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7.5. INTEGRATION OF EXPONENTIAL EXPRESSIONS 187
Figure 7.3: Algorithm 35: IntExp–Polynomial
Algorithm 35 (IntExp–Polynomial)
Input: p = ∑ n
i=−m a iθ i ∈ K[θ], where θ = exp η.
Output: An expression for ∫ pdx, or failed if not elementary
Ans:=0
for i := −m, . . . , −1, 1, . . . , n do
b i := RDE(iη ′ , a i )
if b i =failed
return failed
else Ans:=Ans+b i θ i
c 0 := ∫ a 0dx
# integration in K
if c 0 =failed
return failed
Ans:=Ans+c 0
7.5.2 The Rational Expression Part
The last equation (7.36) of Lemma 11 says that
∫ q
r = (
n
∑
i=k+1
c i n i
)
} {{ }
“correction”
(
q0
η +
r 0
)
+
n∑
i=k+1
c i (log v i − n i η) , (7.37)
} {{ }
proper rational expression
where the v i are monic polynomials (not divisible by θ) of degree n i . The proper
rational expression part is determined by an analogue of Hermite’s algorithm,
and (7.18) is still valid, though we should point out that the justification involved
stating that gcd(r i , r i ′) = 1, where the r i were the square-free factors of r 0 . Since
θ ′ = ηθ, this is no longer true if θ|r i , but this is excluded since such factors were
moved into the Laurent polynomial (Definition 86) part.
Hence the Hermite part of Algorithm 36 is identical to the rational and
logarithmic cases. The Trager–Rothstein part is sligtly more complicated, since
v i ′/v i is no longer a proper rational expression (the cause of the term marked
‘correction” in (7.37). Suppose we have a term s 2 /t 2 (in lowest terms: the
programmer must not forget this check) left after Hermite’s algorithm, and
write
∫
s2
= ∑ c i (log v i − n i η), (7.38)
t 2
(7.38) can be differentiated to
s 2
t 2
= ∑ c i
v ′ i − n iη ′ v i
v i
. (7.39)