Answer
$$\cosh ^{-1} x-\frac{\sqrt{x^{2}-1}}{x}+C$$
Work Step by Step
Given $$ \int \frac{\sqrt{x^{2}-1}}{x^{2}} d x $$
Let $$ x=\cosh t\ \ \to \ dx=\sinh t dt $$
\begin{aligned} \int \frac{\sqrt{x^{2}-1}}{x^{2}} d x &=\int \frac{\sqrt{\cosh ^{2} t-1}}{\cosh ^{2} t} \cdot \sinh t d t \\ &=\int \frac{\sqrt{\sinh ^{2} t}}{\cosh ^{2} t} \cdot \sinh t d t \\ &=\int \frac{\sinh ^{2} t}{\cosh ^{2} t} d t \\ &=\int \tanh ^{2} t d t \\ &=\int\left(1-\operatorname{sech}^{2} t\right) d t \\
&=t-\tanh t +C\\
&= \cosh ^{-1} x-\frac{\sinh t}{\cosh t}+C\\
&=\cosh ^{-1} x-\frac{\sqrt{x^{2}-1}}{x}+C
\end{aligned}
