How can we prove that: $$\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy= -4 \pi ^2 \left(\frac{\pi }{\sqrt{3}}+\log (2)-\frac{\psi ^{(1)}\left(\frac{1}{6}\right)}{2 \sqrt{3} \pi }\right)$$ Where $\psi^{(1)}$ denotes trigamma function. It's J. Borwein's review on experimental mathematics that offers this interesting identity (I've verified it numerically). The literature refer this formula to V.Adamchik, but I haven't find any related source dealing with this kind of integrals. In fact, I've asked this question on this site a year before but no answer was given. Since I have still no solution, I'd like you to give some suggestions again. Any help will be appreciated.
from Hot Weekly Questions - Mathematics Stack Exchange
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