Edit detail for ExampleIntegration2 revision 2 of 2

Here we show how FriCAS can help solving calculus problems. Task is: find integer n, such that x^n/(x^120 - 1) has simple integral.

Theory of integration of rational functions says that integral is sum of constant multiples of logarithms of factors of x^120 - 1. We write x^120 - 1 as product of four factor:

(1) -> )set output tex off
 

)set output algebra on

(x^30 + 1)*(x^30 - 1)*(x^30 + %i)*(x^30 - %i)

         120
   (1)  x    - 1

Parametric form of integral corresopnding to those factors is and its derivative are:

ii := a*log(x^30 - 1) + b*log(x^30 + 1) + c*log(x^30 + %i) + d*log(x^30 -%i)

               30               30                30                30
   (2)  b log(x   + 1) + c log(x   + %i) + d log(x   - %i) + a log(x   - 1)

f := D(ii, x)

   (3)
                                   119                                     89
       (30 d + 30 c + 30 b + 30 a)x    + (30 %i d - 30 %i c - 30 b + 30 a)x
     + 
                                   59                                       29
     (- 30 d - 30 c + 30 b + 30 a)x   + (- 30 %i d + 30 %i c - 30 b + 30 a)x
  /
      120
     x    - 1

We see that f is a linear combination of powers of x divided by x^120 - 1. The only power in rage is x^89. So we need to find a, b, c, d such that coefficients of other powers are zero. To do this we extract coefficients and solve relevant system of equation. As a first step is it convenient to convert f to a rational function with coefficient being complx polynomials, then the other steps are easy:

-- convert to rational function
rf := f::FRAC(POLY(COMPLEX(INT)))

   (4)
                                   119                                     89
       (30 d + 30 c + 30 b + 30 a)x    + (30 %i d - 30 %i c - 30 b + 30 a)x
     + 
                                   59                                       29
     (- 30 d - 30 c + 30 b + 30 a)x   + (- 30 %i d + 30 %i c - 30 b + 30 a)x
  /
      120
     x    - 1

-- extract numerator and coefficients of powers of x
nf := numer(rf)

   (5)
                                 119                                     89
     (30 d + 30 c + 30 b + 30 a)x    + (30 %i d - 30 %i c - 30 b + 30 a)x
   + 
                                   59                                       29
     (- 30 d - 30 c + 30 b + 30 a)x   + (- 30 %i d + 30 %i c - 30 b + 30 a)x

lc := coefficients(univariate(nf, x))

   (6)
   [30 d + 30 c + 30 b + 30 a, 30 %i d - 30 %i c - 30 b + 30 a,
    - 30 d - 30 c + 30 b + 30 a, - 30 %i d + 30 %i c - 30 b + 30 a]

-- solve systme of linear equations
sl := solve([lc(1), lc(2) - 1, lc(3), lc(4)], [a, b, c, d])

               1          1        %i         %i
   (7)  [[a = ---, b = - ---, c = ---, d = - ---]]
              120        120      120        120

Finally we check obtaine result:

ii1 := eval(ii, sl(1))

               30                30                 30              30
        - log(x   + 1) + %i log(x   + %i) - %i log(x   - %i) + log(x   - 1)
   (8)  -------------------------------------------------------------------
                                        120

D(ii1, x)

            89
           x
   (9)  --------
         120
        x    - 1

Problem solved.

Remark: How we knew that four factors will give solution? Trying shows that there are so solution with two factors. Similarly there are no solution with 3 factors. So, solution with four factors is the simplest one.