Does an elementary method exist?

Calculus Level 5

\[\large{\displaystyle \int^{\infty}_{- \infty} \frac{\sin x}{x^2+4x+5}\, dx}\]

If the value of the above integral is equal to \(- \pi^{A} e^{B} \sin C\) where \(A,B\) and \(C\) are integers, find \(A \times B \times C\).

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