Don't use calculus

Algebra Level 3

(x+1x)6(x6+1x6)2(x+1x)3+(x3+1x3)\large \frac{(x+\frac{1}{x})^{6}-(x^{6}+\frac{1}{x^{6}})-2}{(x+\frac{1}{x})^{3}+(x^{3}+\frac{1}{x^{3}})}

Find the minimum value of the expression above for positive xx.


Source: TN RMO 2015
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