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 $2(A+B+C+D)$ .