Greetings,algebraic integrals? To be clear, I am puzzled by the nested radicals within the arctans, ie Sqrt[(x^2 + Sqrt[2] x (-1 + x^3)^(1/4) + Sqrt[-1 + x^3])/x^2]. In comparison, Trager's algorithm in Maple does not return nested radicals.
I was looking at this test on Nasser's website:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/9_Blake_problems/resu1078.htm#x1106-11050003.11.64>
Unless I am mistaken, the result from FriCAS is seemingly not in the minimal extension field. I vaguely recall this is a guarantee for the transcendental case of the Risch-Trager-Bronstein algorithm. Is this not the case for Risch-Trager-Bronstein for
Sam Blake <samuel.th...@gmail.com> wrote:for algebraic integrals? To be clear, I am puzzled by the nested radicals within the arctans, ie Sqrt[(x^2 + Sqrt[2] x (-1 + x^3)^(1/4) + Sqrt[-1 + x^3])/x^2]. In comparison, Trager's algorithm in Maple does not return nested radicals.
Greetings,
I was looking at this test on Nasser's website:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/9_Blake_problems/resu1078.htm#x1106-11050003.11.64>
Unless I am mistaken, the result from FriCAS is seemingly not in the minimal extension field. I vaguely recall this is a guarantee for the transcendental case of the Risch-Trager-Bronstein algorithm. Is this not the case for Risch-Trager-Bronstein
It is not clear which example you have in mind. Note:
- ATM FriCAS makes no attempt to denest root, so if nested root is
created at some stage it has high chance to appear in final result
- FriCAS may discover that there is nontrivial algebraic dependency
in the input and express is using nested root
- things like asin effectively have hidden root which will be
visible in result
- FriCAS takes shorcuts in some places. Shortcuts may lead to
algebraic extentions.
--
Waldek Hebisch
On Saturday, January 22, 2022 at 8:54:05 AM UTC+11, anti...@math.uni.wroc.pl wrote:for algebraic integrals? To be clear, I am puzzled by the nested radicals within the arctans, ie Sqrt[(x^2 + Sqrt[2] x (-1 + x^3)^(1/4) + Sqrt[-1 + x^3])/x^2]. In comparison, Trager's algorithm in Maple does not return nested radicals.
Sam Blake <samuel.th...@gmail.com> wrote:
Greetings,
I was looking at this test on Nasser's website:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/9_Blake_problems/resu1078.htm#x1106-11050003.11.64>
Unless I am mistaken, the result from FriCAS is seemingly not in the minimal extension field. I vaguely recall this is a guarantee for the transcendental case of the Risch-Trager-Bronstein algorithm. Is this not the case for Risch-Trager-Bronstein
sqrt(2)*log(4*(x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2) + 1/2*sqrt(2)*log(4*(x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2)It is not clear which example you have in mind. Note:
- ATM FriCAS makes no attempt to denest root, so if nested root is
created at some stage it has high chance to appear in final result
- FriCAS may discover that there is nontrivial algebraic dependency
in the input and express is using nested root
- things like asin effectively have hidden root which will be
visible in result
- FriCAS takes shorcuts in some places. Shortcuts may lead to
algebraic extentions.
--
Waldek Hebisch
The example is
integrate(x^2*(x^3-4)/(x^3-1)^(3/4)/(x^4+x^3-1),x)
Per Nasser's website, FriCAS returns
2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2) - x - sqrt(2)*(x^3 - 1)^(1/4))/x) + 2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2) + x - sqrt(2)*(x^3 - 1)^(1/4))/x) - 1/2*
Sam Blake <samuel.thomas.blake@gmail.com> wrote:
On Saturday, January 22, 2022 at 8:54:05 AM UTC+11, anti...@math.uni.wroc.pl wrote:
Sam Blake <samuel.th...@gmail.com> wrote:
Greetings,
I was looking at this test on Nasser's website:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/9_Blake_problems/resu1078.htm#x1106-11050003.11.64>
Unless I am mistaken, the result from FriCAS is seemingly notIt is not clear which example you have in mind. Note:
in the minimal extension field. I vaguely recall this is a
guarantee for the transcendental case of the
Risch-Trager-Bronstein algorithm. Is this not the case for Risch-Trager-Bronstein for algebraic integrals? To be clear, I
am puzzled by the nested radicals within the arctans, ie
Sqrt[(x^2 + Sqrt[2] x (-1 + x^3)^(1/4) + Sqrt[-1 + x^3])/x^2].
In comparison, Trager's algorithm in Maple does not return
nested radicals.
- ATM FriCAS makes no attempt to denest root, so if nested root is created at some stage it has high chance to appear in final result
- FriCAS may discover that there is nontrivial algebraic
dependency in the input and express is using nested root
- things like asin effectively have hidden root which will be
visible in result
- FriCAS takes shorcuts in some places. Shortcuts may lead to
algebraic extentions.
The example is
integrate(x^2*(x^3-4)/(x^3-1)^(3/4)/(x^4+x^3-1),x)
Per Nasser's website, FriCAS returns
2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x
+ sqrt(x^3 - 1))/x^2) - x - sqrt(2)*(x^3 - 1)^(1/4))/x) + > > 2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x
+ sqrt(x^3 - 1))/x^2) + x - sqrt(2)*(x^3 - 1)^(1/4))/x) - 1/2*sqrt(2)*log(4*(x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 -
1))/x^2) + 1/2*sqrt(2)*log(4*(x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x +
sqrt(x^3 - 1))/x^2)
This is postprocessing. Actual result from integrator is:
+------+
4| 3
--+ \|x - 1
> %LTJ log(- %LTJ + ---------)
--+ x
4
%LTJ + 1 = 0
That is root sum over roots of order 4 from 1. This gets expanded
to more explicit form. In this case expander messed up things...
antispam@math.uni.wroc.pl schrieb:
Sam Blake <samuel.thomas.blake@gmail.com> wrote:
On Saturday, January 22, 2022 at 8:54:05 AM UTC+11, anti...@math.uni.wroc.pl wrote:
Sam Blake <samuel.th...@gmail.com> wrote:
Greetings,
I was looking at this test on Nasser's website:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/9_Blake_problems/resu1078.htm#x1106-11050003.11.64>
Unless I am mistaken, the result from FriCAS is seemingly notIt is not clear which example you have in mind. Note:
in the minimal extension field. I vaguely recall this is a
guarantee for the transcendental case of the
Risch-Trager-Bronstein algorithm. Is this not the case for Risch-Trager-Bronstein for algebraic integrals? To be clear, I
am puzzled by the nested radicals within the arctans, ie
Sqrt[(x^2 + Sqrt[2] x (-1 + x^3)^(1/4) + Sqrt[-1 + x^3])/x^2].
In comparison, Trager's algorithm in Maple does not return
nested radicals.
- ATM FriCAS makes no attempt to denest root, so if nested root is created at some stage it has high chance to appear in final result
- FriCAS may discover that there is nontrivial algebraic
dependency in the input and express is using nested root
- things like asin effectively have hidden root which will be
visible in result
- FriCAS takes shorcuts in some places. Shortcuts may lead to
algebraic extentions.
The example is
integrate(x^2*(x^3-4)/(x^3-1)^(3/4)/(x^4+x^3-1),x)
Per Nasser's website, FriCAS returns
2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x
+ sqrt(x^3 - 1))/x^2) - x - sqrt(2)*(x^3 - 1)^(1/4))/x) + > > 2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x
+ sqrt(x^3 - 1))/x^2) + x - sqrt(2)*(x^3 - 1)^(1/4))/x) - 1/2*sqrt(2)*log(4*(x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 -
1))/x^2) + 1/2*sqrt(2)*log(4*(x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x +
sqrt(x^3 - 1))/x^2)
This is postprocessing. Actual result from integrator is:
+------+
4| 3
--+ \|x - 1
> %LTJ log(- %LTJ + ---------)
--+ x
4
%LTJ + 1 = 0
That is root sum over roots of order 4 from 1. This gets expanded
to more explicit form. In this case expander messed up things...
The postprocessing designer must have had something in mind.
On the real axis, a term of the form C*ATAN(f(x)) can only take values
from (-C*pi/2, +C*pi/2); antiderivatives involving such a term must
therefore exhibit jump discontinuities if they need to cover a larger
range. Using familiar relations of trigonometry, the ATAN term may be rewritten as C/2*ATAN(f2(x)) and more generally as C/n*ATAN(fn(x)),
which, however, makes matters worse. Integrators should therefore
produce antiderivatives transformed in the reverse manner as long as
this is possible without introducing spurious radical extensions.
To my knowledge, FriCAS is not doing this consistently. In the case at
hand, it is even going one step too far, introducing the unnecessary
nested radical
SQRT((x^2 - SQRT(2)*(x^3 - 1)^(1/4)*x + SQRT(x^3 - 1))/x^2)
supposedly in order to avoid discontinuities. Without this step, the antiderivative simply reads:
INT(x^2*(-4 + x^3)/((-1 + x^3)^(3/4)*(-1 + x^3 + x^4)), x) = SQRT(2)*ATAN(SQRT(2)*x*(-1 + x^3)^(1/4)/(- x^2 + SQRT(-1 + x^3))) - SQRT(2)*ATANH(SQRT(2)*x*(-1 + x^3)^(1/4)/(x^2 + SQRT(-1 + x^3)))
which, apart from two logarithmic poles, is continuous on the real axis nonetheless.
| Sysop: | Keyop |
|---|---|
| Location: | Huddersfield, West Yorkshire, UK |
| Users: | 296 |
| Nodes: | 16 (2 / 14) |
| Uptime: | 59:33:39 |
| Calls: | 6,653 |
| Calls today: | 5 |
| Files: | 12,200 |
| Messages: | 5,331,283 |