Use the Hermite interpolating polynomial of $f$ at distinct nodes $x_{0}, \ldots, x_{n}$ to approximate the integral $\int_{-1}^{1} f(x) d x$.
It seems that the key idea to approach the problem would be the divided-difference method and applying Simpson's rule at the end. However, after some computations and corresponding integrations, no concrete results yield the desired approximation. What could be the major intermediary steps that could accelerate this process?