I write a primitive version patch implementing my ideas,
it works when "g x == sqrt x" in the test, and gives far more
compact result when "g x == sqrt x + x^(1/3)".
I wonder if some part of this patch should be independent
functions:
1. test if a kernel is a constant/algebraic number, algNumIfCan?
2. "deep" univariate, eg. "deepunivariate(x,first kernels sqrt x)"
should return "?^2".
Also, what's the best way to create a non-conflicting random
new kernel?
---- test
f := (2560*x^3 - 400*x^2 - 576*x - 84)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9)
g x == sqrt x -- or g x == sqrt exp x
h == eval(f, x = g x)*D(g x, x)
integrate(h,x)
draw(%,x=0..1)
---- test ends
diff --git a/src/algebra/irexpand.spad b/src/algebra/irexpand.spad
index 6e43ea41..cf485bfd 100644
--- a/src/algebra/irexpand.spad
+++ b/src/algebra/irexpand.spad
@@ -106,6 +106,16 @@
retractIfCan(a)@Union(Q, "failed") case Q => 2 * atan(-b/a)
2 * atan(a/b)
+ uni2(exp : F, kers : List K) : Fraction UP ==
+ maxk := "max"/kers
+ mink := "min"/kers
+ if (#kers ~= 2 or name maxk ~= 'nthRoot or first argument maxk
~= coerce(mink)@F)
+ then return univariate(exp, maxk)
+ newker := kernel('randomkernel)
+ newexp := coerce(newker)@F
+ n := retractIfCan(second argument maxk)@Union(Integer,"failed")::Integer
+ univariate(subst(exp,[maxk, mink]::List K, [newexp,
newexp^n]::List F), newker)
+
-- transforms i log((a + i b) / (a - i b)) into a sum of real
-- arc-tangents using Rioboo's algorithm
-- lk is a list of kernels which are parameters for the integral
@@ -113,8 +123,9 @@
l := setDifference(setUnion(variables numer a, variables numer b),
setUnion(lk, setUnion(variables denom a, variables denom b)))
empty? l => tantrick(a, b)
+ l := select((k:K):Boolean+->not(is?(k,'nthRoot) and null
kernels argument k), l)
k := "max"/l
- ilog0(a, b, numer univariate(a, k), numer univariate(b, k), k::F)
+ ilog0(a, b, numer uni2(a, l), numer uni2(b, l), k::F)
-- transforms i log((a + i b) / (a - i b)) into a sum of real
-- arc-tangents using Rioboo's algorithm
--
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