This is not elementary.
Loiuville shown that if are rational function then the integral is elementary if and only if, there exists a rational function such as .
In this case, let us we have that.
and solve the differencial equation,
This equation has no rational solutions.
Thus the function is not elementary.
(However, it does exists. This is a consequence of the fundamental theorem of calculus. It can be shown that any countinous funtion on a closed interval or on the number line always has a an anti-derivative.)
If you are curious about which integral can and cannot be there is a topic called Differencial Algebra
Furthermore, this is an important function. It is used as an approximation of the binomial distribution (normal curve). I belive (I might be wrong) it is reffered to as
Lasty, there is a nice idenity involving this function,
If you wish I can post a prove of this (you need to be familiar with double integration).