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Indefinite integral is incorrect

asked 2018-10-26 14:08:37 +0200

proy87 gravatar image

updated 2023-01-09 23:59:48 +0200

tmonteil gravatar image

indefinite_integral(sqrt(1+cos(x)**2), x).full_simplify() gives 1/6*sin(x)^3, which is incorrect.

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Where did you get the indefinite_integral function ?

tmonteil gravatar imagetmonteil ( 2018-10-26 14:31:50 +0200 )edit

in 8.5.b0:

sage: integral(sqrt(1+cos(x)**2), x).full_simplify()
FrédéricC gravatar imageFrédéricC ( 2018-10-26 15:16:28 +0200 )edit

@tmonteli, I use sage 8.1

proy87 gravatar imageproy87 ( 2018-10-26 17:02:46 +0200 )edit

Using integral(sqrt(1+cos(x)**2), x).full_simplify() as suggested still results in an answer which is incorrect, doesn't it? This is a nonelementary integral. See here or here

dazedANDconfused gravatar imagedazedANDconfused ( 2018-10-26 18:41:45 +0200 )edit

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answered 2018-10-26 22:40:40 +0200

Emmanuel Charpentier gravatar image

updated 2018-10-27 19:01:01 +0200

slelievre gravatar image

This is a bug. Furthermore, it is not a maxima bug, as it is often the case. Here, Sage truly screws things up itself:

  • Maxima doesn't give a false answer:

    sage: maxima.integrate(sqrt(1+cos(x)^2),x).sage()
    integrate(sqrt(cos(x)^2 + 1), x)
  • When one tries to "ease" the problem,

    • maxima doesn't recognize the "obvious", but does not give a false answer:

      sage: maxima.integrate(sqrt(1-m*sin(x)^2),x).sage()
      integrate(sqrt(-m*sin(x)^2 + 1), x)
    • Sage does:

      sage: integrate(sqrt(1-m*sin(x)^2),x)
      1/4*m*x - 1/8*m*sin(2*x)

BTW: what is expected:

sage: elliptic_e(x,1/2).diff(x)
sqrt(-1/2*sin(x)^2 + 1)

One can easily check that sympy, giac and fricas all fail to integrate, but that none of them gives misleading "answers".

This one does not seem to be related to existing indefinite integral bugs, and is an original, genuine, Sage-specific one. Reported as Trac #26563.

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Asked: 2018-10-26 14:08:37 +0200

Seen: 388 times

Last updated: Oct 27 '18