TITLE: Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$
QUESTION [15 upvotes]: $$\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$$
Is it possible to prove this?

REPLY [2 votes]: Hint: Note that $$\ln^2\cos t+t^2=\ln^2\cos t-\ln^2 e^{it}=(\ln e^{it}\cos t)(\ln e^{-it}\cos t)$$
Then let $ e^{2it}=s$ to obtain a contour integral surrounding the simple pole at $s=0$