Please don't take this personally and keep in mind that your description is vague enough that I have to extrapolate some details and might go off in the wrong direction.
But to be blunt, I disagree that it is non-trivial. From the viewpoint of research mathematics, taken on its own, being able to integrate a certain class of functions isn't interesting in any way, especially if as you say it turns out to just be a linear combination of known results. In fact I could out dozens of such classes using the reverse process of just differentiating random terms and then reversing the calculation. So even if you would find a journal to publish this in, no one would read it.
What matters is context. Mathematicians are generally interested in the greater whole. Two possibilities come to mind in this context, but I fear neither of them is true here.
The proof uses interesting and novel techniques which ideally can be used in lots of different problems. Keep in mind though that novel in this context will not mean a tricky looking substitution (that's 18th century stuff), but some application of deeper theorems or other tricks usually from an entirely different area of mathematics. As a rule of thumb, if your proof is short (i.e. less than say 10 pages) and not heavily relying on citations from somewhere else, this is unlikely.
The class of integrals is in itself interesting. This usually means that it occurs in a lot of completely different places, which would indicate that those places have something in common that has not been explored yet. But even in this context, the solution of the integrals would not be the important part, but rather an analysis of the fact that these integrals seem to occur everywhere.
As a final remark, don't be discouraged, physicists and mathematicians just think differently and as a result have different concepts of what is interesting. I've seen the converse happen as well, where I was able to prove something about a physical problem that looked interesting to me but where the physicists I talked to did not see the point.
If you still want to continue with this, the options I see is to either just post your integral on math.stackexchange to see if someone else has seen something similar or if you want to publish your physics-result anyway, just mention your integrals in the abstract and cross-list the arxiv pre-print to math.CA (which I think should be the most fitting category).