E and pi?

Calculus Level 5

\[\large \int _{ 0 }^{ \infty }{ \dfrac { x-1 }{ { e }^{ x^{ 2 }-1 } } \, dx } =\dfrac { 1 }{ A } (Be-Ce\sqrt { D\pi } )\]

If the equation above holds true for positive integers \(A,B,C\) and \(D\), find \(A+B+C+D\).

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