How can i evaluate
$$\int _0^{\pi }x\sin \left(x\right)\operatorname{Li}_2\left(\cos \left(2x\right)\right)\:dx$$ $$=\frac{\pi ^3}{6}-\frac{\pi ^3}{6\sqrt{2}}-4\pi +6\pi \ln \left(2\right)-\frac{\pi }{2\sqrt{2}}\ln ^2\left(2\sqrt{2}+3\right)-\sqrt{2}\pi \operatorname{Li}_2\left(2\sqrt{2}-3\right)$$
This is what I've done thus far. \begin{align*} &\int _0^{\pi }x\sin \left(x\right)\operatorname{Li}_2\left(\cos \left(2x\right)\right)\:dx\\ &=\frac{\pi }{2}\int _0^{\pi }\sin \left(x\right)\operatorname{Li}_2\left(\cos \left(2x\right)\right)\:dx\\[2mm] &=\frac{\pi }{4}\int _0^{\pi }\sin \left(\frac{x}{2}\right)\operatorname{Li}_2\left(\cos \left(x\right)\right)\:dx+\frac{\pi }{4}\int _{\pi }^{2\pi }\sin \left(\frac{x}{2}\right)\operatorname{Li}_2\left(\cos \left(x\right)\right)\:dx\\[2mm] &=\frac{\pi }{2}\int _0^{\pi }\sqrt{\frac{1+\cos \left(x\right)}{2}}\operatorname{Li}_2\left(-\cos \left(x\right)\right)\:dx=\pi \int _0^{\infty }\frac{\operatorname{Li}_2\left(\frac{t^2-1}{1+t^2}\right)}{\left(1+t^2\right)\sqrt{1+t^2}}\:dt\\[2mm] &=\frac{\pi }{2}\int _1^{\infty }\frac{\operatorname{Li}_2\left(\frac{x-2}{x}\right)}{x\sqrt{x}\sqrt{x-1}}\:dx=\frac{\pi }{2}\int _0^1\frac{\operatorname{Li}_2\left(1-2x\right)}{\sqrt{1-x}}\:dx\\[2mm] &=\pi \zeta \left(2\right)+\frac{\pi }{\sqrt{2}}\int _{-1}^1\frac{\sqrt{1+x}\ln \left(1-x\right)}{x}\:dx \end{align*} But I'm not sure how to proceed with either that polylogarithmic integral on the $5$th line nor the last one.
I'd appreciate any hints or ideas, thanks.