Let \( [\,\cdot\,] \) be the greatest integer function. If \( \alpha = \int_{0}^{64} \left(x^{1/3} - \left[x^{1/3}\right]\right) dx \), then \( \displaystyle \frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta \) is equal to
