16.1 Problem number 22

$$\int \frac{1}{a+b{\sinh }^{-1}(cx)}dx$$

Optimal antiderivative $$\frac{\mathrm{hyperbolicCosineIntegral}\left(\frac{a+b\mathrm{arcsinh}\left(cx\right)}{b}\right)\cosh \left(\frac{a}{b}\right)}{bc}-\frac{\mathrm{hyperbolicSineIntegral}\left(\frac{a+b\mathrm{arcsinh}\left(cx\right)}{b}\right)\sinh \left(\frac{a}{b}\right)}{bc}$$

command

Integrate[(a + b*ArcSinh[c*x])^(-1),x]

Mathematica 13.1 output

$$\int \frac{1}{a+b{\sinh }^{-1}(cx)}dx$$

Mathematica 12.3 output

$$\frac{\cosh \left(\frac{a}{b}\right)\text{Chi}(\frac{a}{b}+{\sinh }^{-1}(cx))-\sinh \left(\frac{a}{b}\right)\text{Shi}(\frac{a}{b}+{\sinh }^{-1}(cx))}{bc}$$