Note that yours is not a valid probability density function.
Indeed, you have $f(x)< 0$, for all $x< -1$. But, by definition, a PDF is always non-negative, ie. $$f(x)\geq 0, \text{ for all } x.$$
In addition,
$$ \int_{-\infty}^{-1} 1/x^3\,dx + \int_{1}^{\infty} 1/x^3\, dx= -\frac{1}{2}+\frac{1}{2}=0. $$ Thus the CDF is also not valid either.