$Evaluate \int 2^{2^{2^x}}{2}^{2^x}{2}^{x}dx$

$\begin{array}{l}I=\int 2^{2^{2^x}}{2}^{2^x}{2}^{x}dx\\ \text{ Let }{2}^{2^{2^x}}=t\text{ or }{2}^{2^{2^x}}{2}^{2^x}{2}^x(\log 2{)}^{3}dx=dt\\ \begin{array}{rl}\therefore I=\int \frac{1}{(\log 2{)}^3}dt& =\frac{1}{(\log 2{)}^3}t+c\end{array}\\ =\frac{1}{(\log 2{)}^3}{2}^{2^{2^x}}+c\end{array}$