From: hebisch@antispam.uni.wroc.pl   
      
   clicliclic@freenet.de wrote:   
   >   
   > Waldek Hebisch schrieb:   
   > >   
   > > clicliclic@freenet.de wrote:   
   > > >   
   > > > Hello!   
   > > >   
   > > > How long does it take the popular integrators to evaluate the following   
   > > > elementary indefinite integrals? Imagine them to appear in terms of some   
   > > > series expansion:   
   > > >   
   > > > INT(1/((1+x+2*x^2)^5*(3-2*x)^(11/2)),x)   
   > > >   
   > > > INT(1/((1+x+2*x^2)^10*(3-2*x)^(21/2)),x)   
   > > >   
   > > > INT(1/((1+x+2*x^2)^20*(3-2*x)^(41/2)),x)   
   > > >   
   > > > and:   
   > > >   
   > > > INT(1/((1+x+2*x^2)^5*(3-2*x+x^2)^(11/2)),x)   
   > > >   
   > > > INT(1/((1+x+2*x^2)^10*(3-2*x+x^2)^(21/2)),x)   
   > > >   
   > > > INT(1/((1+x+2*x^2)^20*(3-2*x+x^2)^(41/2)),x)   
   > > >   
   > > > Does Maple beat FriCAS? Does Rubi beat Mathematica?   
   > > >   
   > > > Martin.   
   > >   
   > > For FriCAS results for 'integrate' are somewhat embarrassing:   
   > > 0.20s, 0.49s, 21.85s, 0.53s, 1.89s and the sixth one would   
   > > take more than few hours.  Core integrator is reasonably fast,   
   > > calling 'lfintegrate' I get: 0.02s, 0.04s, 0.11s, 0.07s,   
   > > 0.20s, 1.02s.  'integrate' tries to make result nicer and   
   > > in the process attempts to factor largish integers which   
   > > takes a lot of time...   
   > >   
   > > Remark: 'lfintegrate' gives correct result only if certain   
   > > assumptions are satified.  'integrate' transforms integrals   
   > > so that 'lfintegrate' can handle them, calls 'lfintegrate'   
   > > and then tries to make the result nicer.  Your integrals   
   > > are simple and well-behaved, so can be sent directly to   
   > > 'lfintegrate'.   
   > >   
   >   
   > If 'integrate' on the last one could be accelerated to about 60 seconds,   
   > there would be no reason for embarassment anymore ...   
      
   Embarassment is due to bad scaling. 'lfintegrate' works reasonably   
   up to power 320 in linear case, that is   
      
   INT(1/((1+x+2*x^2)^320*(3-2*x)^(641/2)),x)   
      
   (which needs 397.71s) and 160 in quadratic case, that is   
      
   INT(1/((1+x+2*x^2)^160*(3-2*x+x^2)^(321/2)),x)   
      
   (which needs 594.07s).  But integer factoring is not going to   
   scale.   
      
   And concerning sixth example: 'integrate' did not finish after   
   few hours.  I am confident that integer factoring routine   
   will finish in finite time so I wrote "more than few hours"   
   but given exponenetial complexity it hard to guess how   
   much time it would need.   
      
   And when it comes to speeding up 'integrate': instead of   
   factoring integers FriCAS should use different method.   
   But it is tricky to eliminate factoring without causing   
   worse results in entirely different calculations.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)