inequality about characteristic function

Question

Let $X$ be a random variable with density $f(x)=|x|^{-3}1_{|x|\ge1}$ and $\phi_{X}(t)=E[e^{itX}]$. Show that $\forall t\in[-1,1] $ $$|\phi_{X}(t)-1-t^2log|t||\le3t^2$$ I noticed that $E[X]=0$, so $|\phi_{X}(t)-1-t^2log|t||=|E[e^{itX}-1-itX-t^2log|t|]|$. But $1,itX$ are the first two terms of the Taylor series of $e^{itX}$. Any ideas?