Non-elementary functions with elementary proofs!

Calculus Level 5

\[\large \text{Si}(x) = - \int_{x}^{\infty} \frac{\sin(t)}{t} \, dt \\ \large \text{Ci}(x) = - \int_{x}^{\infty} \frac{\cos(t)}{t} \, dt\]

The functions \(\text{Si}(x)\) and \(\text{Ci}(x)\) are usually defined as above. Given that

\[ \large \left| \int_{0}^{\infty} \sin(x) \text{Si}(x) \, dx + \int_{0}^{\infty} \cos(x) \text{Ci}(x) \, dx \right| = \frac{A\pi^{B}}C,\]

where \(A,B,C\) are all positive integers with \(A,C\) coprime. Find \(A + B + C\).

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