Fourier transform of $1 - \cos(xe^{-x^2})$

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Is there a closed form expression or maybe an infinite series? If not is there a "good" approximation to it? Even a "good" approximation of the fourier transform close to zero frequency would do. Can I use the taylors series for the cosine function and find the fourier transform term by term? That is the only idea i have right now. Thanks for your help!