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Calculus Level 5

The integral \[ \int_0^1 \frac{\tan^{-1}\sqrt{2+x^2}}{(1+x^2)\sqrt{2+x^2}}\,dx \] can be shown to be equal to \[ \frac{A\pi^B}{C} \] for positive integers \(A,B,C\), where \(A,C\) are coprime. What is the value of \(A+B+C\)?

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