18.1 Problem number 178

$$\int \frac{1}{\sqrt{a+b{\cosh }^{-1}(c+dx)}}dx$$

Optimal antiderivative $$-\frac{{\mathrm{e}}^{\frac{a}{b}}\mathrm{erf}\left(\frac{\sqrt{a+b\mathrm{arccosh}(dx+c)}}{\sqrt{b}}\right)\sqrt{\pi }}{2d\sqrt{b}}+\frac{\mathrm{erfi}\left(\frac{\sqrt{a+b\mathrm{arccosh}(dx+c)}}{\sqrt{b}}\right)\sqrt{\pi }{\mathrm{e}}^{-\frac{a}{b}}}{2d\sqrt{b}}$$

command

Integrate[1/Sqrt[a + b*ArcCosh[c + d*x]],x]

Mathematica 13.1 output

$$\int \frac{1}{\sqrt{a+b{\cosh }^{-1}(c+dx)}}dx$$

Mathematica 12.3 output

$$\frac{e^{-\frac{a}{b}}(e^{\frac{2a}{b}}\sqrt{\frac{a}{b}+{\cosh }^{-1}(c+dx)}\mathrm{\Gamma }(\frac{1}{2},\frac{a}{b}+{\cosh }^{-1}(c+dx))+\sqrt{-\frac{a+b{\cosh }^{-1}(c+dx)}{b}}\mathrm{\Gamma }(\frac{1}{2},-\frac{a+b{\cosh }^{-1}(c+dx)}{b}))}{2d\sqrt{a+b{\cosh }^{-1}(c+dx)}}$$