Evaluate a contour integral around a spiral closed by a segment

Let \[ \beta(t)=te^{it},\qquad 0\le t\le4\pi, \] and let \(\gamma\) be the closed curve obtained by following \(\beta\) and then the line segment from \(4\pi\) to \(0\). Evaluate \[ \int_\gamma \frac{z^2+1}{(z+1)(z+4)}\,dz. \]