I'm trying to compute an integral of spherical Bessel functions. $$ \int_{\mathbb{R}^9} d^3 \vec{x}_1d^3 \vec{x}_2 d^3 \vec{x}_3 j_0(k_1 x_1)j_0(k_2 x_2)j_0(k_3 x_3)j_0(k_4 |\vec{x}_1-\vec{x}_2|)j_0(k_5 |\vec{x}_1-\vec{x}_3|)j_0(k_6 |\vec{x}_2-\vec{x}_3|),$$ where $k_1...k_6$ are real positive numbers. Integrals of the form $ \int_{\mathbb{R}^3} d^3 \vec{x}_1 j_0(k_1 x_1)j_0(k_2 x_1)j_0(k_3 x_1)$ (or even products of four spherical bessel functions) are well studied. I've also been able to solve a first generalisation of the form $$ \int_{\mathbb{R}^6} d^3 \vec{x}_1d^3 \vec{x}_2j_0(k_1 x_1)j_0(k_2 x_2)j_0(k_3 |\vec{x}_1-\vec{x}_2|)j_0(k_4 x_1)j_0(k_5 x_2),$$ by brute force plugging in spherical coordinate definitions and Fubini.
For the Full integral in the first line, however, I'm already stuck by just trying to do the radial part:
Transforming to spherical coordinates we have $$ \int_{\mathbb{R}_+^3} d x_1 d x_2 d x_3 x_1^2x_2^2x_3^2j_0(k_1 x_1)j_0(k_2 x_2)j_0(k_3 x_3)\int d\Omega_1d\Omega_2d\Omega_3j_0(k_4 |\vec{x}_1-\vec{x}_2|)j_0(k_5 |\vec{x}_1-\vec{x}_3|)j_0(k_6 |\vec{x}_2-\vec{x}_3|),$$
Now I suspect that the key is in solving: $$\int d\Omega_1d\Omega_2d\Omega_3j_0(k_4 |\vec{x}_1-\vec{x}_2|)j_0(k_5 |\vec{x}_1-\vec{x}_3|)j_0(k_6 |\vec{x}_2-\vec{x}_3|)$$. Intuitively I would think that since the integral only depends on the angles between the vectors $\vec{x}_1,\vec{x}_2,\vec{x}_3$. With $\vec{x}_1=x_1 (\sin (\text{$\theta_1 $}) \cos (\text{$\phi_1$}),\sin (\text{$\theta_1$}) \sin (\text{$\phi_1$}),\cos (\text{$\theta_1$}))$ I would have hoped for some transformation $$\mu_{ij}=\sin (\text{$\theta_i$}) \sin (\text{$\theta_j$}) \cos (\text{$\phi_i $}-\text{$\phi_j $})+\cos (\text{$\theta_i $}) \cos (\text{$\theta_j $}),$$ but I have not yet been able to find a full working 6D substitution.
I've been stuck on this for way too long, so any idea/intuition is appreciated!