Re: Integrate 'learns'?

• To: mathgroup at smc.vnet.net
• Subject: [mg106322] Re: [mg106269] Integrate 'learns'?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 8 Jan 2010 04:16:31 -0500 (EST)
• References: <201001070728.CAA23688@smc.vnet.net>

```Tony Harker wrote:
>   If I open a clean notebook in Version 7.0 for Microsoft Windows (32-bit)
> 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?[...]

That's exactly right. Sometimes it also uses the "remove two wrong

> [...] Can anyone enlighten me?
>   Tony Harker

Okay, actually Integrate and functions it uses (e.g. Simplify and Limit)
can make use of partial results cached along the way, in beating out
time constraints in later invocations. This is sort of hinted at but
perhaps not stated outright, in the materials at the URL below.

http://library.wolfram.com/infocenter/Conferences/5832/

The upshot is that results can change with further calls. I believe this
has come up on this group in past, but my searching capabilities have
not sufficed to locate relevant prior threads.

I am not seeing the behavior you describe on my machine, by the way.
Versions 7.0.0, 7.0.1, and development all give me the second result and
no message, each time I run the example. I may try it on my machine at
home to see if the platform change makes a difference.

Daniel Lichtblau
Wolfram Research

```

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