Compare the result of
integrate(1/(x^8-1),x)
integrate(8*sqrt(2)/(x^8-1),x)
integrate(a/(x^8-1),x)
The first is more complex than the second one.
Why? Because:
(1) -> factor(x^4+1)
4
(1) x + 1
Type: Factored(Polynomial(Integer))
(2) -> factor(x^4+4)
2 2
(2) (x - 2x + 2)(x + 2x + 2)
Type: Factored(Polynomial(Integer))
The factorization of "x^4+r" depends on r, or else it will
introduce extra algebraic kernels.
In rational function integration, sum logs over roots of
"x^4+r=0" is pretty common, but "lg2func" doesn't deal
with this case. So I manually facor "x^4+r" into quadratics.
This patch improves 1190a 1191a 1207a 1217a 1218a 1228a
of mapleok.input, making 1191a 1207a 1217a continuous.
I have to use "rootSimp" in the patch, I'm not sure if this
will causes trouble elsewhere.
diff --git a/src/algebra/irexpand.spad b/src/algebra/irexpand.spad
index eb90a9c9..eaffa9e4 100644
--- a/src/algebra/irexpand.spad
+++ b/src/algebra/irexpand.spad
@@ -84,6 +84,7 @@
zero?(delta := (b := coefficient(p, 1))^2 - 4 *
(a := coefficient(p, 2)) * (p0 := coefficient(p, 0))) =>
[linear(monomial(1, 1) + (b / a)::UP, lg)]
+ delta := rootSimp delta
e := (q := quadeval(lg, c := - b * (d := inv(2*a)), d, delta)).ans1
lgp := c * log(nrm := (e^2 - delta * (f := q.ans2)^2))
s := (sqr := insqrt delta).sqrt
@@ -152,8 +153,18 @@
-- one? d => [linear(p, lg.logand)]
(d = 1) => [linear(p, lg.logand)]
d = 2 => quadratic(p, lg.logand, x)
- odd? d and
- ((r := retractIfCan(reductum p)@Union(F, "failed")) case F) =>
+ r := retractIfCan(reductum p)@Union(F, "failed")
+ d = 4 and r case F =>
+ -- In this "p := ?^4 + r" case, we manually factor p into
+ -- "?^2 + sqrt(r) - ?*sqrt(2*sqrt(r))" and
+ -- "?^2 + sqrt(r) + ?*sqrt(2*sqrt(r))".
+ upx := monomial(1,1)$UP
+ t : F := rootSimp sqrt(2 * rootSimp sqrt r)
+ -- quadratic will always return 1 result (instead of 2),
+ -- because the sign of delta is fixed.
+ [first quadratic(upx^2 + sqrt(r)::UP - upx*t, lg.logand, x)
+ + first quadratic(upx^2 + sqrt(r)::UP + upx*t, lg.logand, x)]
+ odd? d and r case F =>
pairsum([cmplex(alpha := rootSimp zeroOf p, lg.logand)],
lg2func([lg.scalar,
(p exquo (monomial(1, 1)$UP - alpha::UP))::UP,
--
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