Maclaurin series of an inverse function.

I have a problem I can't seem to really wrap around my head at the moment.

Let $g(x)=2\tan(x)-x$, $|x|\le\cfrac{\pi}{2}$

My task is to find the maclaurin series of the inverse function $g^{-1}(x)$ up till the term in $x^5$.

First, I know that the maclaurin series of $\displaystyle g(x)=x+\cfrac{2}{3}x^3+\cfrac{4}{15}x^5+\ldots$

I also know that $g$ is an odd function.

I tried using $g(g^{-1}(x))=x$ and differentiating from there but I realized that it is extremely tedious and easy to make careless mistakes.

Are there any tricks/new ideas I can use to make this process less painful? Thanks for the help.