Just toying with it a bit, I note:
n = 4 results in something between 1 and 2 half-periods
n = 7 results in something between 2 and 3 half-periods
n = 13 results in something between 4 and 5 half-periods
n = 26 results in something between 8 and 9 half-periods
Seems like this could lead somewhere useful, at least getting bounds on it.
Okay, I haven't quite talked myself into it, yet, but let's see how far it gets me.
For an arbitrary 'n', the half period is . If we restrict the evaluation of the integral to the first half period, the Absolute Values become unnecessary.
Given the half-period of , quite obviously there are such half periods in [0,1].
Somewhat to my surprise, this makes the result independent of 'n'.
It's not obvious what I'm missing. Clearly, the limit it greater than that, but I figured I'd get it out there for ridicule.