Editing Risch algorithm (section)

==Problem examples==
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by [[Henri Cohen (number theorist)|Henri Cohen]] in 1993<ref>{{Cite web |last=Cohen |first=Henri |date=December 21, 1993 |title=A Christmas present for your favorite CAS |url=https://groups.google.com/g/sci.math.symbolic/c/BPOIUsVMuY0/m/2moCKQY_cz4J }}</ref>) has an elementary antiderivative, as [[Wolfram Mathematica]] since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral):<ref>{{Cite web|title=Wolfram Cloud|url=https://www.wolframcloud.com/obj/d9af14f6-3b98-43c4-b996-11dedc9d9f10|access-date=December 11, 2021|website=Wolfram Cloud|language=en}}</ref><ref>This example was posted by Manuel Bronstein to the [[Usenet]] forum ''comp.soft-sys.math.maple'' on November 24, 2000.[https://groups.google.com/d/msg/comp.soft-sys.math.maple/5CcPIR9Ft-Y/xYfGiyJauuoJ]</ref>

: <math> f(x) = \frac{x}{\sqrt{x^4 + 10 x^2 - 96 x - 71}},</math>

namely:

: <math>\begin{align} F(x) = - \frac{1}{8}\ln &\,\Big( (x^6+15 x^4-80 x^3+27 x^2-528 x+781) \sqrt{ x^4+10 x^2-96 x-71} \Big. \\ & {} - \Big .(x^8 + 20 x^6 - 128 x^5 + 54 x^4 - 1408 x^3 + 3124 x^2 + 10001) \Big) + C. \end{align}</math>

But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions,<ref name=":0" /> as [[FriCAS]] also shows. Some [[computer algebra system]]s may here return an antiderivative in terms of ''non-elementary'' functions (i.e. [[elliptic integral]]s), which are outside the scope of the Risch algorithm. For example, Mathematica returns a result with the functions EllipticPi and EllipticF. Integrals in the form <math>\int \frac{x+A}{\sqrt{x^4+ax^3+bx^2+cx+d}}\, dx</math> were solved by [[Pafnuty Chebyshev|Chebyshev]] (and in what cases it is elementary),<ref>{{Cite book |last=Chebyshev |first=P. L. |url=http://archive.org/details/117744684_001 |title=Oeuvres de P.L. Tchebychef |date=1899–1907 |publisher=St. Petersbourg, Commissionaires de l'Academie imperiale des sciences |others=University of California Berkeley |pages=171–200 |language=French}}</ref> but the strict proof for it was ultimately done by [[Yegor Ivanovich Zolotarev|Zolotarev]].<ref name=":0">{{Cite journal|last=Zolotareff|first=G.|date=December 1, 1872|title=Sur la méthode d'intégration de M. Tchébychef|url=https://doi.org/10.1007/BF01442910|journal=Mathematische Annalen|language=fr|volume=5|issue=4|pages=560–580|doi=10.1007/BF01442910|s2cid=123629827 |issn=1432-1807|url-access=subscription}}</ref>

The following is a more complex example that involves both algebraic and [[transcendental function]]s:<ref>{{harvnb|Bronstein|1998}}.</ref>

: <math>f(x) = \frac{x^2+2x+1+ (3x+1)\sqrt{x+\ln x}}{x\,\sqrt{x+\ln x}\left(x+\sqrt{x+\ln x}\right)}.</math>

In fact, the antiderivative of this function has a fairly short form that can be found using substitution  <math>u = x + \sqrt{x + \ln x}</math> ([[SymPy]] can solve it while FriCAS fails with "implementation incomplete (constant residues)" error in Risch algorithm):

: <math>F(x) = 2 \left(\sqrt{x+\ln x} + \ln\left(x+\sqrt{x+\ln x}\right)\right) + C.</math>
Some Davenport "theorems"{{Definition needed|Davenport has not been mentioned to this point in the article, and his name only appears once later, and not in the context of theorems.|date=July 2022}} are still being clarified. For example in 2020 a counterexample to such a "theorem" was found, where it turns out that an elementary antiderivative exists after all.<ref>{{Cite journal |last1=Masser |first1=David |last2=Zannier |first2=Umberto |date=December 2020 |title=Torsion points, Pell's equation, and integration in elementary terms |url=https://www.intlpress.com/site/pub/pages/journals/items/acta/content/vols/0225/0002/a002/ |journal=Acta Mathematica |language=EN |volume=225 |issue=2 |pages=227–312 |doi=10.4310/ACTA.2020.v225.n2.a2 |s2cid=221405883 |issn=1871-2509|doi-access=free |hdl=11384/110046 |hdl-access=free }}</ref>