Answer
$$x \sinh ^{-1} x-\sqrt{x^{2}+1}+C$$
Work Step by Step
Given $$\int \sinh ^{-1} x d x$$
Let
\begin{align*}
u&= \sinh^{-1}x\ \ \ \ \ \ \ \ \ \ \ \ dv=dx\\
du&= \frac{1}{\sqrt{x^{2}+1}} d x\ \ \ \ \ \ \ \ \ \ \ \ dv=x\\
\end{align*}
Then
\begin{aligned}
\int \sinh ^{-1} x d x &=x \sinh ^{-1} x-\int \frac{x}{\sqrt{x^{2}+1}} d x\\
&=x \sinh ^{-1} x-\int x(x^{2}+1)^{-1/2}d x\\
&= x \sinh ^{-1} x-\sqrt{x^{2}+1}+C
\end{aligned}
