The same bit of trivial algebra in two different places?
Well, if you may not have realized, $\frac{\ln(x\pm\sqrt{x^2-1})}{i}+\frac{\pi}2+2\pi n=\arccos(x)$. The derivative for $\arccos(x) \, dx$, as you may already know, is $-\frac1{\sqrt{1-x^2}}$
This relates the derivative of the logarithm you presented with trigonometry/geometry.
A derivation of the $\arccos(x)$ can be found here.