Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$ Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$
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Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$
I cannot think of a change of variable nor other integrating methods. Maybe there is a known method that I am missing. 
 A: See also V. Adamchik's formula $$\int_0^1 \frac{x^{p-1}}{1+x^n}\log \log \frac{1}{x}dx = \frac{\gamma+\log(2n)}{2n}(\psi(\frac{p}{2n})-\psi(\frac{n+p}{2n}))+\frac{1}{2n}(\zeta'(1,\frac{p}{2n})-\zeta'(1,\frac{n+p}{2n}))$$ in http://dx.doi.org/10.1145/258726.258736 .
A: The above integral was long-time known as the Vardi's integral. Quite recently, an interesting research work devoted to this integral was published by Iaroslav Blagouchine. It appears that this integral was first evaluated by Carl Malmsten in 1842 (and not by Ilan Vardi in 1988). Blagouchine describes two different methods for its evaluation: the original Malmsten's method and the contour integration method. In the same paper, numerous integrals similar to the above one are also treated, and it is shown that many of them may be evaluated by the contour integration methods. 
More recent paper by Blagouchine (mentioned above in the context of Malmsten's formula derived in 1846) is also integersting, but it is less directly related to the integral in question than the first one.
