Integration Mania Continued

Calculus Level 5

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

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 $A + B + C + D + E + F + G + H$ .