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

The integral above can be expressed as

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

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

Submit your answer as the sum of products \(AB+CD+EF+GHI \).

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