That's what I thought Allan, but a completely unrelated reasoning made me suspect. Consider the number

, an irrational number, and the number

, another irrational number. If you add

them, [tex]\sqrt{2}+(2-\sqrt{2}) = 2[/tex], which is a rational number. So addition is nasty (as is her twin) and cannot be trusted. Now, take

. Integral of

is non-elementary and the integral of

is non-elementary. If we add them,

, the integral of which is of course elementary.

This question came to my head while I was pondering over this:

No luck yet.