I don't know a precise definition of "elementary", but one thing about the elementary functions (exp,log,sin,cos,arcsin,arccos,+,x,-,/) is that they interact wi

I don't know a precise definition of "elementary", but one thing about the elementary functions (exp,log,sin,cos,arcsin,arccos,+,x,-,/) is that they interact with each other in nice ways: log(ab)=log(a)+log(b) cos(a+b)=cos(a)cos(b)-sin(a)sin(b) sin(log(x))=(x^-i - x^i)i/2 etc. This means that if your integral yields and answer in terms of log, exp or trigonometric functions then there is a good chance that any further algebra you have to do will work out nicely. But erf satisfies no such relations and so an answer in terms of it is often an algebraic dead end.