For example, there is an integration bee and lots of youtube videos tackling hard or impossible integrals with just tools of Calculus I & II. However, a lot of these integrals found on youtube need assistance from Multivariable (Gaussian integral) or Complex Analysis (use residue theorem to solve a real integral) or some other tricks.
Much of what follows is a matter of opinion, and it is kind of stream-of-conciousness. Caveat lector.
I think that courses (particularly informal courses) on "hard integrals" already exist (at least in the United States). These courses are an outgrowth of the graduate school admissions process here. Specifically, many PhD programs in the US require applicants to take (and do well on) the Math Subject GRE. While this examination ostensibly covers the breadth of undergraduate mathematics, it seems to emphasize the calculus sequence quite a bit. As such, GRE prep classes often spend a great deal of their time on techniques of integration, and focus on "tricky" integrals. While such courses are about more than just integration, they are likely the closest thing to a course on "hard integrals" that you will find.
I think that a course in "Computational Techniques" would be an interesting and valuable course to have running around in the curriculum, but I think that it fills a niche that doesn't really exist at the moment. I have argued to colleagues that the undergraduate calculus sequence focuses too much on computational techniques and not enough on theory, and that one way of mitigating this might be to rejigger the curriculum a bit to change this. Instead of teaching three semesters of calculus, then a semester of linear algebra and another (or two) of differential equations, I think it might be worthwhile to condense things a bit, focus on theory, then offer a semester long course on computational techniques.
Such a course would likely cover a lot of ground---I might defer most of the trigonometric integrals to such a course, play around with "integration bee" style problems, and hit the Cartesian-to-spherical integration problems pretty hard. The course should also spend some time on transformational techniques for solving DEs (e.g. the Laplace and Fourier transforms), some of the more clever tricks in linear algebra, and perhaps some of the techniques from complex analysis (though the target audience that I have in mind probably wouldn't have the background for complex, so maybe not).
My feeling is that such a course would appeal primarily to non-math majors (particularly physics majors), but that it could also find an audience among the kind of students that currently enroll in more informal GRE prep classes. However, I can't see there being a great deal of demand in the current environment, as much of the material already covered, albeit spread out over a lot of other undergraduate courses.
In short, my answer to the headline question is a very qualified "Yes?"
I think it would be hard to get the demand for it, especially as a formal course. Even for those students who were interested, there might be more "bang for the buck" to just take a diffyQ course or the like (more advanced material).