Nasser M. Abbasi  wrote:   
   > On 2/22/2018 7:57 AM, clicliclic@freenet.de wrote:   
   > >   
   > > "Nasser M. Abbasi" schrieb:   
   > >>   
   > >> How high a number "n" can your cas integrate this?   
   > >>   
   > >>          1/((1 + x^n)*(1 + x^2))   
   > >>   
   > >> Fricas 1.3.2 and Rubi 4.14.7 go up to n=18. Then after that, they   
   > >> return unevaluated.   
   > >>   
   > >> Mathematica gives an answer for higher than 18, but the result after   
   > >> 18 is in terms of Root objects. I tried up to n=100.   
   > >>   
   > >> Maple also can go higher than n=18, and it also gives results   
   > >> in terms of RootOf. Tried up to n=200. So to remove these Roots, one   
   > >> has to evaluate the answer numerically.  I assume this is why Rubi and   
   > >> Fricas stop at n=18.   
   > >>   
   > >> Any insight why 18 is the limit here? Is it due to some   
   > >> factorization done, which after n=18 produces polynomials that   
   > >> can't be solved exactly for higher order?   
   > >>   
   > >   
   >   
   > > Derive 6.10 evaluates the integrals for n=1 through n=6 quickly. For   
   > > the n=7 integrand, factors containing ATAN functions are produced,   
   > > which bog the computation down. I have not tried to find out if Derive   
   > > would eventually finish, churn on forever, or exhaust its memory in   
   > > this case.   
   > >   
   > > Martin.   
   > >   
   >   
   > Sorry, let me clarify things again. FriCAS hangs at n=29.   
   > I have it running for one hr now at n=29. Will leave it   
   > running and will check on it after I come back from school.   
   >   
   > Here is what I found:   
   > ======================   
   > Rubi:  Up to n=18 OK, then unevaluated after n=18.   
   > Maple: Up to n=18 OK, then uses ROOT objects for n>18. Tried to 200   
   > Mathematica: Up to n=18 OK, then uses ROOT objects for n>18. Tried to 100   
   > FriCAS: Up to n=28 OK, then "hangs" or still trying.....   
   >   
   > So the winner so far is FriCAS on this test, it does it   
   > for n=28, with no root objects. root objects can only be   
   > evaluated numerically to get rid of them.   
      
   To make things clear: FriCAS also uses root objects.  Basically   
   this integral is sum over roots of appropriate polynomials   
   and you need the roots to express it.  One can hide root   
   in RootSum, one can try to hide them inside atan (contains   
   hidden sqrt(-1)) or asin, but hidden or not roots are   
   neccessery and one can not do this integral without them.   
   You may overlook them as FriCAS uses notation like '%%F0'   
   to express them.   
      
   One can argue that the optimal result for such integrals   
   is root sum.  It will simplify when it should, that is   
   when you differentiate it, but will stay in symbolic   
   form when no simplification is possible (that is in   
   normal arithmetic).  And it is easy to understand   
   structure of integral when given as root sum.   
      
   Concerning magic 18: for n = 18 you needs roots of   
   polynomial of degree 8 which contains only even   
   powers, so you can use formulas for solution to quartic   
   to express roots.  When n = 17 we get irreducible polynomial   
   of degree 8 for which FriCAS has no explicit formula for   
   roots, so uses root objects.   
      
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