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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