Powers should be doable exactly using confluent hypergeometric functions. See, e.g., the final case analyzed at https://mathworld.wolfram.com/GaussianIntegral.html.
It may also be possible to return the exact answer for log: https://math.stackexchange.com/questions/1359584/an-integral-involving-a-gaussian-and-a-logarithm
Powers should be doable exactly using confluent hypergeometric functions. See, e.g., the final case analyzed at https://mathworld.wolfram.com/GaussianIntegral.html.
It may also be possible to return the exact answer for
log: https://math.stackexchange.com/questions/1359584/an-integral-involving-a-gaussian-and-a-logarithm