We have:\(\dfrac{y}{x^3-1}-\dfrac{x}{y^3-1}=\dfrac{y}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{x}{\left(y-1\right)\left(y^2+y+1\right)}\)

\(=\dfrac{y}{\left(x-x-y\right)\left(x^2+x+1\right)}-\dfrac{x}{\left(y-x-y\right)\left(y^2+y+1\right)}\)

\(=\dfrac{y}{-y\left(x^2+x+1\right)}+\dfrac{-x}{-x\left(y^2+y+1\right)}=\dfrac{-1}{x^2+x+1}+\dfrac{1}{y^2+y+1}\)

\(=\dfrac{-y^2-y-1+x^2+x+1}{\left(x^2+x+1\right)\left(y^2+y+1\right)}\)

\(=\dfrac{x^2-y^2+x-y}{x^2y^2+xy^2+y^2+x^2y+xy+y+x^2+x+1}\)

\(=\dfrac{\left(x-y\right)\left(x+y\right)+x-y}{x^2y^2+xy\left(x+y\right)+x^2+y^2+x+y+xy+1}\)

\(=\dfrac{2x-2y}{x^2y^2+xy+1-2xy+1+xy+1}\)

\(=\dfrac{2\left(x-y\right)}{x^2y^2+3}\)