This question has no detail answers and I would like to ask the similar question. I deal with the following integral: $$\int_{0}^{1}dx\,x(1-x) \ln\frac{M^2-k^2x(1-x)}{\lambda^2},$$ where one can assume that $M>0$, $\lambda>0$, ... (in other words, I would loke to emphasize that integral is convergent). The "branch point" is $$x_{1,2}=\frac{1}{2}\pm\frac{1}{2}\sqrt{k^2-4M^2}$$ and I don't know how to evaluate this integral. I think about integration in complex plane with suitable contour but don't know how to perform this calculation.
I know the answer with help of Mathematica but I am interested in how to do it "by hands". Also, may be this question this question is helpful for further discussion.