I know that the following integral equation holds

$$\int\_{-\infty}^{\infty}\frac{\log(a^2+x^2)}{b^2+x^2}dx=\frac{2\pi}{b}\log(a+b)$$

for $a,b>0\in\mathbb{R}$. Does it also hold for $a,b\in\mathbb{C}$ ?