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MathGroup Archive 2000

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problem with Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23061] problem with Integrate
  • From: Jack Michel CORNIL <jmcornil at club-internet.fr>
  • Date: Sat, 15 Apr 2000 03:00:21 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

I am reading a paper about classical algorithmic methods 
  for integrating rational function and I am very 
  surprised that Mathematica is unable to give a "good" 
  expression of the following integral 

p = 7*x^13 + 10*x^8 + 4*x^7 - 7*x^6 - 4*x^3 - 4*x^2 + 3*x + 3;

q = x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 - x^2 + 2*x + 1;

Integrate[p/q, x]

1/2*RootSum[#1^14 - 2*#1^8 - 2*#1^7 - 2*#1^4 - 4*#1^3 - 
     #1^2 + 2*#1 + 1 & , 
   (7*log*(x - #1)*#1^13 + 10*log*(x - #1)*#1^8 + 
      4*log*(x - #1)*#1^7 - 7*log*(x - #1)*#1^6 - 
      4*log*(x - #1)*#1^3 - 4*log*(x - #1)*#1^2 + 
      3*log*(x - #1)*#1 + 3*log*(x - #1))/
     (7*#1^13 - 8*#1^7 - 7*#1^6 - 4*#1^3 - 6*#1^2 - #1 + 
      1) & ]

then the Rothstein/Trager method with a calculus 
of resultant gives quickly a good expression with 
two logarithmic functions.

Do I understand that for rational functions Mathematica 
uses only the elementary methods teached to students 
in an introductory course about integration ?

Or is there some option for the function Integrate 
asking Mathematica using other algorithms ?


By the way, I found a bug (I suppose) when using twice 
the function Integrate as follow


y = (-7*Sqrt[2]*x^6 + 7*x^6 + 2*Sqrt[2]*x - 4*x + 2*Sqrt[2] - 3)/
      (2*(x^7 + Sqrt[2]*x^2 + Sqrt[2]*x - x - 1)); 

Integrate[y, x]

-(1/2)*RootSum[#1^7 + Sqrt[2]*#1^2 + Sqrt[2]*#1 - #1 - 
     1 & , (7*Sqrt[2]*log*(x - #1)*#1^6 - 
      7*log*(x - #1)*#1^6 - 2*Sqrt[2]*log*(x - #1)*#1 + 
      4*log*(x - #1)*#1 - 2*Sqrt[2]*log*(x - #1) + 
      3*log*(x - #1))/(7*#1^6 + 2*Sqrt[2]*#1 + Sqrt[2] - 
      1) & ]

Integrate[y, x]

-(1/2)*RootSum[#1^7 + Sqrt[2]*#1^2 + Sqrt[2]*#1 - #1 - 
     1 & , (-7*log*(x - #1)*#1^6 + 7*log*(x - #1)*
       Integrate`A*1[6]*#1^6 + 4*log*(x - #1)*#1 - 
      2*log*(x - #1)*Integrate`A*1[6]*#1 + 
      3*log*(x - #1) - 2*log*(x - #1)*Integrate`A*1[6])/
     (7*#1^6 + 2*Integrate`A*1[6]*#1 + Integrate`A*1[6] - 
      1) & ]

I don't understand the appearing of Integrate`A*1[6] 
in the second expression . 
Such a thing was not existing in Release 3




Jack-Michel CORNIL             VERSAILLES - FRANCE


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