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7.2. INTEGRATION OF RATIONAL EXPRESSIONS 173
where the integral on the right-hand resolves itself as purely a sum of logarithms,
i.e. is the ∑ n ∑ ni
i=1 j=i β i,j,1log(x − α i,j ) term. Then a standard argument from
Galois theory shows that s 1 , t 1 , s 2 and t 2 do not involve any of the α i , i.e.
that the decomposition (7.13) can be written without introducing any algebraic
numbers. If we could actually obtain this decomposition without introducing
these algebraic numbers, we would have gone a long way to solving objection 2
above.
We can perform a square-free decomposition (Definition 32) of r as ∏ ri i,
and then a partial fraction decomposition to write
q
r = ∑ q i
r i i
(7.14)
and, since each term is a rational expression and therefore integrable, it suffices
to integrate (7.14) term-by-term.
Now r i and r i ′ are relatively prime, so, by Bezout’s Identity (2.10), there are
polynomials a and b satisfying ar i + br i ′ = 1. Therefore
∫ ∫
qi qi (ar i + br i ′
ri
i =
)
ri
i (7.15)
∫ ∫
qi a qi br i
′ =
r i−1 +
r i (7.16)
i
i
∫ ∫
qi a (qi b/(i − 1)) ′ ( ) ′ qi b/(i − 1)
= +
−
(7.17)
r i−1
i
(
qi b/(i − 1)
= −
r i−1
i
r i−1
i
)
+
r i−1
i
∫
qi a + (q i b/(i − 1)) ′
, (7.18)
r i−1
i
and we have reduced the exponent of r i by one. When programming this method
one may need to take care of the fact that, while qi
is a proper rational expression,
q ib
r i−1
i
may not be, but the excess is precisely compensated for by the other
term in (7.18).
Hence, at the cost of a square-free decomposition and a partial fraction decomposition,
but not a factorization, we have found the rational part of the
integral, i.e. performed the decomposition of (7.14). In fact, we have done
term will have been split into summands cor-
somewhat better, since the ∫ s 2
t2
responding to the different r i .
r i i
7.2.3 The Ostrogradski–Horowitz Algorithm
Although quite simple, Hermite’s method still needs square-free decomposition
and partial fractions. Horowitz [Hor69, Hor71] therefore proposed to computer
algebraists the following method, which was in fact already known [Ost45], but
largely forgotten 2 in the west. It follows from (7.18) that, in the notation of
name.
2 The author had found no references to it, but apparently it had been taught under that