Numerical integration: Quadrature method which one to use?

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Since “it depends” is the proper answer to a question about what quadrature method to use in evaluating an integral, what are the things that one should consider when making a choice.

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Do you have equally spaced values available? ~~-> Simpson or similar

Can you fix the values where to compute the function? ~~-> use one of the Gaussian methods

Wildly varying function? ~~-> some adaptative method