Integral $\int_{-\infty}^\infty e^{-x^2-\cos^2(x)}\sin^2(x) dx$

Question

I need help with this integral: $$I = \int\limits_{-\infty}^\infty e^{-x^2 -\cos^2(x)}\sin^2(x) dx$$

Unfortunately, neither Mathematica nor WolframAlpha are able to compute it. WolframAlpha gives the approximate value of the integral as $$I \approx 0.394862$$ and the integrand looks something like this: click

I would greatly appreciate, if anyone knew a closed form solution to this.