Will the ILATE rule work?

Calculus Level 5

\[\large \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}}\frac{x^{4}}{1-x^{4}}\cos^{-1}\left(\frac{2x}{1+x^{2}}\right)dx\] The closed form of the integral above is of the form \(\displaystyle \frac{\pi ^{2}}{a}-\frac{\pi }{\sqrt{b}}-\frac{\pi }{c}\ln \left(\frac{\sqrt{b}-1}{\sqrt{b}+1}\right)\).

Find the value of \(a+b+c\).

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