 Fourier transform of the logarithm of one plus x squared

For $u(x)=\frac{1}{2}\log (1+x^2)$, the angular-frequency transform is the tempered distributionThe numerator cancels the singularity at zero and makes this integral absolutely convergent. Differentiating $u$ and using $\widehat{(1+x^2{)}^{-1}}=\pi e^{-|\xi |}$ determines this expression up to a Dirac delta distribution. That delta coefficient is zero: testing with the expanding Gaussian ${\phi }_n(x)=(2\sqrt{\pi }{)}^{-1}{e}^{-x^2/(4n)}$ makes both the proposed expression and $⟨u,{\hat{\phi }}_n⟩$ tend to zero, while ${\phi }_n(0)$ stays fixed. Changing the subtraction convention would change the delta coefficient.