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ExampleIntegration

Edit detail for ExampleIntegration revision 6 of 6

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Editor: page Time: 2022/06/23 17:06:57 GMT+0
Note:

changed:
-definingPolynomial(%%F0)
definingPolynomial(%%E0)

changed:
-eval(i0,%%F0=rhs s(1));
eval(i0,%%E0=rhs s(1));

changed:
-eval(i0,%%F0=rhs s(2));
eval(i0,%%E0=rhs s(2));

changed:
-eval(i0,%%F0 = rhs s(3));
eval(i0,%%E0 = rhs s(3));

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)

$\label{eq7}\frac{{8 \ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}\ {\arctan \left({\frac{{\left({9 \ {\sqrt{23}}\ \%\%E 0}+{\sqrt{23}}\right)}\ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}}{{{10}\ {\sqrt{-{{368}\ {{\%\%E 0}^{2}}}-{{23}\ \%\%E 0}+{81}}}}-{{207}\ {{\%\%E 0}^{2}}}+{{23}\ \%\%E 0}+{9 4}}}\right)}}-{8 \ {\sqrt{{{69}\ {{\%\%E 0}^{2}}}+ 4}}\ {\arctan \left({\frac{{\left({9 \ {\sqrt{23}}\ \%\%E 0}+{\sqrt{23}}\right)}\ {\sqrt{{{6 9}\ {{\%\%E 0}^{2}}}+ 4}}}{{{10}\ {\sqrt{-{{161}\ {{\%\%E 0}^{2}}}-{{46}\ \%\%E 0}+{12}}}}-{{207}\ {{\%\%E 0}^{2}}}+{{23}\ \%\%E 0}+{44}}}\right)}}+{4 \ {\sqrt{23}}\ \%\%E 0 \ {\log \left({{{2 07}\ {{\%\%E 0}^{2}}}-{{23}\ \%\%E 0}+{56}}\right)}}-{4 \ {\sqrt{2 3}}\ \%\%E 0 \ {\log \left({{{207}\ {{\%\%E 0}^{2}}}-{{23}\ \%\%E 0}+{31}}\right)}}+{{\sqrt{23}}\ \%\%E 0 \ {\log \left({-{{2875}\ {{\%\%E 0}^{2}}}-{{2875}\ \%\%E 0}-{500}}\right)}}-{{\sqrt{23}}\ \%\%E 0 \ {\log \left({-{{64975}\ {{\%\%E 0}^{2}}}-{{10350}\ \%\%E 0}+{5825}}\right)}}}{4 \ {\sqrt{23}}}$