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Submitted by : (unknown) at: 2007-11-17T22:26:52-08:00 (9 years ago)
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Axiom Version :
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axiom
)set output tex off
 
axiom
)set output algebra on

axiom
integrate(1/(x^7+x^6+x^5+x^4+x^3+x^2+x+1),x)
(1) +-------------------------------------+ | 2 2 +-+ +-+ (- 2\|- 24%%F1 - 16%%F0 %%F1 - 24%%F0 - 1 - 4\|2 %%F1 - 4\|2 %%F0) * log +-+ +-+ +-+ +-+ ((128\|2 %%F0 + 8\|2 )%%F1 + 8\|2 %%F0 - \|2 ) * +-------------------------------------+ | 2 2 \|- 24%%F1 - 16%%F0 %%F1 - 24%%F0 - 1 + 2 2 2 (- 512%%F0 - 32)%%F1 + (- 512%%F0 - 4)%%F1 - 32%%F0 - 4%%F0 + 3x + 1 + +-------------------------------------+ | 2 2 +-+ +-+ (2\|- 24%%F1 - 16%%F0 %%F1 - 24%%F0 - 1 - 4\|2 %%F1 - 4\|2 %%F0) * log +-+ +-+ +-+ +-+ ((- 128\|2 %%F0 - 8\|2 )%%F1 - 8\|2 %%F0 + \|2 ) * +-------------------------------------+ | 2 2 \|- 24%%F1 - 16%%F0 %%F1 - 24%%F0 - 1 + 2 2 2 (- 512%%F0 - 32)%%F1 + (- 512%%F0 - 4)%%F1 - 32%%F0 - 4%%F0 + 3x + 1 + +-+ 8\|2 %%F1 * log 2 2 3 (1024%%F0 + 64)%%F1 + (1024%%F0 + 8)%%F1 + 1024%%F0 + 32%%F0 + 3x + 3 + +-+ 3 2 +-+ 2 8\|2 %%F0 log(- 1024%%F0 + 64%%F0 - 24%%F0 + 3x - 5) - \|2 log(x + 1) + +-+ +-+ 2\|2 log(x + 1) + 2\|2 atan(x) / +-+ 8\|2
Type: Union(Expression(Integer),...)

Apparently these are "rootOf" expressions:

axiom
%%F0::InputForm
(2) (rootOf (/ (+ (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0)) 1) 2048) %%F0)
Type: InputForm
axiom
%%F1::InputForm
(3) (rootOf
(/
(+
(+
(+
(* 128
(^
(rootOf
(/
(+ (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0)) 1)
2048)
%%F0)
3) )
(* (* 128 %%F1)
(^
(rootOf
(/
(+ (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0)) 1)
2048)
%%F0)
2) ) )
(* (+ (* 128 (^ %%F1 2)) 4)
(rootOf
(/ (+ (+ (+ (* 2048 (^ %%F0 4)) (* 64 (^ %%F0 2))) (* 16 %%F0)) 1) 2048)
%%F0) ) )
(+ (+ (* 128 (^ %%F1 3)) (* 4 %%F1)) 1))
128)
%%F1)
Type: InputForm

We can in fact solve for some of these.

axiom
)set output tex on
 
axiom
)set output algebra off

axiom
definingPolynomial %%F0

\label{eq1}{{{2048}\ {\%\%F 0^4}}+{{64}\ {\%\%F 0^2}}+{{16}\  \%\%F 0}+ 1}\over{2048}(1)
Type: Expression(Integer)
axiom
F0:=radicalSolve(%::UP('%%F0,COMPLEX FRAC INT))

\label{eq2}\begin{array}{@{}l}
\displaystyle
\left[{\%\%F 0 ={-{{1 \over 2}\ {\sqrt{-{{1 \over{16}}\  i}}}}-{{1 \over 8}\  i}}}, \:{\%\%F 0 ={{{1 \over 2}\ {\sqrt{-{{1 \over{1
6}}\  i}}}}-{{1 \over 8}\  i}}}, \: \right.
\
\
\displaystyle
\left.{\%\%F 0 ={-{{1 \over 2}\ {\sqrt{{1 \over{16}}\  i}}}+{{1 \over 8}\  i}}}, \:{\%\%F 0 ={{{1 \over 2}\ {\sqrt{{1 \over{1
6}}\  i}}}+{{1 \over 8}\  i}}}\right] 
(2)
Type: List(Equation(Expression(Complex(Fraction(Integer)))))

axiom
definingPolynomial %%F1

\label{eq3}{\left(
\begin{array}{@{}l}
\displaystyle
{{128}\ {\%\%F 0^3}}+{{128}\  \%\%F 1 \ {\%\%F 0^2}}+ 
\
\
\displaystyle
{{\left({{128}\ {\%\%F 1^2}}+ 4 \right)}\  \%\%F 0}+{{128}\ {\%\%F 1^3}}+{4 \  \%\%F 1}+ 1 
(3)
Type: Expression(Integer)
axiom
F11:=subst(%,`%%F0=rhs(F0.1))::UP('%%F1,?)

\label{eq4}\begin{array}{@{}l}
\displaystyle
{\%\%F 1^3}+{{\left(-{{1 \over 2}\ {\sqrt{-{{1 \over{16}}\  i}}}}-{{1 \over 8}\  i}\right)}\ {\%\%F 1^2}}+ 
\
\
\displaystyle
{{\left({{1 \over 8}\  i \ {\sqrt{-{{1 \over{16}}\  i}}}}+{1 \over{64}}-{{1 \over{64}}\  i}\right)}\  \%\%F 1}+ 
\
\
\displaystyle
{{\left({1 \over{128}}+{{1 \over{128}}\  i}\right)}\ {\sqrt{-{{1 \over{16}}\  i}}}}+{1 \over{512}}-{{1 \over{512}}\  i}
(4)
Type: UnivariatePolynomial(%%F1,Expression(Complex(Fraction(Integer))))

Why doesn't Axiom automatically provide this sort of information?

Perhaps related to #262

Severity: normal => wishlist




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