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$\ds{\int_{0}^{\pi/2}{\ln\pars{\sin\pars{x}}\ln\pars{\cos\pars{x}} \over \tan\pars{x}}\,\dd x:\ {\large ?}}$
\begin{align} &\int_{0}^{\pi/2}{\ln\pars{\sin\pars{x}}\ln\pars{\cos\pars{x}}\over \tan\pars{x}}\,\dd x \,\,\,\stackrel{x\ =\ \arcsin\pars{t}}{=}\,\,\, \int_{0}^{1}{\ln\pars{t}\ln\pars{\root{1 - t^{2}}} \over t\,/\root{1 - t^{2}}}\, {\dd t \over \root{1 - t^{2}}} \\[5mm]= &\ \half\int_{0}^{1}{\ln\pars{t}\ln\pars{1 - t^{2}} \over t}\,\dd t = \half\int_{0}^{1}{\ln\pars{t^{1/2}}\ln\pars{1 - t}\over t^{1/2}}\,\half\,t^{-1/2}\,\dd t \\[5mm] = &\ {1 \over 8}\int_{0}^{1}{\ln\pars{t}\ln\pars{1 - t} \over t}\,\dd t =-\,{1 \over 8}\int_{0}^{1}{{\rm Li}_{1}\pars{t} \over t}\,\ln\pars{t}\,\dd t =-\,{1 \over 8}\int_{0}^{1}{\rm Li}_{2}'\pars{t}\ln\pars{t}\,\dd t \\[5mm]&={1 \over 8}\int_{0}^{1}{{\rm Li}_{2}\pars{t} \over t}\,\dd t ={1 \over 8}\int_{0}^{1}{\rm Li}_{3}'\pars{t}\,\dd t ={1 \over 8}\,{\rm Li}_{3}\pars{1} = \bbox[10px,border:1px dotted navy]{\ds{\zeta\pars{3}}} \approx 0.1503 \end{align}
$\ds{{\rm Li}_{\rm s}\pars{z}}$ is a PolyLogarithm Function: I used some properties of them as reported in the cited link. $\ds{\zeta\pars{s}}$ is the Riemann Zeta Function.