I have been trying, with no luck, to find the closed-form expression of the following integral
$$\int_ 0^{\pi/2} (1-x \cot x)^2 \csc^2 x \log (1-x \cot x) dx \approx -0.1769450352055053460701097297399568$$
that is involved in the high-density expansion of the correlation energy of the uniform electron gas [see Hoffman, Phys. Rev. B 1992, 45:8730–8733].
The previous integral can be recast in a more common form as
$$ \int_0^\infty \left(1-u \tan ^{-1}\left(\frac{1}{u}\right)\right)^2 \log \left(1-u \tan ^{-1}\left(\frac{1}{u}\right)\right)$$
I suspect that the fact that the result is almost identical to the largest root of $106t^2+7863t+1388=0$ with an absolute error of $5.14\times 10^{-14}$ does not have much meaning.
Surprizing or not, a series solution works quite well. $$(1-x \cot (x))^2 \csc ^2(x)\log (1-x \cot (x))=\sum_{n=1}^\infty \frac {a_n+b_n\,\log \left(\frac{x^2}{3}\right)}{c_n}\, x^{2n}$$ The first coefficients are $$\left( \begin{array}{cccc} n & a_n & b_n & c_n \\ 1 & 0 & 1 & 9 \\ 2 & 1 & 7 & 135 \\ 3 & 111 & 404 & 28350 \\ 4 & 1529 & 3870 & 1275750 \\ 5 & 824891 & 1635240 & 2946982500 \\ 6 & 159349639 & 263847000 & 2873307937500 \\ 7 & 1483775683 & 2135385000 & 150848666718750 \\ 8 & 123947770511 & 159246112500 & 76932820026562500 \\ \end{array} \right)$$
Now, using $$\int_0^{\frac \pi 2} \frac {a_n+b_n\,\log \left(\frac{x^2}{3}\right)}{c_n}\, x^{2n}\,dx=$$ $$\left(\frac{\pi }{2}\right)^{2 n+1}\,\,\,\frac{a_n(2n+1)-b_n\left((2 n+1) \log \left(\frac{12}{\pi ^2}\right)+2 \right) }{c_n\,(2n+1)^2 }$$
Computing for different orders $O(x^{2p+2})$, the results $$\left( \begin{array}{cc} p & \text{result} \\ 10 & -0.1769527245716249716398478 \\ 20 & -0.1769450352466270065906185 \\ 30 & -0.1769450352055054501145367 \\ 40 & -0.1769450352055053460703062 \\ 50 & -0.1769450352055053460701097 \\ 60 & -0.1769450352055053460701097 \\ \end{array} \right)$$