I have the following integral, which I am trying to solve:
$\dfrac{1}{2 \sin^2 \alpha} \int_{0}^{\pi} d\theta \int_{0}^{2 \pi} d\phi \dfrac{sin^{3}(\theta) \cos^2(\phi) (cos^2(\alpha) +2 \cos(\alpha) \cos(\alpha/2) \cos(\theta) +\cos^2 (\alpha/2) \cos^2(\theta) - \sin^2 (\alpha/2) \sin^2(\theta) \sin^2(\phi))}{1 + 2 \cos(\alpha/2) \cos(\theta) + \cos^2(\alpha/2) \cos^2(\theta) - \sin^2(\alpha/2) \sin^2(\theta) \sin^2(\phi)}$
I already did the $\phi$ integral, and ended up with $\dfrac{\pi}{2 \sin^2 \alpha} \int_{0}^{\pi} d\theta \sin(\theta) \left(\sin^2(\theta) + 8 \cos(\alpha/2)(\cos(\alpha/2) + \cos(\theta))\left(\dfrac{|\cos(\alpha/2) + \cos(\theta)|}{1+\cos(\alpha/2)\cos(\theta)} - 1\right)\right)$
However according to the paper I am reading, the result from this integral should involve $\ln(\sin(\alpha/2))$, while I am obtaining $\ln(\cot^{2}(\alpha/4))$. Could anyone step in with some advice how to go about the integral?