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ExampleGroebner ←	↑	→ ExpressionInteger

ExampleIntegration

Edit detail for ExampleIntegration revision 5 of 6

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Editor: hemmecke Time: 2015/03/10 22:08:29 GMT+0
Note:

added:

ks := kernels i
[definingPolynomial(k::Expression(Integer)) for k in ks | name operator k = 'rootOf]


added:
\end{axiom}


\begin{axiom}

fricas

f := x/(x^3-x+1)

$\label{eq1}x \over{{{x}^{3}}- x + 1}$

(1)

Type: Fraction(Polynomial(Integer))

fricas

)set output tex off

fricas

)set output algebra on

i := integrate(f, x)

   (2)
           +-----------+
           |      2
         4\|69%%F0  + 4
      *
         atan
                                   +-----------+
                 +--+        +--+  |      2
              (9\|23 %%F0 + \|23 )\|69%%F0  + 4
           /
                   +---------------------------------------------------+
                   |                 2                       2
                10\|(- 207x + 46)%%F0  + (23x - 69)%%F0 + 25x  - 6x - 7
              + 
                         2
                - 207%%F0  + 23%%F0 + 50x - 6
     + 
         +--+                2
       2\|23 %%F0 log(207%%F0  - 23%%F0 + 25x + 6)
     + 
          +--+                          2                       2
       - \|23 %%F0 log((- 207x + 46)%%F0  + (23x - 69)%%F0 + 25x  - 6x - 7)
  /
       +--+
     2\|23

Type: Union(Expression(Integer),...)

fricas

)set output tex on

fricas

)set output algebra off

D(i, x)

$\label{eq2}x \over{{{x}^{3}}- x + 1}$

(2)

Type: Expression(Integer)

fricas

ks := kernels i

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\arctan{\left({{{\left({9 \ {\sqrt{23}}\ \%\%F 0}+{\sqrt{2 3}}\right)}\ {\sqrt{{{69}\ {{\%\%F 0}^{2}}}+ 4}}}\over{{{10}\ {\sqrt{{{\left(-{{207}\ x}+{46}\right)}\ {{\%\%F 0}^{2}}}+{{\left({{23}\ x}-{69}\right)}\ \%\%F 0}+{{25}\ {{x}^{2}}}-{6 \ x}- 7}}}-{{20 7}\ {{\%\%F 0}^{2}}}+{{23}\ \%\%F 0}+{{50}\ x}- 6}}\right)}}, \right. \ \ \displaystyle \left.\:{\sqrt{{{69}\ {{\%\%F 0}^{2}}}+ 4}}, \:{\log \left({{{2 07}\ {{\%\%F 0}^{2}}}-{{23}\ \%\%F 0}+{{25}\ x}+ 6}\right)}, \: \right. \ \ \displaystyle \left.{\log{\left({{{\left(-{{207}\ x}+{46}\right)}\ {{\%\%F 0}^{2}}}+{{\left({{23}\ x}-{69}\right)}\ \%\%F 0}+{{25}\ {{x}^{2}}}-{6 \ x}- 7}\right)}}, \right. \ \ \displaystyle \left.\: \%\%F 0, \:{\sqrt{23}}\right]$

(3)

Type: List(Kernel(Expression(Integer)))

fricas

[definingPolynomial(k::Expression(Integer)) for k in ks | name operator k = 'rootOf]

$\label{eq4}\left[{{{{23}\ {{\%\%F 0}^{3}}}+ \%\%F 0 + 1}\over{23}}\right]$

(4)

Type: List(Expression(Integer))

fricas

definingPolynomial(%%F0)

$\label{eq5}{{{23}\ {{\%\%F 0}^{3}}}+ \%\%F 0 + 1}\over{23}$

(5)

Type: Expression(Integer)

fricas

s:=radicalSolve(%::Fraction Polynomial Integer)

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{\%\%F 0 ={{{{\left(-{{69}\ {\sqrt{- 3}}}+{69}\right)}\ {{\root{3}\of{{-{3 \ {\sqrt{69}}}+{25}}\over{{138}\ {\sqrt{69}}}}}^{2}}}+ 2}\over{{\left({{6 9}\ {\sqrt{- 3}}}+{69}\right)}\ {\root{3}\of{{-{3 \ {\sqrt{69}}}+{25}}\over{{138}\ {\sqrt{69}}}}}}}}, \: \right. \ \ \displaystyle \left.{\%\%F 0 ={{{{\left(-{{69}\ {\sqrt{- 3}}}-{69}\right)}\ {{\root{3}\of{{-{3 \ {\sqrt{69}}}+{25}}\over{{138}\ {\sqrt{69}}}}}^{2}}}- 2}\over{{\left({{6 9}\ {\sqrt{- 3}}}-{69}\right)}\ {\root{3}\of{{-{3 \ {\sqrt{69}}}+{25}}\over{{138}\ {\sqrt{69}}}}}}}}, \: \right. \ \ \displaystyle \left.{\%\%F 0 ={{{{69}\ {{\root{3}\of{{-{3 \ {\sqrt{69}}}+{2 5}}\over{{138}\ {\sqrt{69}}}}}^{2}}}- 1}\over{{69}\ {\root{3}\of{{-{3 \ {\sqrt{69}}}+{25}}\over{{138}\ {\sqrt{69}}}}}}}}\right]$

(6)

Type: List(Equation(Expression(Integer)))

fricas

i0 := integrate(f, x=1..2)

$\label{eq7}{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle 8 \ {\sqrt{{{69}\ {{\%\%F 0}^{2}}}+ 4}}\ \cdot \ \ \displaystyle {\arctan{\left({{{\left({9 \ {\sqrt{23}}\ \%\%F 0}+{\sqrt{23}}\right)}\ {\sqrt{{{6 9}\ {{\%\%F 0}^{2}}}+ 4}}}\over{{{10}\ {\sqrt{-{{368}\ {{\%\%F 0}^{2}}}-{{23}\ \%\%F 0}+{81}}}}-{{207}\ {{\%\%F 0}^{2}}}+{{2 3}\ \%\%F 0}+{94}}}\right)}}$