Two comments: 1. The Risch algorithm is "complete"; i.e., it is an *algorithm*: https://en.wikipedia.org/wiki/Risch_algorithm (I'm assuming that you aren't asking about the computational complexity of the Risch anti-derivative algorithm.) 2. Some closed form integrals may require polynomial root-finding (or root expressing), so you may not like the look of the resulting expression. E.g., you might find floating point approximations to certain numbers instead of a "perfectly precise" answer. At 01:54 PM 12/16/2016, Dan Asimov wrote:
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?
If this question can be made rigorous, how might that be done?
(And if so, what is the rigorous answer, or at least a method of approaching it?)
ÂDan