The idealized integral

Calculus Level 5

\[ \large \int_1^\infty \frac{2x^3-1}{x^6 + 2x^3 + 9x^2 + 1} \, dx \]

If the integral above equals to \( \frac AB \cot^{-1} \left( \frac CD \right) \) for positive integers \(A,B,C,D\) such that \( \gcd(A,B) = \gcd(C,D) = 1 \), evaluate \(2(A+B+C+D) \).