How to evaluate $\int\frac{1}{x^2\ln(x)} dx$

Question

I was trying to evaluate this integral, but I think it is a non-elementary function.

$$I=\int\frac{1}{x^2\ln(x)} dx$$

Do you know any conditions where there is a closed form solution?

Answer 1

Let $x=e^t$ to make $$I=\int \frac {dx}{x^2 \log(x)}=\int \frac{e^{-t}}{t}\,dt=\text{Ei}(-t)$$ where appears the exponential integral function.