integral of $\int_{-\infty}^{+\infty}\frac{\sin^2(x)}{x^2(1+x^2)}dx$ with Plancherel

Question

I should calculate the integral $$\int_{-\infty}^{+\infty}\frac{\sin^2(x)}{x^2(1+x^2)}dx$$ with the Plancherel Formula $\| f\|_2 = (2\pi)^{1/2} \| \mathcal{F}(f) \|_2 $ where $\mathcal{F}(f)$ is the fourier transformation of $f$ and we use the norm in $ L^2(\mathbb{R}) $. I began with trying to find a function, which is either the fourier transformation or the inverse fourier transformation of the square root of the function above, so $\frac{\sin(x)}{x\sqrt{1+x^2}}$. But it seems quite hard to calculate the fourier function (or the inverse one). Is there a trick in this exercise?