It's well known that such integral has no elementary primitive.
Makes sense if you're lookin' for
There are several other theorems that can prove that integrals of various sorts have no elementary primitive. Here is the briefest of brief reviews that barely scratches the surface.
differencial Galois theory address the question which primitive can be solved and which cannot. It looks interesting to me, because algebra is what I study the most, maybe one day I learn more about it.