I had the same question. Not sure, but if you look at the plot of the indefinite integral, you can see it appears to have period 2pi (http://www.wolframalpha.co

I had the same question. Not sure, but if you look at the plot of the indefinite integral, you can see it appears to have period 2pi (http://www.wolframalpha.com/input/?i=integral+log(2-2cos(x)) http://www.wolframalpha.com/input/?i=integral+log(2-2cos(x))). The indefinite integral for log(3-3cos(x)), for example, is not periodic - the value 2 inside the log seems to be a critical point, so I guess it was chosen deliberately to force f(1)=0.