XPost: comp.soft-sys.math.maple   
      
   Axel Vogt schrieb:   
   >   
   > On 19.08.2015 20:19, Peter Luschny wrote:   
   > >   
   > >> I expect the remainder to be handled in the same manner. But I   
   > >> don't see why Derive should not fail to simplify similar   
   > >> expressions whose trigonometric arguments involve larger   
   > >> denominators, as the rule to handle SIN(3*pi/14) - SIN(pi/14) is   
   > >> not generic.   
   > >   
   > > I include some further examples (array of expressions).   
   > >   
   > > case 7:   
   > >   
   > > [...];   
   > >   
   > > case 9:   
   > >   
   > > [...];   
   > >   
   > > case 11:   
   > >   
   > > [...];   
   > >   
   > > Can Rubi handle them?   
   > >   
   > > And: what is the _general_ reduction strategy?   
   > >   
   >   
   > Maple does it, using convert(%, RootOf): simplify(%); gives the   
   > monomials x^k   
      
   Automatic simplification on Derive 6.10 (not Rubi!) reduces Peter's   
   vector expressions as follows:   
      
   [10/7*x^4*COS(1/7*pi)+2/7*COS(1/7*pi)+20/7*x^2*COS(1/7*pi)-20/7*~   
   x^3*COS(1/7*pi)-10/7*x*COS(1/7*pi)-20/7*COS(2/7*pi)*x^2-10/7*COS~   
   (2/7*pi)*x^4+10/7*x*COS(2/7*pi)-2/7*COS(2/7*pi)+20/7*COS(2/7*pi)~   
   *x^3+10/7*x^4*COS(3/7*pi)+2/7*COS(3/7*pi)+20/7*x^2*COS(3/7*pi)-2~   
   0/7*x^3*COS(3/7*pi)-10/7*x*COS(3/7*pi)-1/7+5/7*x-10/7*x^2+10/7*x~   
   ^3-5/7*x^4+x^5,-6/7*x+15/7*x^2-20/7*x^3+15/7*x^4-6/7*x^5+x^6+30/~   
   7*COS(2/7*pi)*x^4-12/7*COS(2/7*pi)*x^5-30/7*x^4*COS(3/7*pi)-30/7~   
   *x^4*COS(1/7*pi)-2/7*COS(3/7*pi)-2/7*COS(1/7*pi)-12/7*x*COS(2/7*~   
   pi)+1/7+2/7*COS(2/7*pi)-30/7*x^2*COS(3/7*pi)-30/7*x^2*COS(1/7*pi~   
   )+40/7*x^3*COS(1/7*pi)+40/7*x^3*COS(3/7*pi)+12/7*x*COS(3/7*pi)+1~   
   2/7*x*COS(1/7*pi)+12/7*x^5*COS(3/7*pi)+12/7*x^5*COS(1/7*pi)-40/7~   
   *COS(2/7*pi)*x^3+30/7*COS(2/7*pi)*x^2]   
      
   [x^5,x^6]   
      
   [x+(4/9*#i)*SIN(1/9*pi)+(4/9*#i)*SIN(2/9*pi)-2/9*COS(2/9*pi)-(4/~   
   9*#i)*SIN(4/9*pi)+2/9*COS(1/9*pi)-2/9*COS(4/9*pi),4/9*x*COS(1/9*~   
   pi)-2/9*COS(1/9*pi)-4/9*COS(2/9*pi)*x+2/9*COS(2/9*pi)-4/9*x*COS(~   
   4/9*pi)+2/9*COS(4/9*pi)+(8/9*#i)*x*SIN(1/9*pi)-(8/9*#i)*SIN(1/9*~   
   pi)+(8/9*#i)*x*SIN(2/9*pi)-(8/9*#i)*SIN(2/9*pi)+(8/9*#i)*SIN(4/9~   
   *pi)-(8/9*#i)*x*SIN(4/9*pi)+x^2,(4/3*#i)*SIN(2/9*pi)*x^2-(8/3*#i~   
   )*x*SIN(1/9*pi)-(8/3*#i)*x*SIN(2/9*pi)-(4/3*#i)*SIN(4/9*pi)+(8/3~   
   *#i)*x*SIN(4/9*pi)+(4/3*#i)*x^2*SIN(1/9*pi)-(4/3*#i)*x^2*SIN(4/9~   
   *pi)+(4/3*#i)*SIN(2/9*pi)+(4/3*#i)*SIN(1/9*pi)-2/3*x*COS(1/9*pi)~   
   +2/3*x^2*COS(1/9*pi)+2/3*COS(2/9*pi)*x-2/3*COS(2/9*pi)*x^2+2/3*x~   
   *COS(4/9*pi)-2/3*x^2*COS(4/9*pi)+x^3,(16/3*#i)*x*SIN(1/9*pi)+(16~   
   /9*#i)*x^3*SIN(1/9*pi)-(16/3*#i)*x*SIN(4/9*pi)-(16/3*#i)*x^2*SIN~   
   (1/9*pi)+(16/3*#i)*x^2*SIN(4/9*pi)-(16/9*#i)*x^3*SIN(4/9*pi)-(16~   
   /3*#i)*SIN(2/9*pi)*x^2+(16/9*#i)*SIN(2/9*pi)*x^3+(16/3*#i)*x*SIN~   
   (2/9*pi)+(16/9*#i)*SIN(4/9*pi)-(16/9*#i)*SIN(2/9*pi)-(16/9*#i)*S~   
   IN(1/9*pi)+8/9*x^3*COS(1/9*pi)-2/9*COS(1/9*pi)-4/3*x^2*COS(1/9*p~   
   i)+4/3*COS(2/9*pi)*x^2+2/9*COS(2/9*pi)-8/9*COS(2/9*pi)*x^3+4/3*x~   
   ^2*COS(4/9*pi)+2/9*COS(4/9*pi)-8/9*x^3*COS(4/9*pi)+x^4,x^5-10/9*~   
   COS(2/9*pi)*x^4+20/9*COS(2/9*pi)*x^3-10/9*x*COS(1/9*pi)+10/9*x*C~   
   OS(4/9*pi)+10/9*COS(2/9*pi)*x+20/9*x^3*COS(4/9*pi)-20/9*x^3*COS(~   
   1/9*pi)-2/9*COS(2/9*pi)+(20/9*#i)*SIN(2/9*pi)-(20/9*#i)*SIN(4/9*~   
   pi)+(20/9*#i)*SIN(1/9*pi)-(80/9*#i)*SIN(2/9*pi)*x^3+(20/9*#i)*SI~   
   N(2/9*pi)*x^4-(20/9*#i)*x^4*SIN(4/9*pi)+(20/9*#i)*x^4*SIN(1/9*pi~   
   )+(40/3*#i)*SIN(2/9*pi)*x^2-(40/3*#i)*x^2*SIN(4/9*pi)+(40/3*#i)*~   
   x^2*SIN(1/9*pi)+(80/9*#i)*x^3*SIN(4/9*pi)-(80/9*#i)*x^3*SIN(1/9*~   
   pi)-(80/9*#i)*x*SIN(2/9*pi)+(80/9*#i)*x*SIN(4/9*pi)-(80/9*#i)*x*~   
   SIN(1/9*pi)+10/9*x^4*COS(1/9*pi)-10/9*x^4*COS(4/9*pi)+2/9*COS(1/~   
   9*pi)-2/9*COS(4/9*pi)]   
      
   [x,x^2,x^3,x^4,x^5]   
      
   [-1/11+x+2/11*COS(5/11*pi)+2/11*COS(1/11*pi)+2/11*COS(3/11*pi)-2~   
   /11*COS(4/11*pi)-2/11*COS(2/11*pi),-2/11*COS(1/11*pi)+4/11*x*COS~   
   (1/11*pi)+2/11*COS(2/11*pi)-4/11*COS(2/11*pi)*x-2/11*COS(3/11*pi~   
   )+4/11*x*COS(3/11*pi)+2/11*COS(4/11*pi)-4/11*x*COS(4/11*pi)-2/11~   
   *COS(5/11*pi)+4/11*x*COS(5/11*pi)+1/11-2/11*x+x^2,2/11*COS(1/11*~   
   pi)+6/11*x^2*COS(1/11*pi)-6/11*x*COS(1/11*pi)-6/11*COS(2/11*pi)*~   
   x^2-2/11*COS(2/11*pi)+6/11*COS(2/11*pi)*x+2/11*COS(3/11*pi)+6/11~   
   *x^2*COS(3/11*pi)-6/11*x*COS(3/11*pi)-6/11*x^2*COS(4/11*pi)-2/11~   
   *COS(4/11*pi)+6/11*x*COS(4/11*pi)+2/11*COS(5/11*pi)+6/11*x^2*COS~   
   (5/11*pi)-6/11*x*COS(5/11*pi)-1/11+3/11*x-3/11*x^2+x^3,-4/11*x+6~   
   /11*x^2-4/11*x^3+x^4-8/11*COS(2/11*pi)*x^3+12/11*COS(2/11*pi)*x^~   
   2-8/11*COS(2/11*pi)*x-12/11*x^2*COS(5/11*pi)+12/11*x^2*COS(4/11*~   
   pi)-12/11*x^2*COS(1/11*pi)-12/11*x^2*COS(3/11*pi)-2/11*COS(5/11*~   
   pi)-2/11*COS(1/11*pi)-2/11*COS(3/11*pi)+2/11*COS(4/11*pi)+8/11*x~   
   ^3*COS(1/11*pi)+8/11*x^3*COS(3/11*pi)+8/11*x^3*COS(5/11*pi)-8/11~   
   *x^3*COS(4/11*pi)+1/11+2/11*COS(2/11*pi)-8/11*x*COS(4/11*pi)+8/1~   
   1*x*COS(3/11*pi)+8/11*x*COS(1/11*pi)+8/11*x*COS(5/11*pi),5/11*x-~   
   10/11*x^2+10/11*x^3-5/11*x^4+x^5+20/11*COS(2/11*pi)*x^3-20/11*CO~   
   S(2/11*pi)*x^2+10/11*COS(2/11*pi)*x-10/11*COS(2/11*pi)*x^4+20/11~   
   *x^2*COS(5/11*pi)-20/11*x^2*COS(4/11*pi)+20/11*x^2*COS(1/11*pi)+~   
   20/11*x^2*COS(3/11*pi)+2/11*COS(5/11*pi)+2/11*COS(1/11*pi)+2/11*~   
   COS(3/11*pi)-2/11*COS(4/11*pi)-20/11*x^3*COS(1/11*pi)-20/11*x^3*~   
   COS(3/11*pi)-20/11*x^3*COS(5/11*pi)+20/11*x^3*COS(4/11*pi)-2/11*~   
   COS(2/11*pi)-1/11+10/11*x^4*COS(5/11*pi)-10/11*x^4*COS(4/11*pi)+~   
   10/11*x^4*COS(3/11*pi)+10/11*x^4*COS(1/11*pi)+10/11*x*COS(4/11*p~   
   i)-10/11*x*COS(3/11*pi)-10/11*x*COS(1/11*pi)-10/11*x*COS(5/11*pi~   
   )]   
      
   [-2*COS(2*pi/11)/11+2*COS(pi/11)/11+2*SIN(5*pi/22)/11-2*SIN(3*pi~   
   /22)/11+2*SIN(pi/22)/11+x-1/11,(2/11-4*x/11)*COS(2*pi/11)+(4*x/1~   
   1-2/11)*COS(pi/11)+(4*x/11-2/11)*SIN(5*pi/22)+(2/11-4*x/11)*SIN(~   
   3*pi/22)+(4*x/11-2/11)*SIN(pi/22)+x^2-2*x/11+1/11,-(6*x^2/11-6*x~   
   /11+2/11)*COS(2*pi/11)+(6*x^2/11-6*x/11+2/11)*COS(pi/11)+(6*x^2/~   
   11-6*x/11+2/11)*SIN(5*pi/22)-(6*x^2/11-6*x/11+2/11)*SIN(3*pi/22)~   
   +(6*x^2/11-6*x/11+2/11)*SIN(pi/22)+x^3-3*x^2/11+3*x/11-1/11,-(8*~   
   x^3/11-12*x^2/11+8*x/11-2/11)*COS(2*pi/11)+(8*x^3/11-12*x^2/11+8~   
   *x/11-2/11)*COS(pi/11)+(8*x^3/11-12*x^2/11+8*x/11-2/11)*SIN(5*pi~   
   /22)-(8*x^3/11-12*x^2/11+8*x/11-2/11)*SIN(3*pi/22)+(8*x^3/11-12*~   
   x^2/11+8*x/11-2/11)*SIN(pi/22)+x^4-4*x^3/11+6*x^2/11-4*x/11+1/11~   
   ,-(10*x^4/11-20*x^3/11+20*x^2/11-10*x/11+2/11)*COS(2*pi/11)+(10*~   
   x^4/11-20*x^3/11+20*x^2/11-10*x/11+2/11)*COS(pi/11)+(10*x^4/11-2~   
   0*x^3/11+20*x^2/11-10*x/11+2/11)*SIN(5*pi/22)-(10*x^4/11-20*x^3/~   
   11+20*x^2/11-10*x/11+2/11)*SIN(3*pi/22)+(10*x^4/11-20*x^3/11+20*~   
   x^2/11-10*x/11+2/11)*SIN(pi/22)+x^5-5*x^4/11+10*x^3/11-10*x^2/11~   
   +5*x/11-1/11]   
      
   So, as anticipated, Derive's rule set cannot fully handle the case of   
   argument denominator 11. Here are the reduction steps for the middle   
   element with argument denominator 9:   
      
   (4/3*#i)*SIN(2/9*pi)*x^2-(8/3*#i)*x*SIN(1/9*pi)-(8/3*#i)*x*SIN(2~   
   /9*pi)-(4/3*#i)*SIN(4/9*pi)+(8/3*#i)*x*SIN(4/9*pi)+(4/3*#i)*x^2*~   
   SIN(1/9*pi)-(4/3*#i)*x^2*SIN(4/9*pi)+(4/3*#i)*SIN(2/9*pi)+(4/3*#~   
   i)*SIN(1/9*pi)-2/3*x*COS(1/9*pi)+2/3*x^2*COS(1/9*pi)+2/3*COS(2/9~   
   *pi)*x-2/3*COS(2/9*pi)*x^2+2/3*x*COS(4/9*pi)-2/3*x^2*COS(4/9*pi)~   
   +x^3   
      
   " SIN(n*pi) -> COS((1/2-n)*pi) "   
      
      
   [continued in next message]   
      
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