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

\label{eq1}\frac{x}{{{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 %%E0 + 4 * atan +------------+ +--+ +--+ | 2 (9 \|23 %%E0 + \|23 )\|69 %%E0 + 4 / +-------------------------------------------------------+ | 2 2 10 \|(- 207 x + 46)%%E0 + (23 x - 69)%%E0 + 25 x - 6 x - 7 + 2 - 207 %%E0 + 23 %%E0 + 50 x - 6 + +--+ 2 2 \|23 %%E0 log(207 %%E0 - 23 %%E0 + 25 x + 6) + +--+ 2 2 - \|23 %%E0 log((- 207 x + 46)%%E0 + (23 x - 69)%%E0 + 25 x - 6 x - 7) / +--+ 2 \|23
Type: Union(Expression(Integer),...)
fricas
)set output tex on
 
fricas
)set output algebra off
D(i, x)

\label{eq2}\frac{x}{{{x}^{3}}- x + 1}(2)
Type: Expression(Integer)
fricas
ks := kernels i

\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[{\arctan{\left({\frac{{\left({9 \ {\sqrt{23}}\  \%\%E 0}+{\sqrt{23}}\right)}\ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}}{{{1
0}\ {\sqrt{{{\left(-{{207}\  x}+{46}\right)}\ {{\%\%E 0}^{2}}}+{{\left({{23}\  x}-{69}\right)}\  \%\%E 0}+{{25}\ {{x}^{2}}}-{6 \  x}- 7}}}-{{207}\ {{\%\%E 0}^{2}}}+{{23}\  \%\%E 0}+{{50}\  x}- 6}}\right)}}, \right.
\
\
\displaystyle
\left.\:{\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}, \:{\log \left({{{2
07}\ {{\%\%E 0}^{2}}}-{{23}\  \%\%E 0}+{{25}\  x}+ 6}\right)}, \: \right.
\
\
\displaystyle
\left.{\log{\left({{{\left(-{{207}\  x}+{46}\right)}\ {{\%\%E 0}^{2}}}+{{\left({{23}\  x}-{69}\right)}\  \%\%E 0}+{{25}\ {{x}^{2}}}-{6 \  x}- 7}\right)}}, \right.
\
\
\displaystyle
\left.\: \%\%E 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[{\frac{{{23}\ {{\%\%E 0}^{3}}}+ \%\%E 0 + 1}{23}}\right](4)
Type: List(Expression(Integer))
fricas
definingPolynomial(%%E0)

\label{eq5}\frac{{{23}\ {{\%\%E 0}^{3}}}+ \%\%E 0 + 1}{23}(5)
Type: Expression(Integer)
fricas
s:=radicalSolve(%::Fraction Polynomial Integer)

\label{eq6}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\%\%E 0 = 
\
\
\displaystyle
{\frac{{{\left(-{{69}\ {\sqrt{- 3}}}+{69}\right)}\ {{\root{3}\of{\frac{-{3 \ {\sqrt{69}}}+{25}}{{138}\ {\sqrt{69}}}}}^{2}}}+ 2}{{\left({{6
9}\ {\sqrt{- 3}}}+{69}\right)}\ {\root{3}\of{\frac{-{3 \ {\sqrt{6
9}}}+{25}}{{138}\ {\sqrt{69}}}}}}}
(6)
Type: List(Equation(Expression(Integer)))

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

\label{eq7}\frac{{8 \ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}\ {\arctan \left({\frac{{\left({9 \ {\sqrt{23}}\  \%\%E 0}+{\sqrt{23}}\right)}\ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}}{{{10}\ {\sqrt{-{{368}\ {{\%\%E 0}^{2}}}-{{23}\  \%\%E 0}+{81}}}}-{{207}\ {{\%\%E 0}^{2}}}+{{23}\  \%\%E 0}+{9
4}}}\right)}}-{8 \ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}\ {\arctan \left({\frac{{\left({9 \ {\sqrt{23}}\  \%\%E 0}+{\sqrt{23}}\right)}\ {\sqrt{{{6
9}\ {{\%\%E 0}^{2}}}+ 4}}}{{{10}\ {\sqrt{-{{161}\ {{\%\%E 0}^{2}}}-{{46}\  \%\%E 0}+{12}}}}-{{207}\ {{\%\%E 0}^{2}}}+{{23}\  \%\%E 0}+{44}}}\right)}}+{4 \ {\sqrt{23}}\  \%\%E 0 \ {\log \left({{{2
07}\ {{\%\%E 0}^{2}}}-{{23}\  \%\%E 0}+{56}}\right)}}-{4 \ {\sqrt{2
3}}\  \%\%E 0 \ {\log \left({{{207}\ {{\%\%E 0}^{2}}}-{{23}\  \%\%E 0}+{31}}\right)}}+{{\sqrt{23}}\  \%\%E 0 \ {\log \left({-{{2875}\ {{\%\%E 0}^{2}}}-{{2875}\  \%\%E 0}-{500}}\right)}}-{{\sqrt{23}}\  \%\%E 0 \ {\log \left({-{{64975}\ {{\%\%E 0}^{2}}}-{{10350}\  \%\%E 0}+{5825}}\right)}}}{4 \ {\sqrt{23}}}(7)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
fricas
eval(i0,%%E0=rhs s(1));
Type: Expression(Integer)
fricas
complexNumeric %

\label{eq8}{0.5657999164 \<u> 5642798549}+{{0.8 E - 19}\  i}(8)
Type: Complex(Float)
fricas
eval(i0,%%E0=rhs s(2));
Type: Expression(Integer)
fricas
complexNumeric %

\label{eq9}{0.5657999164 \<u> 5642798549}-{{0.8 E - 19}\  i}(9)
Type: Complex(Float)
fricas
eval(i0,%%E0 = rhs s(3));
Type: Expression(Integer)
fricas
complexNumeric %

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




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