That's what I thought Allan, but a completely unrelated reasoning made me suspect. Consider the number $\displaystyle \sqrt{2}$, an irrational number, and the number $\displaystyle 2-\sqrt{2}$, another irrational number. If you add

them, [tex]\sqrt{2}+(2-\sqrt{2}) = 2[/Math], which is a rational number. So addition is nasty (as is her twin) and cannot be trusted. Now, take $\displaystyle f(x) = \frac{1}{\log{x}}[/Math] and [Math]g(x) = -\frac{1}{\log{x}}+x^2+3x+4$. Integral of

$\displaystyle f(x)$ is non-elementary and the integral of $\displaystyle g(x)$ is non-elementary. If we add them, $\displaystyle f(x)+g(x) = x^2+3x+4$, the integral of which is of course elementary.

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

$\displaystyle \displaystyle \int\frac{n^2\sin^{2n-1}{nx}\cos{nx}}{\sqrt{\sin^{2n}{nx}+\cot^{n-1}{nx}}}\;{dx}$

No luck yet.