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How to solve the closed forms of the following polylogarithm's integrals
$$\int\limits_0^{\frac{1}{2}} {\frac{{{{\ln }^2}\left( x \right)\ln \left( {1 - x} \right){\rm{L}}{{\rm{i}}_2}\left( x \right)}}{x}dx} ,\int\limits_0^{\frac{1}{2}} {\frac{{\ln \left( x \right)\ln \left( {1 - x} \right){\rm{L}}{{\rm{i}}_3}\left( x \right)}}{x}dx} ,$$ $$\int\limits_0^{\frac{1}{2}} {\frac{{\ln \left( {1 - x} \right){\rm{L}}{{\rm{i}}_4}\left( x \right)}}{x}dx} ,\int\limits_0^{\frac{1}{2}} {\frac{{{\rm{ln}}\left( x \right){\rm{Li}}_2^2\left( x \right)}}{x}dx} .$$

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