Solve in integers: $$ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$$
My idea: $$\Longleftrightarrow (y^3+xy-1)(x^2+x-y)-(x^3-xy+1)(y^2+x-y)=0$$ $$\Longleftrightarrow -x^4-x^3y^2+2x^3y+x^2y^3+2x^2y-x^2+2xy^3-2xy^2-2x-y^4-y^2+2y=0$$ $$\Longleftrightarrow 2xy(x^2+y^2)+2y(x^2+1)+x^2y^3+2y=x^4+y^4+x^3y^2+x^2+y^2+2x$$
following I can't work.