Fourier transform of cos(1/x) (as tempered distribution)

Question

I'm trying to find the (rigorous) Fourier transform of $f(x)=\cos(1/x).$ Since $f$ is not in $L^1$ nor $L^2$, some singularity should appear, and I need to use the Fourier transform in $S'(\mathbb R)$. Thus, I'm looking for the computation of $$T_f(\hat \phi)=\int_{\mathbb R}\cos(1/x)\hat \phi(x)dx,$$ for $\phi \in S(\mathbb R)$, where $\hat \phi$ is the Fourier transform of $\phi$. Can someone help in the computation or give a reference in the literature?