I am trying to find closed form for this expression: $$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 - i\sqrt 2 } \right)} \right)$$
This is the remaining terms from a problem I encountered, this expression has its closed form as follow: $$ - \frac{{{\pi ^2}}}{4} + {\log ^2}(2) + \frac{{3{{\log }^2}(3)}}{4} - \log (3)\log (2) + \frac{1}{2}{\tan ^{ - 1}}\left( {\frac{5}{{\sqrt 2 }}} \right){\tan ^{ - 1}}\left( {2\sqrt 2 } \right)$$ The numerical check matched. I tried to use all dilogarithm identities but none of them work.
May I ask for some hints? Thank you for reading, this is not a homework, I am solving a problem and can't find a way to obtain this closed form, so I need some hints, please don't vote for closing, I think this may be helpful for others.