Question

Quantifiably Difficult Antiderivatives

Answer 1

One can get arbitrarily awful by just increasing the exponent of your classical example.

Answer 2

To increase the complexity as stated, pick something with a large exponent that requires many loops through integration by parts…. Say
Sec^9(x) but you can expand it out by increasing the exponent more…

Answer 3

This reminds me of the system Hardy once gave when evaluating the proofs of difficult integrals in terms of number of “marks” which he assigned by the number of “inversions” in the proof.