By using the generating function for Bessel functions I have discovered the following identity:
\begin{eqnarray} &&\int\limits_{[0,2 \pi] \times [-\frac{\pi}{2},\frac{\pi}{2} ]^2} e^{\imath x \sum\limits_{j=1}^3\left( \cos(2 \phi_j) + \cos(4 \phi_j) + \cos(2 \phi_j + 4 \phi_{(j+1)\%3})\right)} \prod\limits_{j=1}^3 d \phi_j =\\ && 2 \pi^3 \sum\limits_{(n_j)_{j=4}^9 \in Z^6} \imath^{-\sum\limits_{j=4}^6 n_j} \prod\limits_{p=1}^3 J_{n_{p+6}+2(n_{p+3}+n_{f(p-1)+6})}(x) \cdot \prod\limits_{p=4}^9 J_{n_p}(x) \end{eqnarray} where $J_n(x)$ are Bessel functions and $f(p)=p \cdot 1_{p\neq 0} + 3 \cdot 1_{p=0}$. Is it possible to find a closed form expression for the infinite sum on the right hand side?