S log(x + (x^2 -1)^1/2)dx
One approach is to use integration by parts:
$\displaystyle \int u\, dv = uv - \int v \, du$.
Let:
$\displaystyle u = \ln (x + \sqrt{x^2 - 1}) \Rightarrow du = \frac{1}{\sqrt{x^2 - 1}}\, $ (after a modest amount of work),
$\displaystyle dv = dx \Rightarrow v = x$.
For the other five other approaches (where the rest easily follows), await a reply by Krizalid .....