MathAction ExampleIntegration

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fricas

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

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

(1)

Type: Fraction(Polynomial(Integer))

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)set output tex off

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)set output algebra on

i := integrate(f, x)

   (2)
             +--------------+
             |         2
             |- 69 %%E0  - 4
         (-  |--------------  - %%E0)
            \|      23
      *
                             +--------------+
                             |         2
                             |- 69 %%E0  - 4            2
         log((207 %%E0 + 23) |--------------  - 207 %%E0  + 23 %%E0 + 50 x - 6)
                            \|      23
     + 
           +--------------+
           |         2
           |- 69 %%E0  - 4
         ( |--------------  - %%E0)
          \|      23
      *
                             +--------------+
                             |         2
                             |- 69 %%E0  - 4            2
       log((- 207 %%E0 - 23) |--------------  - 207 %%E0  + 23 %%E0 + 50 x - 6)
                            \|      23
     + 
                          2
       2 %%E0 log(207 %%E0  - 23 %%E0 + 25 x + 6)
  /
     2

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

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)set output tex on

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)set output algebra off

D(i, x)

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

(2)

Type: Expression(Integer)

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ks := kernels i

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\log{\left({{{\left({{207}\ \%\%E 0}+{23}\right)}\ {\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}}-{{207}\ {{\%\%E 0}^{2}}}+{{2 3}\ \%\%E 0}+{{50}\ x}- 6}\right)}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{\log{\left({{{\left(-{{207}\ \%\%E 0}-{23}\right)}\ {\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}}-{{207}\ {{\%\%E 0}^{2}}}+{{2 3}\ \%\%E 0}+{{50}\ x}- 6}\right)}}, \right. \ \ \displaystyle \left.\:{\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}, \: \right. \ \ \displaystyle \left.{\log \left({{{207}\ {{\%\%E 0}^{2}}}-{{23}\ \%\%E 0}+{{25}\ x}+ 6}\right)}, \: \%\%E 0 \right]$

(3)

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

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[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))

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definingPolynomial(%%E0)

$\label{eq5}\frac{{{23}\ {{\%\%E 0}^{3}}}+ \%\%E 0 + 1}{23}$

(5)

Type: Expression(Integer)

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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)))

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

$\label{eq7}\frac{{{\left(-{\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}- \%\%E 0 \right)}\ {\log \left({{{\left({{43700}\ \%\%E 0}+{8 050}\right)}\ {\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}}-{{43700}\ {{\%\%E 0}^{2}}}+{{8050}\ \%\%E 0}+{10400}}\right)}}+{{\left({\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}+ \%\%E 0 \right)}\ {\log \left({{{\left({{23000}\ \%\%E 0}+{5750}\right)}\ {\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}}-{{23000}\ {{\%\%E 0}^{2}}}+{{5750}\ \%\%E 0}+{3500}}\right)}}+{{\left(-{\sqrt{\frac{-{{6 9}\ {{\%\%E 0}^{2}}}- 4}{23}}}+ \%\%E 0 \right)}\ {\log \left({{{\left(-{{23000}\ \%\%E 0}-{5750}\right)}\ {\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}}-{{23000}\ {{\%\%E 0}^{2}}}+{{5750}\ \%\%E 0}+{3500}}\right)}}+{{\left({\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}- \%\%E 0 \right)}\ {\log \left({{{\left(-{{43700}\ \%\%E 0}-{8050}\right)}\ {\sqrt{\frac{-{{69}\ {{\%\%E 0}^{2}}}- 4}{23}}}}-{{43700}\ {{\%\%E 0}^{2}}}+{{8050}\ \%\%E 0}+{104 00}}\right)}}+{2 \ \%\%E 0 \ {\log \left({{{21850}\ {{\%\%E 0}^{2}}}-{{4025}\ \%\%E 0}+{3550}}\right)}}-{2 \ \%\%E 0 \ {\log \left({{{11500}\ {{\%\%E 0}^{2}}}-{{2875}\ \%\%E 0}+{1375}}\right)}}}{4}$