I need to find some examples of a function $\xi \left( x\right) > 0$, where $x \in [x_{0},\infty ),x_{0}>e $, such that the integral $$I =\int \frac{1}{x-\frac{x}{\ln \left( x\right) }+\xi \left( x\right) }dx$$ has a (relatively simple) closed-form solution and $$\lim_{x\rightarrow \infty }\frac{\xi \left( x\right) }{\frac{x}{\ln \left( x\right) }}=0.$$ Just in case, if $\xi ( x) \equiv 0$, then $I =\ln ( x) - \ln ( \ln (x) +1)$. If you can offer some $\xi \left( x\right)$ with these properties, it would be great. Thank you.
2026-04-13 16:11:07.1776096667
An integrand of indefinite integral
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