Terms, terms and terms

Calculus Level 5

\[ \large \int_{-\infty}^\infty \dfrac{x^4}{x^8 + 2x^6 - 4x^4 + 8x^2 + 16} \, dx \]

If the value of the integral above is equal to \[ \dfrac \pi a \sqrt{ \dfrac{b-\sqrt{12}}c }, \], where \(a,b\) and \(c\) are positive integers greater than 1, find \((a+b+c)^2 \).

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