Re: Integrate 'learns'?

*To*: mathgroup at smc.vnet.net*Subject*: [mg106337] Re: Integrate 'learns'?*From*: Szabolcs Horvát <szhorvat at gmail.com>*Date*: Fri, 8 Jan 2010 04:19:21 -0500 (EST)*References*: <hi42e4$n46$1@smc.vnet.net>

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.