\displaystyle
\frac{dz}{2-\frac{1}{z}-z}=\frac{dz}{\frac{2z-1-z^2}{z}}=-\frac{zdz}{\left( z-1\right)^2}=-\frac{z\pm 1}{\left( z-1\right)^2}dz=
-\frac{1}{2}\frac{2\left( z-1\right)dz}{\left( z-1\right)^2}+\frac{dz}{\left( z-1\right)^2}