I would need help finding a closed form for this integral:
$$\int_2^\infty \frac{\ln(y-1)}{\ln(y)} - 1 +\frac{1}{y \ln(y)} dy$$
My try: I tried using series expansion with no result (I get a series with no closed form). Mathematica gives an approximated value, but no closed form either.
Any advice? Thank you.
The given integral equals $$\int_{\log 2}^{+\infty}\frac{e^x \log(1-e^{-x})+1}{x}\,dx$$ that can be written as $$ \int_{\log 2}^{+\infty} -\sum_{n\geq 2}\frac{e^{-(n-1)x}}{nx}\,dx\stackrel{\mathcal{L}}{=}-\int_{0}^{+\infty}\sum_{m\geq 1}\frac{ds}{2^{s+m}(s+m)(m+1)}$$ So it is pretty simple to approximate even with great accuracy, but I won't bet a penny on a simple closed form.