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In general Simpson's rule is more efficient, if by efficient you mean having a smaller error for the same number of intervals.
The error for the trapezoidal rule is:
So the error has an upper bound dependent on the extreme values of the second derivative. This error can be made as small as wanted by increasing n (the number of intervals).
The error for Simpson's rule is:
Here the upper bound depends on the extreme value of the fourth derivative. Also notice that as n increases, the error reduces by `n^4` compared to `n^2` for the trapezoidal rule.
Simpson's Rule uses a quadratic to model the function on the interval, while the trapezoidal rule uses a linear function. So the more "curved" the function is, the better Simpson's rule will be compared to the trapezoidal rule.
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