Integrate this one!

Calculus Level 5

$\large{\displaystyle \int^{\infty}_{0} \frac{(12-x^2)}{(x^2+4)^3} \cos (2x) dx=\frac{A \pi^{B}}{C e^{D}}}$

where $$A,B,C,D$$ are integers and $$A,C$$ being coprime.

Find $$A+B+C+D$$

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