Not So Rational Integration

Calculus Level 5

$\large I = \int_0^1 \dfrac{(2x-1)\ln x}{1-x^2} \, dx$

Find the value of the closed form of the above integral $I$ .
Submit your answer as $\lfloor 24I \rfloor$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.