$\displaystyle \int \frac{ln|x|} {x\sqrt{1+ln|x|}} $?
Krizalid thinks a few levels above the average person when it comes to integrals. here is a more conventional substitution for mortals.
$\displaystyle \int \frac u{\sqrt{1 + u}}~du$
Let $\displaystyle t = 1 + u \implies \boxed{u = t - 1}$
$\displaystyle \Rightarrow dt = du$
So our integral becomes:
$\displaystyle \int \frac {t - 1}{\sqrt{t}}~dt$
which we can do using the power rule (after dividing $\displaystyle \sqrt{t}$ into each of the terms in the numerator)