\[\int_{0}^{\infty}\sin^{-1}\left(\dfrac{e^{-x}}{\sqrt{2}}\right)\, dx=\dfrac{\pi^P}Q \ln R + \dfrac GR \]

The equation above holds true for positive integers \(P,Q,R\) with \(R\) being a prime. Find \(P+Q+R\)

\[\] **Notation:** \(\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 \) denotes the Catalan's constant.

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