It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/u-substitution/often tricky).
There are all kinds of anecdotes alluding to this fact (see e.g. this nice one from Feynman). Another consequence of this is that differentiation is well automatable within CAS but integration is often not.
We know that there is a deep symmetry based on the Fundamental theorem of calculus, yet there seems to be another fundamental structural asymmetry. What is going on here...and why?
Some peope asked for clarification, so I try to give it. The main objection to the question is that asymmetry between two inverse operations is more the rule than the exception in math so they are not very surprised by this behaviour.
There is no doubt about that - but, and that is a big but, there is always a good reason for that kind of behaviour! E.g. multiplying prime numbers is obviously easier than factoring the result since you have to test for the factors doing the latter. Here it is understandable how you define the original operation and its inverse.
With symbolic differentiation and integration the case doesn't seem to be that clear cut - this is why there are so many good discussions taking place in this thread (which by the way please me very much). It is this Why at the bottom of things I am trying to understand.
Thank you all again!