So undergraduates have the calculus tools to write an integral so convoluted it admits no closed-form solution. But most such integrals are uninteresting or already useful even without closed-forms, such as the error or gamma functions.
It's immediately clear that few if any mathematicians focus on difficult-to-solve "elementary" integrals, because typically these integrals do not answer important questions. Moreover, with more advanced tools, e.g. from geometry or functional analysis, it is often possible to get their functional behavior within some arbitrary precision.
What I am interested in is whether there are unsolved indefinite integrals, accessible to an undergraduate out of calculus, which underlie the results of modern open problems, where the open problems are either interesting, or of some minor or major importance.
Approaches to solving the integrals may be (realistically should be) significantly advanced, but a calculus student should be able to understand the problem statement.