I'm trying to integrate this: $$f_{jkl}\langle{\hat{a},\hat{b},\hat{c}}\rangle=\frac{1}{4\pi} \int d{\Omega}_n P_j(\hat{a}.\hat{n})P_k(\hat{b}.\hat{n})P_l(\hat{c}.\hat{n})$$ where $\hat{n}$ is a unit vector, $d\Omega_n$ denotes integration over the unit vector $\hat{n}$. $\hat{a},\hat{b}\, \text{and} \, \hat{c}$ form an equilateral triangle with $\hat{a}.\hat{b}=\hat{b}.\hat{c}=\hat{c}.\hat{a}= cos(\alpha)$.
I have to integrate this to find $f_{jkl} ({\hat{a},\hat{b},\hat{c}})$ as a function of $\cos(\alpha)$.
I know that integration of three Legendre polynomials can be calculated using Wigner-3j symbols. But here argument of each Legendre polynomials are different, so how to integrate this expression?