$\int \log(x+e^x) \mathbb{d}x$, or When every CAS fails

Question

No computer algebra system -- at least to my knowledge -- managed to either compute the integral $\int \log(x+e^x)\space \mathbb{d}x$, in terms of any known functions, or even just prove that it is not elementary.

Although it is pretty much obvious that no classical trick would help with getting the antiderivative, it is much less obvious -- to me at least --, as to why is this case so special that modern CASes either freeze (Wolfram), honestly report something like "Implementation incomplete (constant residues)" (Axiom), or a weird answer that is plain wrong if you try differentiating it (Mathcad).

In fact, there is a big family of functions that all crash the modern CASes, in form of $\int f(x+g(x)) \mathbb{d}x$, eg. $\sqrt{x+\cos(x)}$, $\sqrt[3]{x+\sin(x)}$. Mathcad even gives the wrong answer for the integrals of the latter two radicals.

So, my two questions are:

• Are any of those inetegrals expressable in terms of any known functions?
• Why do the modern computer algebra systems fail to prove these are not elementary/...okay, liouvillian?

If anyone reading this happen to know how Risch's algorithm works, I'd love to hear from him or her as to how Risch's algorithm is supposed to tackle this problem. I mean, proving the non-elementarity, what would be the right field extension tower for that, what would follow from it, and how would the algorithm show that this integral is not elementary/liouvillian/... -- or is it?

Answer 1

I was mistaken. Only algebraic examples like $\sqrt{x+e^x}$ cause Axiom to fail. As for the [ $\log(x+e^x)$ ] case, even sympy's risch_integrate proves the non-elementariness successfully. Wolfram still gets confused over this one though.

Axiom successfully proves non-elementariness of $\int\log(x+e^x)\mathbb{d}x$ with basic integrate command.

Sympy fails using general integrate function, but succeeds if explicitly asked to use risch_integrate. Right during the process of Hermite reduction, to be precise. Many thanks to Aaron Meurer (the author of risch_integrate) who explained what went on under the hood as he traced how the example was processed.

With the algebraic example, $\sqrt{x+e^x}$, Axiom honestly admits that "implementation incomplete: constant residues". Well... that algebraic case is the hardest one, is a known fact anyway.

Unfortunately, my math knowledge is nowhere near what is required to try and trace what happened with the algebraic example.