x = 3/2 (y-y^3/3)

Wolfram alpha gives this: http://www.wolframalpha.com/input/?i=inverse++(+3/2(x-x^3/3)+)&asynchronous=false&equal=Submit

How to derive this I'm not sure, but because the function is not bijection actually it does not have the inverse. To find the inverse of the bijective part i'll try to find out.