TITLE: Integral of Combination Log and Inverse Trig Function
QUESTION [34 upvotes]: Does the following integral have a closed-form ?: 

\begin{equation}
\int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right)
\,{\rm d}x
\end{equation}

This integral has been posted in Integral and Series a week ago but it remains unsolved. So, I decide to post it here.
Could anyone here please help me to find the closed-form preferably with elementary ways ( high school methods ) ?.
Any help would be greatly appreciated. Thank you.

REPLY [5 votes]: Using this an other possible solution is 
$$\int_0^1\frac{\ln x}{1+x}\arccos(x)\,dx\, = 2\,G\ln2 + \frac{\pi}{8}\ln^2 2-\frac{5\pi^3}{32} + 4 \, \Im \left[ \operatorname{Li}_3 \left( \frac{1+i}{2} \right) \right].$$
From here we can only give something more, if we find an answer for this question. I think without something really new method we don't have chance to answer it. Maybe if I get some surprising answer to this question at complex arguments, then we can go further, but I really don't have hope.