For $x\in\mathbb{R}$ and $z\in\mathbb{C}\setminus\mathbb{R}$, how can I integrate
$$\int_{-\infty}^{\infty}\frac{du}{\pi}\frac{1}{z-x+\left(z+x\right)u^{2}}.$$
I believe the answer is $\text{sgn}\left(\text{Re} z\right)\frac{1}{\sqrt{z^{2}-x^{2}}}$. I am having trouble obtaining the sign function, which I think comes from branch cuts involving square root and log.