How can NIntegrate
be extended with custom implementation of integration rules?
This answerThis answer of the question "Monte Carlo integration with random numbers generated from a Gaussian distribution""Monte Carlo integration with random numbers generated from a Gaussian distribution" shows customization or a general rule ("MonteCarloRule"). This question is about defining new rules.
A related question is "How to implement custom NIntegrate integration strategies?""How to implement custom NIntegrate integration strategies?".