Caveat: My opinions only. I have not discussed this with other moderators.
IMHO it is questions like this recent beauty that cause more resentment towards questions about integrals. Several answers, certainly nothing that has not been covered zillion times on the site - yet not a single answerer or passer by put in the effort to find a reasonable duplicate. It looks like high-volume freshman questions (not forgetting high school) simply disregard all rules about duplicates.
Enough of that. Back to the question at hand.
I don't think that collecting these integrals to a single CW question would work. That would be a bad question, and, unless I am mistaken, the techniques are relatively diverse.
It could serve as a list of useful links - collecting pointers to techniques for the benefit of whoever wants to learn the tricks of the trade. However, such a list needs maintenance, i.e. a curator. If Ron Gordon or some other active user volunteers to maintain the list, I would certainly support the idea. That may be too much of a commitment, and the interested users can find these questions without such help. So it is not clear it is worth the trouble.
However, there may be some valid concerns and misgivings about these questions. As I have had such feelings myself, let me share them and my conclusions with you. These are impressions of a casual outsider observer.
- Sometimes I could not shake the feeling that those integrals were posted as challenges rather than genuine questions about a problem somebody had encountered. May be straight out of Gradshtein-Ryzhik or some such tome? I guess that this worry originates from Anastasia's past suggestion of actually running an integration contest. The idea was that the contest organizer would hand out bounties as prizes. This was shot down in a meta discussion at the time. So the concern was whether the idea has survived in the form
of a continuously running integration bee :-). But I agree with the sentiment of Normal Human's answer (if I got it right). I don't really see any harm coming from such questions. At least not if these highly upvoted opinions are observed. Let's not forget that similar opportunities are available in other areas of math as well.
- These tough integrals seem to be mostly about honing a skill set rather than reaching for new conceptual heights. But, again, A) where's the harm to the site, and B) similar opportunities exist elsewhere as well. It is those repetitive integrations by parts, standard substitutions, and questions about partial fractions expansions that should (IMVHO) fall under there are only 40 calculus questions.
- There could be a concern that such questions and their answers all come from a smallish group of users, and thus they might silently form a voting ring. Perish this thought! Right away! As a diamond-bearer I can see some stats. Sure, the active integrators have upvoted each other, but the scale is not any different from what you see within any other group of users sharing an interest in some tag.
TL; DR; My misgivings seem to originate from jealousy. Boo-hoo. This is not my cup of tea, so these people should not gain so much rep from working out these esoteric integrals either. If we let the green-eyed monster form site policies I think a downward spiral is inevitable.
So if you don't like these integrals, just upvote other stuff instead.