Showing $\frac{1}{z^3+z}$ to be analytic

Question

$f(z)$ is analytic if $u_x = v_y$ and $u_y = -v_x$ where $u(z)$ and $v(z)$ are the real and complex part, respectively, of $f(z)$. However, $\frac{1}{z^3+z} = \frac{1}{(x+iy)^3+x+iy}$ doesn't seem to have a simple composition of a real and a complex part. Using polar form doesn't seem to make it easier either. Is there perhaps another way of showing it? It seems it should be an easier way, e.g. if $g(z) = z^3+z$ and $h(z) = 1/z$ are analytic, is their composition, $h(g(z))$, necessarily also analytic?