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

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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