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Calculus Level 5

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 $A + B + C?$