$\begin{array}{l} \sqrt[3]{{1 - 27\sqrt[3]{{26}} + 9{{\sqrt[3]{{26}}}^2}}} + \sqrt[3]{{26}}\\ = \sqrt[3]{{{3^3} - {{3.3}^2}.\sqrt[3]{{26}} + 3.3\sqrt[3]{{26}} - 26}} + \sqrt[3]{{26}}\\ = \sqrt[3]{{{{\left( {3 - \sqrt[3]{{26}}} \right)}^3}}} + \sqrt[3]{{26}} = 3 - \sqrt[3]{{26}} + \sqrt[3]{{26}} = 3 \in \mathbb{Z} \end{array}$
