Let's say you have a library of functions you know how to integrate and differentiate. Something that happens a lot is that you end up wanting to do integrate or differentiate a product like x -> f(x)g(x) or composition like x -> f(g(x)) of two or more of the functions you know about.

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.