How to take this integral? $\int \frac{dx}{x\ln x\ln (\ln x)}$

Question

Have this small, but beautiful integral:

$$\int \frac{dx}{x\ln x\ln (\ln x)}$$

Answer 1

Hint

You should notice that :

$$\frac{d}{dx}\left(\ln(\ln(x))\right)=\frac{\frac 1x}{\ln(x)}=\frac{1}{x\ln(x)}$$

Answer 2

Well, you can do repeated substitution of $\ln$. Substituting $u=\ln(x)$ gives: $$\int \frac{1}{u\cdot \ln{u}}~du$$ You can substitute $v=\ln{u}$, and from here on, it should be extremely easy.