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184 CHAPTER 7. CALCULUS
Figure 7.1: Algorithm 33: IntLog–Polynomial
Algorithm 33 (IntLog–Polynomial)
Input: p = ∑ n
i=0 a iθ i ∈ K[θ].
Output: An expression for ∫ pdx, or failed if not elementary
Ans:=0
for i := n, n − 1, . . . , 1 do
c i := ∫ a idx # integration in K: (7.34)
if c i =failed or c i /∈ K[θ]
return failed
Write c i = b i + (i + 1)b i+1 : b i ∈ K and b i+1 constant
Ans:=Ans+b i θ i + b i+1 θ i+1
a i−1 := a i−1 − iθ ′ b i # correction from (7.35)
c 0 := ∫ a 0dx
# integration in K
if c 0 =failed
return failed
Ans:=Ans+c 0
7.4.2 The Rational Expression Part
Now for the rational expression part, where we have to integrate a proper rational
expression, and the integral will be a proper rational expression plus a sum
of logarithms with constant coefficients — see (*) in Algorithm 34. The proper
rational expression part is determined by an analogue of Hermite’s algorithm,
and (7.18) is still valid.
The Trager–Rothstein algorithm is still valid, with that additional clause
that the roots of P (λ), which don’t depend on θ since P (λ) is a resultant,
actually be constants. To see why this is necessary, consider ∫ 1
log xdx. Here
q 1 = 1, r 1 = θ, and P (λ) = Res(1−λ/x, θ, θ) = 1−λ/x (made monic, i.e. λ−x).
This would suggest a contribution to the integral of x log log x, which indeed
gives log x as one term on differentiation, but also a term of log log x, which is
not allowed, since it is not present in the integrand. Note that ∫ 1
x log xdx gives
P = λ − 1 and an integral of log log x, which is correct.
7.5 Integration of Exponential Expressions
Throughout this section, we let θ = θ n be a (transcendental) exponential over
K = C(x, θ 1 , . . . , θ n−1 ), so that θ ′ = η ′ θ. We should note that this choice
is somewhat arbitrary, since θ −1 = θ satisfies θ ′ = −η ′ θ and K(θ) ≡ K(θ).
Hence negative powers of θ are just as legitimate as positive powers, and this
translates into a difference in the next result: rather than writing expressions as
“polynomial + proper rational expressions”, we will make use of the following