For diffentiation there are two ridiculously powerful theorems which says that if you know how to differentiate a bunch of functions you can also differentiate any product or composition of these functions.
For a random other map there is no reason for this to be the case, integration is essentially "as expected" in that except in some specific circumstances knowing how to integrate f and g doesn't tell you how to integrate their product or composition.
It also turns out we can’t even verify, in general, the result of integration:
Suppose you are given two functions, f(x) and G(x), and are told that G(x) is an antiderivative of f(x). So then you let g(x) = G’(x), the derivative of G(x).
Now if G(x) is truly an antiderivative of f(x) then we must have g(x) = f(x) but unfortunately the problem of determining whether two functions are equal is undecidable (a consequence of the halting problem).
Also, implementing it likely would be a challenge. The Risch algorithm (https://en.wikipedia.org/wiki/Risch_algorithm, https://mathworld.wolfram.com/RischAlgorithm.html) ‘only’ handles “rational functions, radicals, logarithms, and exponential functions”, but may never have been fully implemented (https://mathoverflow.net/questions/374089/does-there-exist-a...)
Formally, I'm not sure how to show that's related to the relative difficilties of the two operarions, or if it actually is related in the first place, but intuitively the explanation is appealing.
https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(diffe...
So it's important to keep in mind the difference between integral and anti-derivative. And note that as soon you say "within an epsilon" you're already on the other side of "not a lot of info" vs "infinite info". So global, local doesn't matter, especially since eg an analytic function is completely determined by it's derivatives at a single point (good ol Taylor's theorem).
I wonder if you could relate it to the difference between local and global sections on bundles and connect “global sections don’t always exist” to “global data is harder.”
>The SymPy (Symbolic Python) CAS also intends to incorporate Rubi’s integration rules. The code for Rubi is in the sympy/integrals/rubi subdirectory of the SymPy source-code on GitHub.
AFAIK, this is the only recent benchmark-type study that compares modern CAS software (Mathematica vs Maxima vs FriCAS vs etc).
Unfortunately the answer was in terms of hypergeometric functions and was scarcely more useful than the unevaluated integral. In any case I found Rubi to be very impressivein terms of results.
Digging into the code wasn't much fun, as it's just a colossal Mathematica rules engine, like reading Gradshteyn and Ryzhik, but with square brakets everywhere.
https://12000.org/my_notes/kamek/kamke_differential_equation...