These are notes which may be helpful for implementors who wish to implement the algorithms in Symbolic Integration I: Transcendental Functions by Manuel Bronstein. These have been noted during the implementation of various algorithms in Symtegration.
Polynomial pseudo-division
In section 1.2, page 9, the algorithm for polynomial pseudo-division does not apply when the degree of \(A\) is already smaller than \(B\).
Greatest common divisor
Section 1.3 gives algorithms which derives the greatest common divisor for two polynomials. It was noted in theorem 1.1.1 in section 1.1 that greatest common divisors for polynomials are unique up to multiplication by units. For polynomials where the coefficients are a field, all non-zero constants are units, and the book uses the convention of using the monic greatest common divisor, i.e., the greatest common divisor polynomial where the coefficients have been scaled so that the leading coefficient is 1, in many of the algorithms.
For example, the Hermite reduction algorithm in section 2.2 on page 44 may result in a spurious constant factor multiplying the remaining rational function \(h\) to be integrated when using the greatest common divisor computed by the Euclidean algorithms in section 1.3 directly. The greatest common divisor must be made monic for the Hermite reduction to give correct results.
Resultant
Theorem 1.4.1 in section 1.4, page 18, notes that for two polynomials \(A = a \prod_{i=1}^{n} (x - \alpha_i)\) and \(B = b \prod_{i=1}^{m} (x - \beta_i)\), the resultant is
\[ R = a^m b^n \prod_{i=1}^n \prod_{j=1}^m (\alpha_i - \beta_j) \]
Note that it is \(a^m b^n\) in the above. It is not \(a^n b^m\). The exponent for \(a\) is the number of roots for \(B\), and the exponent for \(b\) is the number of roots for \(A\). Take care not to switch the exponents around. It can be difficult to notice the mistake if this happens if one is not aware of the possibility. This is not the formula one would use while integrating, but it may well be what one might use to test an implementation computing the resultant.