$ \int_0^{2\pi}{ \sqrt{ 1 - \sin{ \theta } \sin{ 2\theta } + \cos\theta \cos{2\theta} } %sqrt \; d\theta } %int $
I tried removing the $2\theta$ terms, choosing the identity $ \cos{2\theta} = 1 - 2 \sin^2{\theta} $, but this results in the unsavory dish:
$ \int_0^{2\pi}{ \sqrt{ 1 - 4\sin^2{ \theta } \cos{ \theta } + \cos{\theta} } %sqrt \; d\theta } %int $
What is a reasonable next step, then? Is removing $2\theta$ terms a good strategy for this problem?