TITLE: Vardi's Integral: $\int_{\pi/4}^{\pi/2} \ln (\ln(\tan x))dx $
QUESTION [15 upvotes]: Prove that:
$\displaystyle\int_{\pi/4}^{\pi/2} \ln (\ln(\tan x))dx =\frac{\pi}{2}\ln \left( \frac{\sqrt{2\pi} \Gamma \left(\dfrac{3}{4} \right)}{\Gamma \left(\dfrac{1}{4} \right)}\right)$
I know that the Vardi's Integral can be evaluated in terms of derivatives of Hurwitz Zeta Function. I would like to see a method which uses differentiation under the integral sign.

REPLY [5 votes]: As a matter of fact, there is a very nice paper by Iaroslav Blagouchine in which the Vardi's integral, as well as numerous integrals akin to it, are treated in details (more correctly they should be called Malmsten's integrals).