There exist positive integers $A$, $B$, $C$ such that
$\int_1^{\sqrt{2}} \sqrt{\frac{4x^2+1}{x^2-1}} \frac{dx}{x} \; = \; \tfrac{1}{A}\pi + \ln B - \tan^{-1}C.$
What is the value of $A + B + C?$

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