Density function without the inverse

Question

Let $Y$ a random variable such that $$f_{Y}(t)=2t^{-3}\mathbb{I}_{t\geq 1}.$$ I would like to find the density function of $$Y^{2}\frac{1-e^{-Y^{2}}}{1+Y^{4}}.$$

There is no explicit inverse function of $$t\mapsto t^{2}\frac{1-e^{-t^{2}}}{1+t^{4}}...$$

Any Idea ?