I never thought I was an integration hotshot (probably partly because my high school class valedictorian is currently one of the top four people in the state in calculus in Mu Alpha Theta competitions), but I know of a couple more that I'd like to add.
First, e^(-(x^2)). Try whatever you want with it as an indefinite integral; nothing works. It is, however, approximable with Taylor and MacLaurin expansions, as well as more mundane methods.
Another that is at least significantly difficult and possibly impossible is sqrt(x)sec(x). I've only tested this on my TI-89, which is much more simplistic than any computer mathematics software one would try these on. If anyone finds this to be either possible or impossible, /msg me, and I'll update this.
When I started writing this w/u, I thought I might have found a heuristic for at least some impossible integrals, but sqrt(tan(x)) disproves it. My false heuristic is that f(x) has one or more jump discontinuities and a bound on the domain. Impossible integrals that meet this heuristic still may be far more common than those that do not, although this prediction is to the zeroth approximation.