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Calculus Level 5

There exist positive integers AA, BB, CC such that 124x2+1x21dxx  =  1Aπ+lnBtan1C. \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?A + B + C?

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