\[ \int \frac {1}{\sqrt {a+b \sinh ^{-1}(c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{\frac {a}{b}} \erf \! \left (\frac {\sqrt {a +b \arcsinh \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {\pi }}{2 d \sqrt {b}}+\frac {\erfi \! \left (\frac {\sqrt {a +b \arcsinh \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {a}{b}}}{2 d \sqrt {b}} \]
command
Integrate[1/Sqrt[a + b*ArcSinh[c + d*x]],x]
Mathematica 13.1 output
\[ \int \frac {1}{\sqrt {a+b \sinh ^{-1}(c+d x)}} \, dx \]
Mathematica 12.3 output
\[ \frac {e^{-\frac {a}{b}} \left (\sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )-e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )}{2 d \sqrt {a+b \sinh ^{-1}(c+d x)}} \]