Challenge · Medium problem

For $x>0$, find the minimum value of $\dfrac{\left(x+\frac{1}{x}\right)^{6}-\left(x^{6}+\frac{1}{x^{6}}\right)-2}{\left(x+\frac{1}{x}\right)^{3}+\left(x^{3}+\frac{1}{x^{3}}\right)}$.