<p>I'm glad I got over having fun with these functions that when integrated, can't be expressed as finite sums of elementary functions. If you're curious about these types of functions, Mathematica and Maple should support them. I did once find a nested series for the integral of x^x dx, but it only converged from 0 to 1.</p>
<p>My question for Kirbus is: did you need this for a class or for a physical application or something... or just curious?</p>