An integral has been pushed me over the edge for several weeks. It reads as: $$\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$

I tried to calculate the surface integral inside using spherical coordinates, but it seems that I couldn't do any further calculation since the integrand function is something like $$e^{-\big(k_1(\varphi)\sin^2\theta+k_2(\varphi)\cos^2\theta+k_3(\varphi)\sin\theta\cos\theta\big)}\sin\theta .$$ Then I tried to use variable substitution to compute, similarly, I didn't get anything useful. I was also trying to use Maple to compute, but it didn't work at all. My original intention is to prove that the formula $$\displaystyle e^{-\frac{1}{2}|x|^2}\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$ is bounded.

I would be grateful if you could give me a definite result.