Relation of not solvable indefinite Integrals to the Galois theory?

Question

Is there any simple explanation around why the following indefinite integral has not any solution? Is it related to the Galois theory? If yes, How? $$ \int\frac{1}{1+e^{-x}} dx = Li_2(e^{-x})+x\log(e^x+1) + c $$

Answer 1

There are algebraic explanations for the nonexistence of elementary indefinite integrals. See How can you prove that a function has no closed form integral?

There is also a Galois theory of differential equations, which includes indefinite integration. It is called Differential Galois theory.

For a recent book, see Lectures on Differential Galois Theory by Magid.