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!