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7.4. INTEGRATION OF LOGARITHMIC EXPRESSIONS 183
Then we can re-arrange (7.30 bis) as
(
q0
p + q k∑
r = c i v ′ ) ′ n∑
p′ i
c i v i
′
0 +
+ +
, (7.32)
v
i=1 i
r 0 v i
i=k+1
} {{ } } {{ }
in K(c 1 , . . . , c n )[θ] proper rational expression
and the decomposition of the right-hand side proves the result.
This means that it is sufficient to integrate the polynomial part (which we
will do in Algorithm 33) and the rational part (which we will do in Algorithm
34) separately, and that failure in either part indicates that the whole expression
does not have an elementary integral. In other words, there is no crosscancellation
between these parts.
7.4.1 The Polynomial Part
Let us turn first to the polynomial part. Assume that p = ∑ n
i=0 a iθ i and
p 0 = ∑ m
i=0 b iθ i . The polynomial parts of equation (7.32) then say that
n∑
m∑ m∑
a i θ i = b ′ iθ i + ib i θ ′ θ i−1 +
i=0
i=0
i=0
k∑
i=1
c i v ′ i
v i
} {{ }
independent of θ
. (7.33)
Hence n = m, except for the special case n = m − 1 and b m constant. If we
consider coefficients of θ n (assuming n > 0) we have
a n = b ′ n + (n + 1)b n+1 θ ′ .
We can integrate this formally (recalling that b n+1 is constant) to get
∫
a n = b n + (n + 1)b n+1 θ. (7.34)
But a n ∈ K, and, by the Risch induction hypothesis (page 182), we have an
integration algorithm for K. In fact, not any integral will do: we want an
answer in K itself, apart possibly from a new logarithmic term of θ, which will
determine b n+1 . If ∫ a n contained any other logarithms, then multiplying them
by θ n would give us a new logarithm with a non-constant coefficient, which is
not allowed by Liouville’s Principle (Theorem 32).
Hence the contribution to ∫ p is b n θ n + b n+1 θ n+1 . However,
(
bn θ n + b n+1 θ n+1) ′
= an θ n + nb n θ ′ θ n−1 , (7.35)
so we should subtract nb n θ ′ from a n−1 . Of course, b n is only determined “up
to a constant of integration b n θ”. When we come to integrate a n−1 , we may
get a term nb n θ, which determines this. The process proceeds until we come
to integrate a 0 , when any new logarithms are allowed, and the constant of
integration here is that of the whole integration. The corresponding algorithm
is given in Figure 7.1.