\[ \large \int_{-\frac1{\sqrt3}}^{\frac1{\sqrt3}} \dfrac{x^4}{1-x^4} \text{arccos} \left( \dfrac{2x}{1+x^2} \right) \, dx \]

The integral above can be expressed as

\[ - \pi ^ A \left [ \dfrac B{\sqrt C} - \dfrac{\pi }{D} - \dfrac EF \ln \left |\dfrac{\sqrt G+ H}{\sqrt G - H} \right | \right ] , \]

where \(A,B,C,D,E,F,G,H\) are all positive integers with \(C,G\) square-free and \(E, F\) coprime.

Submit your answer as the sum \( A + B + C + D + E + F + G + H \).

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