Step-by-step Solution:

Let: \[ 3^{3^{3^x}} = t \] Now, applying logarithmic differentiation: \[ 3^{3^{3^x}} \log{3} \cdot 3^{3^x} \cdot \log{3} \cdot 3^x \cdot \log{3} \, dx = dt \] This simplifies to: \[ 3^{3^{3^x}} \, dx = \frac{dt}{(\log 3)^3} \] Hence, the integration becomes: \[ \int \frac{dt}{(\log 3)^3} = \frac{t}{(\log 3)^3} = \frac{3^{3^{3^x}}}{(\log 3)^3} \]