Complicated definite integral

Calculus Level 4

01cos(x)ln(cos(x))+xsin(x)x2cos(x)dx \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 lnf(1)\ln f(1) , where f(x)f(x) is an elementary function, then what is the value of f(π)f(\pi)?

Note: Don't use CAS!

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