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Re: Integrate 'learns'?


On 2010.01.07. 8:28, 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? Can anyone enlighten me?
>

I believe that this is because intermediate results are cached (see 
ClearSystemCache[]) and because some computations are time constrained 
(i.e. some transformations are tried only for a limited time).  Caching 
results speeds up computations, and thus may influence the result. 
Notice that the second evaluation is also faster.

I get the first result you cite only when the system is under load (and 
also got it when running the computation right after reboot).  I think 
people with fast computers won't get it at all.


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