I was computing integrals using WolframAlpha and I found for many complex numbers $p, q, r$ which are not real numbers, this integral always converged to $0$:
$$\int_{-\infty}^{\infty} \frac{1}{(x-p)(x-q)(x-r)} dx$$
I tried to prove this and I got a partial fraction decomposition of this function i.e.
(1) for different three $p, q, r$ $$\frac{1}{(p-q)(p-r)}\frac{1}{x-p}+\frac{1}{(q-p)(q-r)}\frac{1}{x-q}+\frac{1}{(r-p)(r-q)}\frac{1}{x-r}$$
(2) for doubled $p$ and single $q$ $$\frac{1}{p-q}\frac{1}{(x-p)^2}-\frac{1}{(p-q)^2}\frac{1}{x-p}+\frac{1}{(p-q)^2}\frac{1}{x-q}$$
(3) and the rest is of course $$\frac{1}{(x-p)^3}$$
and then for (3) I could prove my assumption. But when it comes to (1) and (2) it was not very easy for me. Does anyone know about this integral? Is my assumption correct? Thanks in advance.
Edit And I'm not also sure whether I really can use the normal riemann integral method to this function, because it has complex coefficients.