Answer
$$-\frac{1}{\sqrt{x^{2}-1}}-\sec ^{-1} x+C$$
Work Step by Step
Given $$\int \frac{d x}{x\left(x^{2}-1\right)^{3 / 2}} $$
Let
$$ x=\sec u \ \ \ \ \ \ \ dx=\sec u\tan u du$$
Then
\begin{align*}
\int \frac{d x}{x\left(x^{2}-1\right)^{3 / 2}} &= \int \frac{\sec u\tan u du}{\sec u \left(\sec^{2}u-1\right)^{3 / 2}} \\
&= \int \frac{ \tan u du}{ \left(\tan^{2}u\right)^{3 / 2}} \\
&=\int \cot^2u du\\
&=\int (\csc^2u-1)du\\
&=-\cot u -u+C\\
&=-\frac{1}{\sqrt{x^{2}-1}}-\sec ^{-1} x+C
\end{align*}
