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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)}}
(7)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
fricas
eval(i0,%%F0=rhs s(1));
Type: Expression(Integer)
fricas
complexNumeric %

\label{eq8}{0.5657999164 \<u> 5642798548}+{{0.8 E - 19}\  i}(8)
Type: Complex(Float)
fricas
eval(i0,%%F0=rhs s(2));
Type: Expression(Integer)
fricas
complexNumeric %

\label{eq9}{0.5657999164 \<u> 5642798548}-{{0.8 E - 19}\  i}(9)
Type: Complex(Float)
fricas
eval(i0,%%F0 = rhs s(3));
Type: Expression(Integer)
fricas
complexNumeric %

\label{eq10}0.5657999164 \<u> 5642798584(10)
Type: Complex(Float)




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