Integral or sum of $a^x(1-a^x)(1-a^x)$

Question

I'm intested in learning how to get a sum in difference calculus. I'm willing to settle, at least for now, in learning how to integrate the following integral in calculus:

$\int(a^x(1-a^x)(1-a^x))dx$

The sum that I'm really interested in learning how to get, hopefully using difference calculus, is: $\sum_x{a^x(1-a^x)(1-a^x)}$

Even better still, extend this technique to include more $(1-a^x)$, and add limits of integration or summation.

_EDIT_

I'd like to do so without multiplying everything out, if possible

Answer 1

Well, integration is easy: just multiply everything out to get a sum where each term has the form $a^{kx} = e^{(k \ln a)x}$, then integrate term by term.