\[ \int \frac {1}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {{\mathrm e}^{\frac {a}{b}} \erf \! \left (\frac {\sqrt {a +b \arcsinh \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {\pi }}{b^{\frac {3}{2}} d}+\frac {\erfi \! \left (\frac {\sqrt {a +b \arcsinh \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {a}{b}}}{b^{\frac {3}{2}} d}-\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{b d \sqrt {a +b \arcsinh \! \left (d x +c \right )}} \]
command
Integrate[(a + b*ArcSinh[c + d*x])^(-3/2),x]
Mathematica 13.1 output
\[ \int \frac {1}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \, dx \]
Mathematica 12.3 output
\[ \frac {e^{-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \left (-e^{a/b} \left (e^{2 \sinh ^{-1}(c+d x)}+1\right )+e^{\frac {2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+e^{\sinh ^{-1}(c+d x)} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )\right )}{b d \sqrt {a+b \sinh ^{-1}(c+d x)}} \]