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==Constant undecidability==
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:::''Also, the Risch algorithm is not an "algorithm" literally, because it needs as a part to check if some expression is equivalent to zero. And for a common meaning of what an "elementary function" is it's not known whether such an algorithm exists or not''
:: That sentence needs to be put into the intro if its correct, it needs to be sourced, and if its true we need to stop calling it an algorithm. [[User:Fresheneesz|Fresheneesz]] ([[User talk:Fresheneesz|talk]]) 07:40, 18 November 2008 (UTC)
Basically, these problems arise in the constant field. So you can have some function c*f(x), and c == 0 will be undecidable. Therefore, the Risch Algorithm requires that the constant field be computable. For example, for Q, the rational numbers it ''is'' decidable if a == 0, but this also the case, for example, in Q(a), the field or rational functions on a with rational coefficients, where a is some variable not dependent on x, the integration variable, or even Q(a1, a2, …, an), where the ai do not depend on each other or on x. The actual result of course does not apply to the function field, because abs() is non-elementary. Even if you get a function that is really 0, and try to Risch integrate it, you will get as your result a function that is really an element of the constant field. For example, the integral of sin(x)^2 + cos(x)^2 - 1 is (integrating term wise), x/2 - sin(x)*cos(x)/2 + sin(x)*cos(x)/2 + x/2 - x, which is of course 0. But the algorithm never needed to know or care that the integrand was really identically 0 to get to that. Also, you should know that basic things like the division algorithm, which are essential to algorithms like the Risch Algorithm or even Euclid's gcd algorithm, do not work correctly if they cannot determine zero-equivalence in their coefficients. --<
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: simplify(convert(int(x/sqrt(x^4+10*x^2-96*x-71),x),radical));
: The answer also contains "EllipticF" and "EllipticPi". So Maple also does not understand that antiderivative for x/sqrt(x^4+10*x^2-96*x-71) can be written using elementary
: Do you agree with my argumentation?
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:: What the original writer of the article was ''trying'' to say is probably correct. What the article is really saying (i.e. that no system can find an antiderivative---no mention of elementary functions) is incorrect (Mathematica does find an antiderivative, but not in terms of elementary functions). This article needs a big cleanup, but I dare not touch it because this is a very complicated mathematical topic that I know nothing about (I came here to learn a little about it :) ). If you are familiar with the topic, it would be great if you cleaned it up a bit (and made it more precise)! <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/83.166.219.47|83.166.219.47]] ([[User talk:83.166.219.47|talk]]) 15:41, 6 June 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:::Just FYI, this is how complex this is: https://math.stackexchange.com/questions/681893/how-to-integrate-int-fracx-sqrtx410x2-96x-71dx Stackexchange cannot be used on wikipedia though. [[Special:Contributions/109.252.90.67|109.252.90.67]] ([[User talk:109.252.90.67|talk]]) 03:54, 28 November 2021 (UTC)
::::This craziness can be finally solved by Mathematica 13: https://www.wolframcloud.com/obj/d9af14f6-3b98-43c4-b996-11dedc9d9f10 [[Special:Contributions/2A00:1370:812D:5272:8FD:D915:7268:189C|2A00:1370:812D:5272:8FD:D915:7268:189C]] ([[User talk:2A00:1370:812D:5272:8FD:D915:7268:189C|talk]]) 16:55, 11 December 2021 (UTC)
== Unsolved Integral ==
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--[[Special:Contributions/91.39.86.160|91.39.86.160]] ([[User talk:91.39.86.160|talk]]) 10:22, 3 October 2008 (UTC)
:I profiled the code and it seems that it is indeed the rich algorithm that computes this one (i.e., it is not some special case in the code). This is interesting, because SymPy cannot even do the other one on this page (maybe because it is not so easy in the elementary case and it doesn't know about the special functions that Maple and Mathematica produce). I will update the article. <
== Issues with section [[Risch algorithm#Description|Description]]? ==
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::::: But if you are looking for more examples of hard integrals, you might try some of the other ones from Bronstein's integration tutorial (see the references of this article, it's freely downloadable on his website).
::::: And by the way, I got the integral with x + 1 to work just fine.<
== Decidability unknown for elementary functions ==
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But isn’t <math>|x|=\sqrt{x^2}</math> elementary presentable? --[[User:Chricho|Chricho ∀]] ([[User talk:Chricho|talk]]) 23:07, 26 June 2012 (UTC)
:The problem is that there are 2 square roots to each number, and the choice between them cannot be controlled. So <math>\sqrt{x^2}</math> could also mean <math>x</math>, or <math>-x</math>--[[Special:Contributions/77.126.235.228|77.126.235.228]] ([[User talk:77.126.235.228|talk]]) 14:34, 13 May 2013 (UTC)
::I agree with Chriko. Nowadays the <math>\sqrt{}</math> symbol is most widely used to mean "the positive square root of". An example of this is the quadratic formula. If the symbol were used to signify both the positive and negative square root we wouldn't need the plus or minus sign before it. Also <math>\sqrt{x}</math> would not be a function if that symbol were used for both the positive and negative square root. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/98.232.93.206|98.232.93.206]] ([[User talk:98.232.93.206|talk]]) 05:22, 20 October 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
== A New Algorithm ==
It's possible we need a new algorithm. Since this topic about symbolic integration algorithms is connected to differential Galois theory, maybe we need one that uses Galois groups and it's generated permutations. Galois group would be relatively easy to implement, perhaps using set from C++ would be a good idea. We also need to generate a constructor which tests that E is an extension to F. For more information look at [[Galois group|this]] [[User:Gave232|Gave232]] ([[User talk:Gave232|talk]]) 22:49, 15 July 2013 (UTC)Gave232
== Inaccurate statement ==
"the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions."
Unless they are including complex logarithms, you also need to consider inverse tangents, which are the result of irreduceable quadratic factors in the denominator.
For example, the integral of (x^3+x-1)/(x^4+x^2) is ln|x|+1/x+arctanx+C
[[Special:Contributions/98.232.93.206|98.232.93.206]] ([[User talk:98.232.93.206|talk]]) 05:14, 20 October 2013 (UTC)
: Complex logarithms are included. They are also allowed in the definition of an [[elementary function]]. — [[User:Pt|Pt]] <sub>[[User_talk:Pt|(T)]]</sub> 13:20, 20 April 2015 (UTC)
== Missing Algorithm ==
An adequate description of the Risch Algorithm seems to be missing ... from the article about the Risch Algorithm. Bah, minor detail. [[Special:Contributions/88.130.29.105|88.130.29.105]] ([[User talk:88.130.29.105|talk]]) 07:27, 18 April 2015 (UTC)
:Can you use AGI level GPT 4 to create small outline of the algorithm? [[Special:Contributions/2A00:1370:8184:1CE9:15B1:85BD:FE6A:1AB6|2A00:1370:8184:1CE9:15B1:85BD:FE6A:1AB6]] ([[User talk:2A00:1370:8184:1CE9:15B1:85BD:FE6A:1AB6|talk]]) 20:48, 24 March 2023 (UTC)
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