hmmm... this is interesting.I never had it in school that you cannot integrate the function exp(-x^2) analytically .And then I solved some similar integral, of  exp(-x^3/3)... and got an answer and wondered why everybody told me this cannot be solved. Nope. You actually can.  Here is the solution of  the integral of exp(-x^2) :https://www.wolframalpha.com/input/?i=integral+of+exp%28-t%5E2%29+from+-inf+to+xthe integral is 1/2*sqrt(pi)*(erf(x)+1)which you actually can get by yourself also with the definition of incomplete gamma functions and integration by parts. You just have to know the incomplete gamma functions. and substitute -x^2 by -t. If you do that you get and integral that is almost the very definition of the incomplete gamma function. https://en.wikipedia.org/wiki/Incomplete_gamma_functionIt seems you have to be precise what you mean with analytic. analytic is all trigonometric functions and exp plus some sqrt and all their possible combinations....Gamma functions and all kind of other integrals not included.So no need to integrate this one numerically if you have incgamma or erf at hand. And I guess there is a reason why you call it error function. And neither do you need tables anymoreThis is one example where analytic math has sort of stopped before the incomplete gamma funciton and other integral defined functions and a like have been invented. I guess this is actually a case where people would vigorously disagree just because they learned it otherwise without defining precisely what means actually analytic in this case.I always wondered why on earth people think it until I came across this video by a german prof actually shading a little light on it. It seems this is only due to the historic definition of analytichttps://www.youtube.com/watch?v=l6w868U8C-Mhttps://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) For an enghineer I would say this statement doesnt really make sense. Anyhow calculating gammas and gammainc functions, also hypergeometric functions is another story in itself, especially when multiplying special functions like exp and erf with each other to get them, that much I remember from calculating something also called F1.Anyway... incomplete gamma functions are already seriously cool. Since you can calculate them within standard math libraries and they are the general case of many other special funcitons like error function, exponential integral, I think even airy. For many solutions of variable coefficient second order odes it was useful in many cases. And wolfram alpha knows how to handle it pretty well ^ (try for example integral of exp(-1/3*x^3))####[]