For the indefinite integral of $$ \dfrac{x^2}{\sqrt{x^2 -1}}$$ we can use the substitution of $x = \cosh u$ where $u \geq 0$ for the interval $x \in \left[ 1, + \infty \right)$ but which substitution would we use for the interval $x \in \left( -\infty, \ -1 \right]$ ?
@MartinSleziak