An interesting definite integral

Calculus Level 2

\[\int_{1}^{e}\ln (x)+\frac{1}{2}\left ( \ln (x)+\frac{1}{3}\Big ( \ln (x)+\tfrac{1}{4}\big( \ln (x)+\tfrac{1}{5} {\small(\cdots)}\, \big) \Big ) \right )\, dx = \, ?\]

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