Complicated definite integral

Calculus Level 4

\[ \int_0^1 \frac{\cos\left(x\right)\ln\left(\cos\left(x\right)\right)+x\sin\left(x\right)}{x^{2}\cos\left(x\right)}\,dx \]

If the definite integral above can be expressed in the form \(\ln f(1) \), where \(f(x)\) is an elementary function, then what is the value of \(f(\pi)\)?

Note: Don't use CAS!

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