Sure, there definitely are a class of functions that have elementary symbolic primitives that we can find that certain software cannot.
When it comes to something like engineering though, this skill is completely moot, as we need a numerical output. And doing symbolic gymnastics to reach something that can be cobbled out of elementary functions is unnecessary, as we then need to evaluate these functions anyway, which is done using convergent series.
On the purer side, my indifference towards (especially indefinite) integration is just that the mathematics is not very deep at all. We are using our bag of tricks to solve first order single variable ODE's of the form f'=g. With some fancy tricks we might be able to get an elementary f, but for most g we cannot. But so what? We can still obtain pretty much all the properties of f without an elementary symbolic expression for it, indeed most special functions are usually DEFINED by some sort of differential equation and we can still study them and use them.
Definite integration can be somewhat nicer, as then we can exploit symmetries and such to tackle things that we can't antidifferentiate. It's kind of like finding clever ways to sum infinite series.
They can be pleasing to solve, and they certainly can be difficult, but in themselves they just aren't all that interesting or deep to me. (And the kind of difficulty that comes with some of them is not the kind that satisfies me, even if I do solve them.)
