Answer
$$\frac{1}{2}\left[x \sqrt{x^{2}-1}-\cosh ^{-1} x\right]+C$$
Work Step by Step
Given $$\int \sqrt{x^{2}-1} d x $$
Let $$x=\cosh t\ \ \to \ dx=\sinh udu$$
\begin{aligned} \int \sqrt{x^{2}-1} d x &=\int \sqrt{\cosh ^{2} t-1} \sinh t d t \\ &=\int \sqrt{\sinh ^{2} t} \sinh t d t \\ &=\int \sinh t \cdot \sinh t d t \\ &=\int \sinh ^{2} t d t \\ &=\frac{1}{2}\left[\int \cosh 2 t d t-\int 1 d t\right] \\ &=\frac{1}{2}\left[\frac{1}{2} \sinh 2 t-t\right]+C\\
&=\frac{1}{2}\sinh t\cosh t-\frac{1}{2}t+C\\
&=\frac{1}{2}\left[x \sqrt{x^{2}-1}-\cosh ^{-1} x\right]+C
\end{aligned}
