Fractional Parts in 2019

Calculus Level 5

$\displaystyle \int_{0}^{1}{\left(\left\{ \frac{1}{x}\right\} \{ 2x \}\right) \frac{dx}{x} }$

The above integral is equal to $A+\ln{B}-C\gamma,$ where $A,B,C$ are all positive integers.

Find $A+B+C.$

Note: $\{ \}$ denotes the fractional part and $\gamma$ denotes the Euler-Mascheroni constant.