How to show that the following integral is infinite

Question

For an application I want to show the following claim: Let $x\in(0,1)$ and $y\in[0,1]$ be two parameters, then $$\int_{\mathbb{R}}\frac{1+\cos(\pi x (y-0.5))|z|^{x}}{1+2\cos(\pi x (y-0.5))|z|^x+|z|^{2x}}\, dz=\infty$$ holds true. I have tried to use the facts, that $\lambda:=\pi x (y-0.5)\in(-\pi/2,\pi/2)(\rightarrow \cos(\lambda)\in(0,1))$ and $$\int_{1}^{\infty}z^\beta\,dz=\infty\quad\text{iff}\quad \beta>-1$$ but I didn't get it.