How to find interval where function $f(x)=x+\frac{1}{x^{3}}$ is one to one\injective? How to find interval where function is one to one $f(x)=x+\frac{1}{x^{3}}$ ?(graphically or algebraically analytically)
 Let $f(x) =f(y)$ this gives $ (x-y)(\frac{(xy)^{3}-y^{2}-x^{2}+xy}{(xy)^{3}})=0 $
I don't know how to proceed further.

Note that $f'(x) = 1-\frac{3}{x^4}$.  For $|x| < 3^{1/4}$,$f'<0$.  So, $f$ is one-to-one in this region (with $x = 0$ excluded), as you can see in the graph provided by @greedoid.  
