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Calculus Level 5

The integral 01tan12+x2(1+x2)2+x2dx \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 AπBC \frac{A\pi^B}{C} for positive integers A,B,CA,B,C, where A,CA,C are coprime. What is the value of A+B+CA+B+C?

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