# A conjectured closed form of ∞∫0x−1√2x−1 ln(2x−1)dx\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx

Consider the following integral:

I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using Mathematica), but without success.

However, if WolframAlpha is provided with a numerical approximation $\,\mathcal{I}\approx 3.2694067500684...$, it returns a possible closed form:

Further numeric caclulations show that this value is correct up to at least $10^3$ decimal digits. So, I conjecture that this is the exact value of $\mathcal{I}$.

Question: Is this conjecture correct?

Sub $u=\log{(2^x-1)}$. Then $x=\log{(1+e^u)}/\log{2}$, $dx = (1/\log{2}) (du/(1+e^{-u})$. The integral then becomes