Integral of constant divided by polynomial and logarithm

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$$\int_e^\infty\frac{2}{x-\ln(x)}\mathrm dx$$

I'm not sure how to integrate. What are your hints?

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Hint:

$$\int_{e}^{\infty}\frac{2}{x-\ln{(x)}}\,\mathrm{d}x>\int_{e}^{\infty}\frac{1}{x}\,\mathrm{d}x$$