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All of us mathematicians after some time (and trial-and-error, of course) we are able to guess with reasonable accuracy whether or not a given function is elementary integrable (test yourself: $$\int\frac1{x\sin\bigl(\frac1x\bigr)}\,dx\quad\style{font-family:inherit;}{\text{vs.}}\quad\int\frac1{x^2\sin\bigl(\frac1x\bigr)}\,dx\ ;$$

surely the readers can give a lot more challenging and interesting examples).

I would like to know what is the most comprehensive work (survey, book, whatever) dealing with the theory of integration in elementary terms. I know about the pioneering work of Liouville, as well as the classic paper by Rosenlicht, but what else? what about allowing certain "VIP" non-elementary functions (erf function, for example)?

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    $\begingroup$ Manuel Bronstein, Symbolic Integration, published by Springer in 2004, may be of some use. $\endgroup$ – Gerry Myerson Jan 27 '13 at 7:19
  • $\begingroup$ Some nice recent results are mentioned (with references) here. $\endgroup$ – Andrés E. Caicedo Jun 13 '16 at 15:32
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Here is the list of references I've seen on the topic of elementary integration. It is eclectic, and not intended to be complete, but contains most of what seem to be the relevant benchmarks, and several expository accounts in varying degrees of detail. Sadly, I am not familiar with Liouville's or Ostrowski's original papers. (Perhaps I'll use this as an excuse to track them down.)

  1. MR0223346 (36 #6394). Rosenlicht, Maxwell. Liouville's theorem on functions with elementary integrals. Pacific J. Math. 24 (1968), 153–161.

  2. MR0237477 (38 #5759). Risch, Robert H. The problem of integration in finite terms. Trans. Amer. Math. Soc. 139 (1969), 167–189.

  3. MR0269635 (42 #4530). Risch, Robert H. The solution of the problem of integration in finite terms. Bull. Amer. Math. Soc. 76, (1970), 605–608.

  4. MR0321914 (48 #279). Rosenlicht, Maxwell. Integration in finite terms. Amer. Math. Monthly 79 (1972), 963–972.

  5. MR0409427 (53 #13182). Risch, Robert H. Implicitly elementary integrals. Proc. Amer. Math. Soc. 57 (1), (1976), 1–7.

  6. MR0536040 (81b:12029). Risch, Robert H. Algebraic properties of the elementary functions of analysis. Amer. J. Math. 101 (4), (1979), 743–759.

  7. MR0815235 (87a:12009). Richtmyer, R. D. Integration in finite terms: a method for determining regular fields for the Risch algorithm. Lett. Math. Phys. 10 (2-3), (1985), 135–141.

  8. Matthew P Wiener. Elementary integration and $x^x$. Sci.Math post. February 21, 1995. (The pdf version was typed by Apollo Hogan).

  9. Manuel Bronstein. Symbolic Integration Tutorial. "Course notes of an ISSAC (International Symposium on Symbolic and Algebraic Computation) '98 tutorial."

  10. MR1960772 (2004c:12010). Van der Put, Marius; Singer, Michael F. Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328. Springer-Verlag, Berlin, 2003. xviii+438 pp. ISBN: 3-540-44228-6.

  11. MR2106657 (2005i:68092). Bronstein, Manuel. Symbolic integration. I. Transcendental functions. Second edition. With a foreword by B. F. Caviness. Algorithms and Computation in Mathematics, 1. Springer-Verlag, Berlin, 2005. xvi+325 pp. ISBN: 3-540-21493-3.

  12. Brian Conrad. Integration in elementary terms. Unpublished note. (2011?).

  13. Moshe Kamensky. Differential Galois theory. "An introduction to Galois theory of linear differential equations."

Bronstein's book in particular is highly recommended.

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FriCAS software:

http://fricas-wiki.math.uni.wroc.pl/FriCASIntegration

http://fricas-wiki.math.uni.wroc.pl/FriCASSpecialIntegration

Type there your FriCas code into the box at the bottom of the webpage:

$\text{\begin{axiom}}$
...
$\text{\end{axiom}}$

and press the button "Preview".

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