TITLE: An integral related to the Euler gamma function
QUESTION [11 upvotes]: The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral
$$\eta=2\int\limits_0^1 \ln{(\Gamma(x))}\cdot \sin{(2\pi x)}\,dx=        0.7687478924\ldots$$
in terms of the known mathematical constants as $\pi,\,e,\,\gamma,\;\ln{\pi},\,
\ln{2},\,\Gamma{(1/4)}\ldots$? It is shown in the paper that 
$$\frac{\ln{1}}{1}-\frac{\ln{3}}{3}+\frac{\ln{5}}{5}-\ldots=\pi\ln{(\Gamma{(1/4)})}-\frac{\pi^2}{4}\eta-\frac{\pi}{2}\ln{\pi}-\frac{\pi}{4}\ln{2}.$$ So the question actually asks whether one can give a closed-form expression for the series in the l.h.s. in terms of the known mathematical constants.

REPLY [2 votes]: In fact, both the experimental result by Joro and the (very nice) asymptotic analysis by A. Shamov, above, are correct. I'd like to advice those more interested readers that I've included this topic and some generalizations in my most recent work on integrals and series involving the log-Gamma function (to appear in arXiv website). F. M. S. Lima