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Re: Integrate 'learns'?

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  • Subject: [mg106331] Re: Integrate 'learns'?
  • From: Bill Rowe <readnews at>
  • Date: Fri, 8 Jan 2010 04:18:13 -0500 (EST)

On 1/7/10 at 2:28 AM, a.harker at (Tony Harker) wrote:

>If I open a clean notebook in Version 7.0 for Microsoft Windows
>and enter Integrate[x/(3 Sin[x]),{x,\[Pi]/4,\[Pi]/2}] the
>result (after a
>warning message) is (8*Catalan - I*Pi^2 + Pi*(-Log[1 - (-1)^(1/4)] +
>Log[1 + (-1)^(1/4)]) -
>(4*I)*(PolyLog[2, -(-1)^(1/4)] - PolyLog[2, (-1)^(1/4)]))/12
>and if I then repeat the command I get no error and (5*Catalan)/6 -
>((23*I)/288)*Pi^2 + (Pi*ArcTanh[(-1)^(1/4)])/6 +
>((2*I)/3)*PolyLog[2, (-1)^(1/4)]
>which seems to be Mathematica's final answer.

>I am happy that the results are equivalent, but puzzled about what has
>been saved, and where, to generate this difference. Did Mathematica ask
>the audience or phone a friend? Can anyone enlighten me?


In[2]:= $Version

Out[2]= 7.0 for Mac OS X x86 (64-bit) (February 19, 2009)

with a fresh session I get the second result with no warning message.

I believe Integrate like several other built-in functions caches
intermediate computations. The second time the same code is
executed, this cache of intermediate results are available. As a
consequence, the time to get the second result will be shorter.
And in this case have a different form.

You didn't state what the warning message said. My guess is that
if you were to change some of the options for Integrate, you
would get the second result with no warning message the first
time you executed the code. But since I don't get the warning
message on my system, I've no way to determine if this is really
the case or what options should be changed.

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