Sam Blake  wrote:   
   > Greetings,   
   >   
   > I was hoping the Risch-Trager-Bronstein algorithm of FriCAS and/or AXIOM   
   would compute the following integral in a reasonable amount of time   
   >   
   > -> integrate((x+(4+x^2)^(1/2))^(-1/9)*(-2+2^(1/3)*(x+(4+x^2)^(   
   /2))^(2/3))^(1/3),x)   
   >   
   > Was it wishful thinking? My computer has been running FriCAS 1.3.6 for over   
   an hour. The answer to this integral does not require any logarithms, so it   
   should be computable with the algebraic extensions of Hermite reduction, right?   
      
   Yes, it should be done by Hermite reduction.  However, this is quite   
   large example.  First difficulty is that by FriCAS evaluation rules   
   one gets dependent roots.  After exliminating dependent roots by hand   
   we get extention of constants of degree 3 and at level of functions   
   root of order 3 containing root of order 9, so degree 27 on level   
   of functions and degree 81 together.  Cost of arithmetic grows   
   slightly faster than quadratically with degree (having things in   
   nested form increases time needed for operations).  Current FriCAS   
   algorithms form matrices of dimension proportional to degree   
   and FriCAS needs to compute Smith form of such a matrix (with   
   polynomial entries).   
      
   As an extra explanation, your input is "sparse", but FriCAS algorithm   
   are essentially "dense".  There are many examples in algebraic   
   computations that "sparse" input leads to "dense" solutions,   
   and trying only "sparse" candidates for solutions would not   
   solve problem.  So to have completeness FriCAS must use "dense"   
   methods.   
      
   Better implementation cond speed up calculations quite a lot   
   (maybe hundreds of times, meybe thousends of times).  But   
   as you increase degree you will quickly exceed capacity of   
   any implementation using currently known methods.   
      
   BTW: In 4GB of memory current FriCAS runs out of memory after   
   few hours.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)