\int \frac{1}{\ln x} \, dx = \int e^t \, \frac{dt}{t} = \int \sum_{n=0}^{\infty} \frac{t^{n-1}}{n!} \, dt = \sum_{n=0}^{\infty} \frac{t^n}{n \cdot n!} + \mathcal{C} =\sum_{n=0}^{\infty} \frac{\ln^n x}{n \cdot n!}+\mathcal{C}.