Answer
$$ \displaystyle\int \dfrac{\ln\left(2x+1\right)}{x{\cdot}\left(x+1\right)}\, \mathrm d x = -\ln \left(x+1\right)\,\ln \left(2\,x+1\right)+{\it li}_{2}(2\, \left(x+1\right))+\ln \left(-2\,x-1\right)\,\ln \left(x+1\right)- {\it li}_{2}(-2\,x) $$
Explanation
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