Re: Integrate 'learns'?

*To*: mathgroup at smc.vnet.net*Subject*: [mg106324] Re: [mg106269] Integrate 'learns'?*From*: "Tony Harker" <a.harker at ucl.ac.uk>*Date*: Fri, 8 Jan 2010 04:16:54 -0500 (EST)*References*: <201001070728.CAA23688@smc.vnet.net> <4B4637A8.6030505@wolfram.com>

Many thanks, Daniel and Bob. I, too, thought this must have come up before but failed to locate relevant threads. What you say about not seeing my first-described behaviour is consistent with what Szabolcs Horv=E1t told me: "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." Sounds as if it's time for me to upgrade my hardware! Tony ]-> -----Original Message----- ]-> From: Daniel Lichtblau [mailto:danl at wolfram.com] ]-> Sent: 07 January 2010 19:36 ]-> To: Tony Harker ]-> Cc: mathgroup at smc.vnet.net ]-> Subject: Re: [mg106269] Integrate 'learns'? ]-> ]-> 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 answers" option. ]-> ]-> ]-> > [...] 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 ]-> ]->

**References**:**Integrate 'learns'?***From:*"Tony Harker" <a.harker@ucl.ac.uk>