Does an elementary method exist?

Calculus Level 5

sinxx2+4x+5dx\large{\displaystyle \int^{\infty}_{- \infty} \frac{\sin x}{x^2+4x+5}\, dx}

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

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