How can I teach Maple to simplify these expressions?
I thought this would be peanuts for Maple
(especially as it is peanuts for the competitor).

[1], -(2/5)*cos((2/5)*Pi)+(2/5)*cos((1/5)*Pi)+x-1/5

[2], (2/5)*cos((2/5)*Pi)-(4/5)*cos((2/5)*Pi)*x-(2/5)*cos((1/5)*Pi)+(4/5)*cos((1/5)*Pi)*x-(2/5)*x+x^2+1/5

[3], -(6/5)*cos((2/5)*Pi)*x^2+(6/5)*cos((2/5)*Pi)*x-(2/5)*cos((2/5)*Pi)+(6/5)*cos((1/5)*Pi)*x^2-(6/5)*cos((1/5)*Pi)*x+(2/5)*cos((1/5)*Pi)+(3/5)*x-(3/5)*x^2+x^3-1/5

[4], (2/5)*cos((2/5)*Pi)-(8/5)*cos((2/5)*Pi)*x^3+(12/5)*cos((2/5)*Pi)*x^2-(8/5)*cos((2/5)*Pi)*x-(2/5)*cos((1/5)*Pi)+(8/5)*cos((1/5)*Pi)*x^3-(12/5)*cos((1/5)*Pi)*x^2+(8/5)*cos((1/5)*Pi)*x-(4/5)*x+x^4+1/5-(4/5)*x^3+(6/5)*x^2

[5], (x-1)*(1+2*cos((2/5)*Pi)*x^2-2*cos((2/5)*Pi)*x^3-2*cos((2/5)*Pi)*x-2*cos((1/5)*Pi)*x^2+2*cos((1/5)*Pi)*x^3+2*cos((1/5)*Pi)*x+2*x^2+x^4)

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How can I teach Maple to simplify these expressions?
I thought this would be peanuts for Maple
(especially as it is peanuts for the competitor).

Depends on what one wants do have ... If L denotes
the list of your equations then for example
convert(L, radical):
simplify(%);
2 3 4 4 3 2
[x, x , x , x , (x - 1) (x + x + x + x + 1)]

Neat! I tried simplify and simplify-trig without success.
Why do they not work?

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On Friday, August 14, 2015 at 4:02:34 PM UTC+2, Axel Vogt wrote:

On 14.08.2015 15:00, Peter Luschny wrote:

How can I teach Maple to simplify these expressions?
I thought this would be peanuts for Maple
(especially as it is peanuts for the competitor).

...

Depends on what one wants do have ... If L denotes
the list of your equations then for example

convert(L, radical):
simplify(%);

OK. So what about these?

[1] -1/7+x-(2/7)*cos((2/7)*Pi)+(2/7)*cos((3/7)*Pi)+(2/7)*cos((1/7)*Pi)

[2] (4/7)*x*cos((1/7)*Pi)-(2/7)*cos((1/7)*Pi)-(4/7)*x*cos((2/7)*Pi)+(2/7)*cos((2/7)*Pi)+(4/7)*x*cos((3/7)*Pi)-(2/7)*cos((3/7)*Pi)+1/7-(2/7)*x+x^2

[3] (2/7)*cos((1/7)*Pi)+(6/7)*x^2*cos((1/7)*Pi)-(6/7)*x*cos((1/7)*Pi)-(2/7)*cos((2/7)*Pi)-(6/7)*cos((2/7)*Pi)*x^2+(6/7)*x*cos((2/7)*Pi)+(2/7)*cos((3/7)*Pi)+(6/7)*x^2*cos((3/7)*Pi)-(6/7)*x*cos((3/7)*Pi)-1/7+(3/7)*x-(3/7)*x^2+x^3

[4] -(2/7)*cos((1/7)*Pi)-(12/7)*x^2*cos((1/7)*Pi)+(8/7)*x*cos((1/7)*Pi)+(8/7)*x^3*cos((1/7)*Pi)-(8/7)*cos((2/7)*Pi)*x^3+(2/7)*cos((2/7)*Pi)+(12/7)*cos((2/7)*Pi)*x^2-(8/7)*x*cos((2/7)*Pi)-(2/7)*cos((3/7)*Pi)-(12/7)*x^2*cos((3/7)*Pi)+(8/7)*x*cos((3/7)*Pi)+(
8/7)*x^3*cos((3/7)*Pi)+1/7-(4/7)*x+(6/7)*x^2-(4/7)*x^3+x^4

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convert(L, radical):
simplify(%);

OK. So what about these?

Be warned, if you can solve this case (n=7) I will send in
the next case (n=9) and Maple will never return form it.

So we see a pattern: n=3 works with simplify, n=5 only with
your special radical method, n=7 fails even with the radical
method and n=9 is ending in nirvana.

So there clearly is some insufficient implementation of these
simplifications.

Are these cases too complex? Or to specialized?

Unfortunately the answer is no. We are talking about
simplifications of sum of roots of unity (which your solution
shows). Something I expect a good CAS can handle.

Hi Maple developers, please take note. You mastered the
even cases, now go for the odd cases also.

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XPost: sci.math.symbolic

On 14.08.2015 18:45, Peter Luschny wrote:

On Friday, August 14, 2015 at 4:02:34 PM UTC+2, Axel Vogt wrote:

On 14.08.2015 15:00, Peter Luschny wrote:

How can I teach Maple to simplify these expressions?
I thought this would be peanuts for Maple
(especially as it is peanuts for the competitor).

...

Depends on what one wants do have ... If L denotes
the list of your equations then for example

convert(L, radical):
simplify(%);

OK. So what about these?

[1] -1/7+x-(2/7)*cos((2/7)*Pi)+(2/7)*cos((3/7)*Pi)+(2/7)*cos((1/7)*Pi)

[2] (4/7)*x*cos((1/7)*Pi)-(2/7)*cos((1/7)*Pi)-(4/7)*x*cos((2/7)*Pi)+(2/7)*cos((2/7)*Pi)+(4/7)*x*cos((3/7)*Pi)-(2/7)*cos((3/7)*Pi)+1/7-(2/7)*x+x^2

[3] (2/7)*cos((1/7)*Pi)+(6/7)*x^2*cos((1/7)*Pi)-(6/7)*x*cos((1/7)*Pi)-(2/7)*cos((2/7)*Pi)-(6/7)*cos((2/7)*Pi)*x^2+(6/7)*x*cos((2/7)*Pi)+(2/7)*cos((3/7)*Pi)+(6/7)*x^2*cos((3/7)*Pi)-(6/7)*x*cos((3/7)*Pi)-1/7+(3/7)*x-(3/7)*x^2+x^3

[4] -(2/7)*cos((1/7)*Pi)-(12/7)*x^2*cos((1/7)*Pi)+(8/7)*x*cos((1/7)*Pi)+(8/7)*x^3*cos((1/7)*Pi)-(8/7)*cos((2/7)*Pi)*x^3+(2/7)*cos((2/7)*Pi)+(12/7)*cos((2/7)*Pi)*x^2-(8/7)*x*cos((2/7)*Pi)-(2/7)*cos((3/7)*Pi)-(12/7)*x^2*cos((3/7)*Pi)+(8/7)*x*cos((3/7)*Pi)

+(8/7)*x^3*cos((3/7)*Pi)+1/7-(4/7)*x+(6/7)*x^2-(4/7)*x^3+x^4


evalf[20](L): fnormal(%): identify(%); # to have a guess

2 3 4
[x, x , x , x ]

convert(L, RootOf): # nun aber in echt ...
simplify(%);
2 3 4
[x, x , x , x ]

I think it is also "what is intended by simplify (and should trig
survive)?" Thus I included sci.math.symbolic for further answers.

PS: would you mind to post as list

PPS: well, it may break down at some degree

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Am Freitag, 14. August 2015 23:45:53 UTC+2 schrieb Axel Vogt:

evalf[20](L): fnormal(%): identify(%); # to have a guess

2 3 4
[x, x , x , x ]

convert(L, RootOf): # nun aber in echt ...
simplify(%);
2 3 4
[x, x , x , x ]

I think it is also "what is intended by simplify (and should trig
survive)?"

Thank you Axel! "what is intended by simplify?" Exactly what you showed.

PS: would you mind to post as list

I think we need no further examples. As I said in my last post
we are talking about sums of roots of unity (and their reciprocals)
and I strongly believe these formulas should be handled by Maple
whiteout further ingenuity on the side of the user.

Otherwise I can save my money and use Wolfram Alpha ...

http://www.wolframalpha.com/input/?i=%284%2F7%29*x*cos%28%281%2F7%29*Pi%29-%282%2F7%29*cos%28%281%2F7%29*Pi%29-%284%2F7%29*x*cos%28%282%2F7%29*Pi%29%2B%282%2F7%29*cos%28%282%2F7%29*Pi%29%2B%284%2F7%29*x*cos%28%283%2F7%29*Pi%29-%282%2F7%29*cos%28%283%2F7%
29*Pi%29%2B1%2F7-%282%2F7%29*x%2Bx%5E2+

See 'Alternate forms'.

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Hi Martin!

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:

[(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)-(20/7)*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)+(12/7)*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];

case 9:

[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)*sin((1/9)*Pi)+(8/9)*x^3*cos((1/9)*Pi)-(2/9)*cos((1/9)*Pi)-(4/3)*x^2*cos((1/9)*Pi)+(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*cos((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)*sin((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)];

case 11:

[-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)+(8/11)*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)*cos((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)*Pi)-(10/11)*x*cos((3/11)*Pi)-(10/11)*x*cos((1/11)*Pi)-(10/11)*x*cos((5/11)*Pi)];

Can Rubi handle them?

And: what is the _general_ reduction strategy?

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XPost: sci.math.symbolic

Axel Vogt schrieb:

On 14.08.2015 18:45, Peter Luschny wrote:

OK. So what about these?

[1] -1/7+x-(2/7)*cos((2/7)*Pi)+(2/7)*cos((3/7)*Pi)+(2/7)*cos((1/7)*Pi)

[2] (4/7)*x*cos((1/7)*Pi)-(2/7)*cos((1/7)*Pi)-(4/7)*x*cos((2/7)*Pi)+(2/7)*cos((2/7)*Pi)+(4/7)*x*cos((3/7)*Pi)-(2/7)*cos((3/7)*Pi)+1/7-(2/7)*x+x^2

[3] (2/7)*cos((1/7)*Pi)+(6/7)*x^2*cos((1/7)*Pi)-(6/7)*x*cos((1/7)*Pi)-(2/7)*cos((2/7)*Pi)-(6/7)*cos((2/7)*Pi)*x^2+(6/7)*x*cos((2/7)*Pi)+(2/7)*cos((3/7)*Pi)+(6/7)*x^2*cos((3/7)*Pi)-(6/7)*x*cos((3/7)*Pi)-1/7+(3/7)*x-(3/7)*x^2+x^3

[4] -(2/7)*cos((1/7)*Pi)-(12/7)*x^2*cos((1/7)*Pi)+(8/7)*x*cos((1/7)*Pi)+(8/7)*x^3*cos((1/7)*Pi)-(8/7)*cos((2/7)*Pi)*x^3+(2/7)*cos((2/7)*Pi)+(12/7)*cos((2/7)*Pi)*x^2-(8/7)*x*cos((2/7)*Pi)-(2/7)*cos((3/7)*Pi)-(12/7)*x^2*cos((3/7)*Pi)+(8/7)*x*cos((3/7)*

Pi)+(8/7)*x^3*cos((3/7)*Pi)+1/7-(4/7)*x+(6/7)*x^2-(4/7)*x^3+x^4

evalf[20](L): fnormal(%): identify(%); # to have a guess

2 3 4
[x, x , x , x ]

convert(L, RootOf): # nun aber in echt ...
simplify(%);
2 3 4
[x, x , x , x ]

I think it is also "what is intended by simplify (and should trig
survive)?" Thus I included sci.math.symbolic for further answers.

PS: would you mind to post as list

PPS: well, it may break down at some degree

Derive 6.10 doesn't need any teaching: your quadruple expression

[-1/7+x-2/7*COS(2/7*pi)+2/7*COS(3/7*pi)+2/7*COS(1/7*pi),4/7*x*CO~ S(1/7*pi)-2/7*COS(1/7*pi)-4/7*x*COS(2/7*pi)+2/7*COS(2/7*pi)+4/7*~ x*COS(3/7*pi)-2/7*COS(3/7*pi)+1/7-2/7*x+x^2,2/7*COS(1/7*pi)+6/7*~ x^2*COS(1/7*pi)-6/7*x*COS(1/7*pi)-2/7*COS(2/7*pi)-6/7*COS(2/7*pi~ )*x^2+6/7*x*COS(2/7*pi)+2/7*COS(3/7*pi)+6/7*x^2*COS(3/7*pi)-6/7*~ x*COS(3/7*pi)-1/7+3/7*x-3/7*x^2+x^3,-2/7*COS(1/7*pi)-12/7*x^2*CO~ S(1/7*pi)+8/7*x*COS(1/7*pi)+8/7*x^3*COS(1/7*pi)-8/7*COS(2/7*pi)*~ x^3+2/7*COS(2/7*pi)+12/7*COS(2/7*pi)*x^2-8/7*x*COS(2/7*pi)-2/7*C~ OS(3/7*pi)-12/7*x^2*COS(3/7*pi)+8/7*x*COS(3/7*pi)+8/7*x^3*COS(3/~ 7*pi)+1/7-4/7*x+6/7*x^2-4/7*x^3+x^4]

is automatically simplified to

[x,x^2,x^3,x^4]

within a fraction of a second. These are the reduction steps for the
first expression:

-1/7+x-2/7*COS(2/7*pi)+2/7*COS(3/7*pi)+2/7*COS(1/7*pi)

" COS(n*pi) -> SIN((1/2-n)*pi) "

-1/7+x-2*SIN(3*pi/14)/7+2*COS(3*pi/7)/7+2*COS(pi/7)/7

" COS(n*pi) -> SIN((1/2-n)*pi) "

-1/7+x-2*SIN(3*pi/14)/7+2*SIN(pi/14)/7+2*COS(pi/7)/7

" SIN(3*pi/14)-SIN(pi/14) -> COS(pi/7)-1/2 "

-1/7+x-2*(COS(pi/7)-1/2)/7+2*COS(pi/7)/7

" one final step "

x

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.

Martin.

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On 14.08.2015 15:00, Peter Luschny wrote:

How can I teach Maple to simplify these expressions?
I thought this would be peanuts for Maple
(especially as it is peanuts for the competitor).

...

Depends on what one wants do have ... If L denotes
the list of your equations then for example

convert(L, radical):
simplify(%);

2 3 4 4 3 2
[x, x , x , x , (x - 1) (x + x + x + x + 1)]

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Maple does it, using convert(%, RootOf): simplify(%); gives the monomials x^k

Thank you dear Axel.

The method "convert(%, RootOf)" seems to be the
right one since, as I said above, we are talking
about simplifications of roots of unity.

However what sense makes a method which even a Maple
expert finds only after several days of search and
which takes "the world's most powerful math engine"
for a simple case such as for n=7 for the list of the
first twelve expressions 457.5 seconds to find while a program
that high-school kids in the last millennium used for
their homework solves in a fraction of a second?

Do not misunderstand me: I appreciate your investigations
because unfortunately I do not possess Derive 6.10. Still
I insist that Maple needs to be improved in this matter.

--- SoupGate-Win32 v1.05
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XPost: sci.math.symbolic

On 19.08.2015 20:19, Peter Luschny wrote:

Hi Martin!

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:

[(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)-(20/7)*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)+(12/7)*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];

case 9:

[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)*sin((1/9)*Pi)+(8/9)*x^3*cos((1/9)*Pi)-(2/9)*cos((1/9)*Pi)-(4/3)*x^2*cos((1/9)*Pi)+(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*cos((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)*sin((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)];

case 11:

[-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)+(8/11)*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)*cos((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)*Pi)-(10/11)*x*cos((3/11)*Pi)-(10/11)*x*cos((1/11)*Pi)-(10/11)*x*cos((5/11)*Pi)];

Can Rubi handle them?

And: what is the _general_ reduction strategy?

Maple does it, using convert(%, RootOf): simplify(%); gives the monomials x^k

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

XPost: sci.math.symbolic

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) "

8*#i*x*SIN(4*pi/9)/3+4*#i*x^2*SIN(pi/9)/3-4*#i*x^2*SIN(4*pi/9)/3~ +4*#i*SIN(2*pi/9)/3+4*#i*SIN(pi/9)/3-2*x*COS(pi/9)/3+2*x^2*COS(p~ i/9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2~ *x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*p~ i/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(n*pi) -> COS((1/2-n)*pi) "

8*#i*x*COS(pi/18)/3+4*#i*x^2*SIN(pi/9)/3-4*#i*x^2*SIN(4*pi/9)/3+~ 4*#i*SIN(2*pi/9)/3+4*#i*SIN(pi/9)/3-2*x*COS(pi/9)/3+2*x^2*COS(pi~ /9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2*~ x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi~ /9)-8*x*SIN(pi/9)/3)+x^3

" SIN(n*pi) -> COS((1/2-n)*pi) "

-4*#i*x^2*COS(pi/18)/3+#i*(8*x*COS(pi/18)/3+4*x^2*SIN(pi/9)/3)+4~ *#i*SIN(2*pi/9)/3+4*#i*SIN(pi/9)/3-2*x*COS(pi/9)/3+2*x^2*COS(pi/~ 9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2*x~ ^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/~ 9)-8*x*SIN(pi/9)/3)+x^3

" SIN(z)+SIN(w) -> 2*SIN(z/2+w/2)*COS(z/2-w/2) "

#i*(4*x^2*SIN(pi/9)/3+(8*x/3-4*x^2/3)*COS(pi/18)+8*SIN(pi/6)*COS~ (pi/18)/3)-2*x*COS(pi/9)/3+2*x^2*COS(pi/9)/3+2*x*COS(2*pi/9)/3-2~ *x^2*COS(2*pi/9)/3+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*#i*CO~ S(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(pi/6) -> 1/2 "

#i*(4*x^2*SIN(pi/9)/3+(8*x/3-4*x^2/3)*COS(pi/18)+4*COS(pi/18)/3)~ -2*x*COS(pi/9)/3+2*x^2*COS(pi/9)/3+2*x*COS(2*pi/9)/3-2*x^2*COS(2~ *pi/9)/3+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3~ +#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" COS(z)-COS(w) -> -2*SIN(z/2-w/2)*SIN(z/2+w/2) "

2*(-2*x^2/3+2*x/3)*SIN(pi/18)*COS(2*pi/3)+#i*(4*x^2*SIN(pi/9)/3-~ (4*x^2/3-8*x/3-4/3)*COS(pi/18))+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi~ /9)/3-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(~
pi/9)/3)+x^3

" COS(n*pi) -> SIN((1/2-n)*pi) "

-4*x*(1-x)*SIN(pi/6)*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-(4*x^2/3~ -8*x/3-4/3)*COS(pi/18))+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*~ #i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)~
+x^3

" SIN(pi/6) -> 1/2 "

2*x*(x-1)*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-(4*x^2/3-8*x/3-4/3)~ *COS(pi/18))+2*x*COS(4*pi/9)/3-2*x^2*COS(4*pi/9)/3-4*#i*COS(pi/1~ 8)/3+#i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" COS(n*pi) -> SIN((1/2-n)*pi) "

2*x*(x-1)*SIN(pi/18)/3+2*x*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-4*~ (x^2-2*x-1)*COS(pi/18)/3)-2*x^2*COS(4*pi/9)/3-4*#i*COS(pi/18)/3+~ #i*((4*x^2/3-8*x/3)*SIN(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" COS(n*pi) -> SIN((1/2-n)*pi) "

2*x^2*SIN(pi/18)/3-2*x^2*SIN(pi/18)/3+#i*(4*x^2*SIN(pi/9)/3-4*(x~ ^2-2*x-1)*COS(pi/18)/3)-4*#i*COS(pi/18)/3+#i*((4*x^2/3-8*x/3)*SI~ N(2*pi/9)-8*x*SIN(pi/9)/3)+x^3

" SIN(z)+SIN(w) -> 2*SIN(z/2+w/2)*COS(z/2-w/2) "

#i*(2*(4*x^2/3-8*x/3)*SIN(pi/6)*COS(pi/18)-4*x*(x-2)*COS(pi/18)/~
3)+x^3

" SIN(pi/6) -> 1/2 "

#i*(4*x*(x-2)*COS(pi/18)/3+4*x*(2-x)*COS(pi/18)/3)+x^3

" one final step "

x^3

Examples of Derive's rule SIN(3*pi/14) - SIN(pi/14) -> COS(pi/7) - 1/2
extended to higher denominators are SIN(5*pi/22) - SIN(3*pi/22) +
SIN(pi/22) -> COS(2*pi/11) - COS(pi/11) + 1/2 and SIN(5*pi/26) -
SIN(3*pi/26) + SIN(pi/26) -> COS(3*pi/13) - COS(2*pi/13) + COS(pi/13) -
1/2. Generalization to arbitrary denominators seems straightforward.
Numerical approximation to high precision could be an acceptable
alternative in many situations.

Martin.

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