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? $$ \int\frac{1}{1+e^{-x}} dx = Li_2(e^{-x})+x\log(e^x+1) + c $$
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1It should be: $$ \int\frac{1}{1+e^{-x}} dx = Ln(e^x+1) + c $$ – Mariusz Iwaniuk Jul 26 '16 at 11:51
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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.
