Simplify: $\;$ $2 \left(x^2 + \sqrt{x^4 - 1}\right) \left[\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}} + \sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}\right]^{-2}$
$2 \left(x^2 + \sqrt{x^4 - 1}\right) \left[\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}} + \sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}\right]^{-2}$ $\;\;\; \cdots \; (1)$
Consider the expression $\;\;$ $\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}}$
$= \left[\dfrac{\left(x^2 + 1\right) \left(x^2 + 1\right)^{\frac{1}{2}}}{x}\right]^{\frac{1}{3}}$
$= \left[\dfrac{\left(x^2 + 1\right)^{\frac{3}{2}}}{x}\right]^{\frac{1}{3}}$
$= x^{\frac{-1}{3}} \left(x^2 + 1\right)^{\frac{1}{2}}$ $\;\;\; \cdots \; (2)$
Consider the expression $\;\;$ $\sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}$
$= \left[\dfrac{\left(x^2 - 1\right) \left(x^2 - 1\right)^{\frac{1}{2}}}{x}\right]^{\frac{1}{3}}$
$= \left[\dfrac{\left(x^2 - 1\right)^{\frac{3}{2}}}{x}\right]^{\frac{1}{3}}$
$= x^{\frac{-1}{3}} \left(x^2 - 1\right)^{\frac{1}{2}}$ $\;\;\; \cdots \; (3)$
In view of expressions $(2)$ and $(3)$, the expression $\;\;$ $\left[\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}} + \sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}\right]^{-2}$ $\;\;$ becomes
$\left[x^{\frac{-1}{3}} \left(x^2 + 1\right)^{\frac{1}{2}} + x^{\frac{-1}{3}} \left(x^2 - 1\right)^{\frac{1}{2}}\right]^{-2}$
$= \left[x^{\frac{-1}{3}} \left(\sqrt{x^2 + 1} + \sqrt{x^2 - 1}\right)\right]^{-2}$
$= \left[\dfrac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{x^{\frac{1}{3}}}\right]^{-2}$
$= \left[\dfrac{x^{\frac{1}{3}}}{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}\right]^2$
$= \dfrac{x^{\frac{2}{3}}}{x^2 + 1 + x^2 - 1 + 2 \sqrt{\left(x^2 + 1\right) \left(x^2 - 1\right)}}$
$= \dfrac{x^{\frac{2}{3}}}{ 2 \left(x^2 + \sqrt{x^4 - 1}\right)}$ $\;\;\; \cdots \; (4)$
$\therefore \;$ In view of expression $(4)$, expression $(1)$ becomes
$2 \left(x^2 + \sqrt{x^4 - 1}\right) \times \dfrac{x^{\frac{2}{3}}}{2 \left(x^2 + \sqrt{x^4 - 1}\right)}$
$= x^{\frac{2}{3}} = \sqrt[3]{x^2}$