{b}\right ) \cosh \! \left (\frac {a}{b}\right )}{b c}-\frac {\hyperbolicSineIntegral \! \left (\frac {a +b \arcsinh \left (c x \right )}{b}\right ) \sinh \! \left (\frac {a}{b}\right )}{b c} \]
command
Integrate[(a + b*ArcSinh[c*x])^(-1),x]
Mathematica 13.1 output
\[ \int \frac {1}{a+b \sinh ^{-1}(c x)} \, dx \]
Mathematica 12.3 output
\[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c} \]