Integrate $\int_{3-2\sqrt{2}}^{3+2\sqrt{2}} \frac{\ln x}{x\sqrt{-x^2+6x-1}}$ How to integrate $$\int_{3-2\sqrt{2}}^{3+2\sqrt{2}} \frac{\ln x}{x\sqrt{-x^2+6x-1}}\, dx$$
I believe the exact value of this integral cannot be found , however there could be a way I am not aware of. If not, is it possible to put into a familiar form using other functions such as $Li(x)$ for example. 
After some attempts i am still lost and Mathematica is not too helpful.
Thank you kindly for your help and time.

Substitute $x=\frac1{3-2\sqrt2\cos t}$ to get
\begin{align}
& \int_{3-2\sqrt{2}}^{3+2\sqrt{2}} \frac{\ln x}{x\sqrt{-x^2+6x-1}}\, dx\\
= & \int_0^\pi \ln(3-2\sqrt2 \cos t) dt\\
=& \int_0^\pi [\ln2+\ln(1+  \frac12-\frac2{\sqrt2 }\cos t) ]dt=\pi\ln2
\end{align}
where $\int_0^\pi \ln(1+ a^2-2a\cos t)dt=0$ with $a=\frac1{\sqrt2}$ is used.
Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$
