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Integrate_use_rootsof Calculator

Integrate_use_rootsof

-- Option variable: integrate_use_rootsof Default value: false

When integrate_use_rootsof is true and the denominator of a rational function cannot be factored, integrate returns the integral in a form which is a sum over the roots (not yet known) of the denominator.

integrate_use_rootsof: false;
 integrate (1/(1+x+x^5), x);
 integrate_use_rootsof: true;
 integrate (1/(1+x+x^5), x);

For example, with integrate_use_rootsof set to false, integrate returns an unsolved integral of a rational function in noun form:

          (%i1) integrate_use_rootsof: false$
          (%i2) integrate (1/(1+x+x^5), x);
                  /  2
                  [ x  - 4 x + 5
                  I ------------ dx                            2 x + 1
                  ]  3    2                2            5 atan(-------)
                  / x  - x  + 1       log(x  + x + 1)          sqrt(3)
          (%o2)   ----------------- - --------------- + ---------------
                          7                 14             7 sqrt(3)

Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function:

          (%i3) integrate_use_rootsof: true$
          (%i4) integrate (1/(1+x+x^5), x);
                ====        2
                \       (%r4  - 4 %r4 + 5) log(x - %r4)
                 >      -------------------------------
                /                    2
                ====            3 %r4  - 2 %r4
                                3    2
                %r4 in rootsof(x  - x  + 1)
          (%o4) ----------------------------------------------------------
                         7

2 x + 1 2 5 atan(-------) log(x + x + 1) sqrt(3) - --------------- + --------------- 14 7 sqrt(3)

Alternatively the user may compute the roots of the denominator separately, and then express the integrand in terms of these roots, e.g., 1/((x - a)*(x - b)*(x - c)) or 1/((x^2 - (a+b)*x + a*b)*(x - c)) if the denominator is a cubic polynomial. Sometimes this will help Maxima obtain a more useful result.

(%o1)                                true
(%i2) 

Integrate_use_rootsof Example

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