I am interested in computing a rather nasty integral of the following form:
$$ 4\int_0^x \mathrm{d}\theta \frac{2|l_m\cos(\theta-\theta_m)|\log|l_m\cos(\theta-\theta_m)|-\sum_{j\in\{1,2,3\}}|l_j\cos(\theta-\theta_j)|\log|l_j\cos(\theta-\theta_j)|}{\prod_{j\in\{1,2,3\}}|l_j\cos(\theta-\theta_j)|} $$
where $m=\mathrm{argmax}_{m\in\{1,2,3\}}|l_m\cos(\theta-\theta_m)|$.
Curiously, when integrated to $x=2\pi$, this integral yields $\frac{(2\pi)^2}{A}$, where $A$ is the area of the triangle with angles $\theta_{1,2,3}$ and sides $l_{1,2,3}$ (assuming a valid triangle and verified numerically). Any clues on how to approach this analytically, for any $x\leq2\pi$?
