I have created a file of 10335 random testcases of exp-log
functions. This should be relatively easy test: with
high probablity such functions are transcendental. Also
pobablity of hitting problematic case is relatively low.
However, it seems that some system have trouble with such
easy cases. Already exmple number 9, that is integral of
(((-2)*x^3+34*x^2+392*x+800)*log((((-25)*x+(-100))*log(x)+(x^2+x))/(25*x+100))+((-6)*x^3+102*x^2+1176*x+2400))/((25*x^3+200*x^2+400*x)*log(x)+((-1)*x^4+(-5)*x^3+(-4)*x^2))
is returned unevaluated by maxima 5.44 and Maple 15.
It works in FriCAS and Mathematica 12.0.0.
So, can your system do easy integrals?
antispam@math.uni.wroc.pl schrieb:
I have created a file of 10335 random testcases of exp-log
functions. This should be relatively easy test: with
high probablity such functions are transcendental. Also
pobablity of hitting problematic case is relatively low.
However, it seems that some system have trouble with such
easy cases. Already exmple number 9, that is integral of
(((-2)*x^3+34*x^2+392*x+800)*log((((-25)*x+(-100))*log(x)+(x^2+x))/(25*x+100))+((-6)*x^3+102*x^2+1176*x+2400))/((25*x^3+200*x^2+400*x)*log(x)+((-1)*x^4+(-5)*x^3+(-4)*x^2))
is returned unevaluated by maxima 5.44 and Maple 15.
It works in FriCAS and Mathematica 12.0.0.
So, can your system do easy integrals?
Derive 6.10 has no problem with that one:
INT((((-2)*x^3 + 34*x^2 + 392*x + 800)*LN((((-25)*x + -100)*LN(x) +
(x^2 + x))/(25*x + 100)) + ((-6)*x^3 + 102*x^2 + 1176*x + 2400))/
((25*x^3 + 200*x^2 + 400*x)*LN(x) + ((-1)*x^4 + (-5)*x^3 + (-4)*x^2)),
x)
LN((x*(x + 1) - 25*(x + 4)*LN(x))/(x + 4))^2 + LN((25*(x + 4)*LN(x) -
x*(x + 1))/(x + 4))*(6 - 4*LN(5))
This answer is as good as instantaneous. I haven't tried any of the
other integrands yet.
So, can your system do easy integrals?
You can find test file at
http://www.math.uni.wroc.pl/~hebisch/fricas/rand3c.input
On 9/7/2021 7:39 PM, antispam@math.uni.wroc.pl wrote:
So, can your system do easy integrals?
You can find test file at
http://www.math.uni.wroc.pl/~hebisch/fricas/rand3c.input
I plugged in your file to the independent CAS integration test program
and this is the result.
<https://www.12000.org/my_notes/CAS_integration_tests/index.htm> <https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
There are 10,335 integrals in this file. This is the
largest file now in CAS integration tests and took the program
10 days running 24 hrs to complete. There are now
a total of 210 files in the CAS integration test suite.
Here is summary of the result copied from the above link
CAS systems used
==================
1. Mathematica 12.3.1 (64 bit) on windows 10.
2. Rubi 4.16.1 on Mathematica 12.3.1 on windows 10.
3. Maple 2021.1 (64 bit) on windows 10.
4. Maxima 5.45 on Linux. (via sagemath 9.4)
5. Fricas 1.3.7 on Linux (via sagemath 9.4)
6. Giac/Xcas 1.7 on Linux. (via sagemath 9.4)
7. Sympy 1.8 under Python 3.8.8 using Anaconda distribution.
8. Mupad using Matlab 2021a with Symbolic Math Toolbox Version 8.7
under windows 10 (64 bit)
This is the final result of percentage passed
system % passed
======== ===========
1. Fricas 99.92
2. Maple 97.49
3. Mathematica 97.37
4. Sympy 95.43
5. Maxima 93.17
6. Mupad 90.13
7. Giac 85.09
8. Rubi 63.38
The following is the grading result (notice that Mupad is not
currently graded, a pass is given B grade automatically else an F
grade is given).
System %A grade % B grade %C grade %F grade
======= ======== ========= ===== =======
Mathematica 88.51 7.80 0.81 2.63
Fricas 81.60 18.18 0.15 0.08
Maple 79.98 9.78 7.73 2.51
Sympy 77.10 18.16 0.17 4.57
Maxima 63.97 23.97 5.23 6.83
Giac 63.83 20.98 0.28 14.91
Rubi 49.55 12.19 1.64 36.62
Mupad N/A 90.13 0.00 9.87
The following is performance table
system Mean time (sec) Normalized mean
to solve one integral size of antiderivative
======== =========== =======================
Mathematica 0.36 4.02
Maple 0.50 8.38
Fricas 0.59 1.53
Maxima 0.68 2.49
Rubi 0.74 2.95
Giac 0.83 2.47
Sympy 1.77 1.42
Mupad 2.93 6.05
[...]
"Nasser M. Abbasi" schrieb:
On 9/7/2021 7:39 PM, antispam@math.uni.wroc.pl wrote:
So, can your system do easy integrals?
You can find test file at
http://www.math.uni.wroc.pl/~hebisch/fricas/rand3c.input
I plugged in your file to the independent CAS integration test program
and this is the result.
<https://www.12000.org/my_notes/CAS_integration_tests/index.htm> <https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
There are 10,335 integrals in this file. This is the
largest file now in CAS integration tests and took the program
10 days running 24 hrs to complete. There are now
a total of 210 files in the CAS integration test suite.
Here is summary of the result copied from the above link
CAS systems used
==================
1. Mathematica 12.3.1 (64 bit) on windows 10.
2. Rubi 4.16.1 on Mathematica 12.3.1 on windows 10.
3. Maple 2021.1 (64 bit) on windows 10.
4. Maxima 5.45 on Linux. (via sagemath 9.4)
5. Fricas 1.3.7 on Linux (via sagemath 9.4)
6. Giac/Xcas 1.7 on Linux. (via sagemath 9.4)
7. Sympy 1.8 under Python 3.8.8 using Anaconda distribution.
8. Mupad using Matlab 2021a with Symbolic Math Toolbox Version 8.7
under windows 10 (64 bit)
This is the final result of percentage passed
system % passed
======== ===========
1. Fricas 99.92
2. Maple 97.49
3. Mathematica 97.37
4. Sympy 95.43
5. Maxima 93.17
6. Mupad 90.13
7. Giac 85.09
8. Rubi 63.38
The following is the grading result (notice that Mupad is not
currently graded, a pass is given B grade automatically else an F
grade is given).
System %A grade % B grade %C grade %F grade
======= ======== ========= ===== =======
Mathematica 88.51 7.80 0.81 2.63
Fricas 81.60 18.18 0.15 0.08
Maple 79.98 9.78 7.73 2.51
Sympy 77.10 18.16 0.17 4.57
Maxima 63.97 23.97 5.23 6.83
Giac 63.83 20.98 0.28 14.91
Rubi 49.55 12.19 1.64 36.62
Mupad N/A 90.13 0.00 9.87
The following is performance table
system Mean time (sec) Normalized mean
to solve one integral size of antiderivative
======== =========== ======================= Mathematica 0.36 4.02
Maple 0.50 8.38
Fricas 0.59 1.53
Maxima 0.68 2.49
Rubi 0.74 2.95
Giac 0.83 2.47
Sympy 1.77 1.42
Mupad 2.93 6.05
[...]
I haven't checked how many of the exp-log integrands involve algebraic numbers or functions.
The declining performance from Fricas to Maple
to Mathematica to Sympy to Maxima may reflect this fraction, or it may
just reveal deficiencies of their non-algebraic Risch implementations.
clicliclic@freenet.de <nobody@nowhere.invalid> wrote:
"Nasser M. Abbasi" schrieb:
On 9/7/2021 7:39 PM, antispam@math.uni.wroc.pl wrote:
So, can your system do easy integrals?
You can find test file at
http://www.math.uni.wroc.pl/~hebisch/fricas/rand3c.input
I plugged in your file to the independent CAS integration test
program and this is the result.
<https://www.12000.org/my_notes/CAS_integration_tests/index.htm> <https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
There are 10,335 integrals in this file. This is the
largest file now in CAS integration tests and took the program
10 days running 24 hrs to complete. There are now
a total of 210 files in the CAS integration test suite.
Here is summary of the result copied from the above link
CAS systems used
==================
1. Mathematica 12.3.1 (64 bit) on windows 10.
2. Rubi 4.16.1 on Mathematica 12.3.1 on windows 10.
3. Maple 2021.1 (64 bit) on windows 10.
4. Maxima 5.45 on Linux. (via sagemath 9.4)
5. Fricas 1.3.7 on Linux (via sagemath 9.4)
6. Giac/Xcas 1.7 on Linux. (via sagemath 9.4)
7. Sympy 1.8 under Python 3.8.8 using Anaconda distribution.
8. Mupad using Matlab 2021a with Symbolic Math Toolbox Version 8.7
under windows 10 (64 bit)
This is the final result of percentage passed
system % passed
======== ===========
1. Fricas 99.92
2. Maple 97.49
3. Mathematica 97.37
4. Sympy 95.43
5. Maxima 93.17
6. Mupad 90.13
7. Giac 85.09
8. Rubi 63.38
The following is the grading result (notice that Mupad is not
currently graded, a pass is given B grade automatically else an F
grade is given).
System %A grade % B grade %C grade %F grade
======= ======== ========= ===== =======
Mathematica 88.51 7.80 0.81 2.63
Fricas 81.60 18.18 0.15 0.08
Maple 79.98 9.78 7.73 2.51
Sympy 77.10 18.16 0.17 4.57
Maxima 63.97 23.97 5.23 6.83
Giac 63.83 20.98 0.28 14.91
Rubi 49.55 12.19 1.64 36.62
Mupad N/A 90.13 0.00 9.87
The following is performance table
system Mean time (sec) Normalized mean
to solve one integral size of antiderivative ======== =========== ======================= Mathematica 0.36 4.02
Maple 0.50 8.38
Fricas 0.59 1.53
Maxima 0.68 2.49
Rubi 0.74 2.95
Giac 0.83 2.47
Sympy 1.77 1.42
Mupad 2.93 6.05
[...]
I haven't checked how many of the exp-log integrands involve
algebraic numbers or functions.
FriCAS tells me that only 31...
The declining performance from Fricas to Maple
to Mathematica to Sympy to Maxima may reflect this fraction, or it
may just reveal deficiencies of their non-algebraic Risch
implementations.
It seems that there are deficiences in implementation of
transcendental part of Risch algorithm. Or, possibly effect of
incompletness of whatever alternative they use...
antispam@math.uni.wroc.pl schrieb:
clicliclic@freenet.de <nobody@nowhere.invalid> wrote:
"Nasser M. Abbasi" schrieb:
On 9/7/2021 7:39 PM, antispam@math.uni.wroc.pl wrote:
So, can your system do easy integrals?
You can find test file at
http://www.math.uni.wroc.pl/~hebisch/fricas/rand3c.input
I plugged in your file to the independent CAS integration test
program and this is the result.
<https://www.12000.org/my_notes/CAS_integration_tests/index.htm> <https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
There are 10,335 integrals in this file. This is the
largest file now in CAS integration tests and took the program
10 days running 24 hrs to complete. There are now
a total of 210 files in the CAS integration test suite.
Here is summary of the result copied from the above link
CAS systems used
==================
1. Mathematica 12.3.1 (64 bit) on windows 10.
2. Rubi 4.16.1 on Mathematica 12.3.1 on windows 10.
3. Maple 2021.1 (64 bit) on windows 10.
4. Maxima 5.45 on Linux. (via sagemath 9.4)
5. Fricas 1.3.7 on Linux (via sagemath 9.4)
6. Giac/Xcas 1.7 on Linux. (via sagemath 9.4)
7. Sympy 1.8 under Python 3.8.8 using Anaconda distribution.
8. Mupad using Matlab 2021a with Symbolic Math Toolbox Version 8.7 under windows 10 (64 bit)
This is the final result of percentage passed
system % passed
======== ===========
1. Fricas 99.92
2. Maple 97.49
3. Mathematica 97.37
4. Sympy 95.43
5. Maxima 93.17
6. Mupad 90.13
7. Giac 85.09
8. Rubi 63.38
The following is the grading result (notice that Mupad is not
currently graded, a pass is given B grade automatically else an F
grade is given).
System %A grade % B grade %C grade %F grade
======= ======== ========= ===== ======= Mathematica 88.51 7.80 0.81 2.63
Fricas 81.60 18.18 0.15 0.08
Maple 79.98 9.78 7.73 2.51
Sympy 77.10 18.16 0.17 4.57
Maxima 63.97 23.97 5.23 6.83
Giac 63.83 20.98 0.28 14.91
Rubi 49.55 12.19 1.64 36.62
Mupad N/A 90.13 0.00 9.87
The following is performance table
system Mean time (sec) Normalized mean
to solve one integral size of antiderivative ======== =========== ======================= Mathematica 0.36 4.02
Maple 0.50 8.38
Fricas 0.59 1.53
Maxima 0.68 2.49
Rubi 0.74 2.95
Giac 0.83 2.47
Sympy 1.77 1.42
Mupad 2.93 6.05
[...]
I haven't checked how many of the exp-log integrands involve
algebraic numbers or functions.
FriCAS tells me that only 31...
The declining performance from Fricas to Maple
to Mathematica to Sympy to Maxima may reflect this fraction, or it
may just reveal deficiencies of their non-algebraic Risch implementations.
It seems that there are deficiences in implementation of
transcendental part of Risch algorithm. Or, possibly effect of incompletness of whatever alternative they use...
Perhaps those accidental algebraics would then better be removed from
the suite, leaving 10,335 - 31 = 10,304 integrands?
Also please test on Integrate from Wolfram Math. 13.0.0,
since it has IntegrateAlgebraic indide.
On 12/27/2021 2:27 AM, Валерий Заподовников wrote:
The long expression posted previously that Maxima could not integrate can be expanded and much of it is then integrated.Also please test on Integrate from Wolfram Math. 13.0.0,There should be a new build of CAS independent integration
since it has IntegrateAlgebraic indide.
tests which will have Mathematica V 13.0. But waiting for
Maple 2022 and sagemath 9.6 and most important for the next
version of Rubi to be released with its new test input files
with new integrals added.
This is because it takes about 2 months and lots of effort and
time to run all these tests, and do not want to do this now
and then have to do it again few months later.
Hopefully sometime next year.
--Nasser
The long expression posted previously that Maxima could not integrate
can be expanded and much of it is then integrated.
v:
(((-2)*x^3+34*x^2+392*x+800)*log((((-25)*x+(-100))*log(x)+(x^2+x))/(25*x+100))+((-6)*x^3+102*x^2+1176*x+2400))/((25*x^3+200*x^2+400*x)*log(x)+((-1)*x^4+(-5)*x^3+(-4)*x^2));
((-2*x^3+34*x^2+392*x+800)*log(((-25*x-100)*log(x)+x^2+x)/(25*x+100))-6*x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*log(x)-x^4-5*x^3-4*x^2)
There's one piece of the expansion that doesn't come out in the wash,
integrate((3*x^6-119*x^5-608*x^4+27320*x^3+339296*x^2+1411200*x+1920000)/((150*x^2+1200*x+2400)*log(x)-6*x^3-30*x^2-24*x),x)
So this after tossing out what I thought were extraneous to come up
with a simple "bug report" I came to the problem
integrate( 1/(log(x)+x), x) which Maxima 5.45.1 apparently
cannot do.
Also in Maxima, risch(...) returns unchanged, which suggests that
this is not integrable in terms of elementary functions, but I don't
really trust that.
In Mathematica 13, the integral also returns unchanged.
I do not have a recent version of Maple or any version of Fricas.
It seems to me that one can generate increasingly more challenging
examples in a systematic fashion that would illustrate points of
failure more effectively than trying out random algebraic tree
generation.
For instance, irreducible polynomials of increasing degrees; one,
two, ... more logarithmic extensions, exponential extensions, both,
..
RJF
So can you publish 8 failures for FriCAS?? please!
On 12/29/2021 10:18 AM, Валерий Заподовников wrote:
So can you publish 8 failures for FriCAS?? please!
If you mean the 8 integrals FriCAS did not solve in
the 10,335 integrals test you are replying to, then
this is all given in the above link of the message you
are replying to?
Here it is again:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
They are numbers { 833, 975, 2446, 3878, 5066, 5285, 5642, 8482 }
Here they are
integrate(((4*exp(x^2)-4*x)*log(exp(x)*exp(x^2)-exp(x)*x)*log(log(exp(x)*exp(x^2)-exp(x)*x))+(8*x+4)*exp(x^2)-
4*x-4)*(exp(x)*log(log(exp(x)*exp(x^2)-exp(x)*x)))^(2/9)/(9*exp(x^2)-9*x)/log(exp(x)*exp(x^2)-exp(x)*x)/log(lo
g(exp(x)*exp(x^2)-exp(x)*x)),x, algorithm="giac")
----------------
[...]
integrate((6*exp(4)*log(x)-2*exp(4)^2+(6-4*x)*exp(4)-2*x^2+6*x)*exp(1/3*log((-3*x*log(x)+x*exp(4)+x^2)/(x+exp(
4)))-1)*exp(2*exp(1/3*log((-3*x*log(x)+x*exp(4)+x^2)/(x+exp(4)))-1))/((9*x*exp(4)+9*x^2)*log(x)-3*x*exp(4)^2-6
*x^2*exp(4)-3*x^3),x, algorithm="fricas")
Exception raised: TypeError >> Error detected within library code: do_alg_rde: unimplemented kernel
----------------------------------
[...]
integrate(((-log(exp(x)*x)+x+9)*((-25*log(exp(x)*x)+200)/x)^(1/2)-2*log(exp(x)*x)+16)*exp(((-25*log(exp(x)*x)+
200)/x)^(1/2))/(2*x^2*log(exp(x)*x)-16*x^2),x, algorithm="fricas")
Exception raised: TypeError >> Error detected within library code: do_alg_rde: unimplemented kernel
-------------------------------
Here it is again:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
They are numbers { 833, 975, 2446, 3878, 5066, 5285, 5642, 8482 }
Here they are
integrate(((4*exp(x^2)-4*x)*log(exp(x)*exp(x^2)-exp(x)*x)*log(log(exp(x)*exp(x^2)-exp(x)*x))+(8*x+4)*exp(x^2)-
4*x-4)*(exp(x)*log(log(exp(x)*exp(x^2)-exp(x)*x)))^(2/9)/(9*exp(x^2)-9*x)/log(exp(x)*exp(x^2)-exp(x)*x)/log(lo
g(exp(x)*exp(x^2)-exp(x)*x)),x, algorithm="giac")
----------------
This one was calling GIAC, not FriCAS. Also no result or error message
is shown.
[...]
integrate((6*exp(4)*log(x)-2*exp(4)^2+(6-4*x)*exp(4)-2*x^2+6*x)*exp(1/3*log((-3*x*log(x)+x*exp(4)+x^2)/(x+exp(
4)))-1)*exp(2*exp(1/3*log((-3*x*log(x)+x*exp(4)+x^2)/(x+exp(4)))-1))/((9*x*exp(4)+9*x^2)*log(x)-3*x*exp(4)^2-6
*x^2*exp(4)-3*x^3),x, algorithm="fricas")
Exception raised: TypeError >> Error detected within library code: do_alg_rde: unimplemented kernel
----------------------------------
[...]
integrate(((-log(exp(x)*x)+x+9)*((-25*log(exp(x)*x)+200)/x)^(1/2)-2*log(exp(x)*x)+16)*exp(((-25*log(exp(x)*x)+
200)/x)^(1/2))/(2*x^2*log(exp(x)*x)-16*x^2),x, algorithm="fricas")
Exception raised: TypeError >> Error detected within library code: do_alg_rde: unimplemented kernel
-------------------------------
I am seeing those "do_alg_rde: unimplemented kernel" messages for the
first time, where I take "rde" to mean Risch Differential Equation. Apparently, the rde-solver of FriCAS may fail for mixed transcendental- algebraic integrands. Indeed, the first integrand involves:
(- x*(3*LN(x)/(x + #e^4) - 1))^(1/3)
while the second one contains:
SQRT((8 - LN(x))/x - 1)
Martin.
"Nasser M. Abbasi" schrieb:
On 12/29/2021 10:18 AM, ?????????????? ???????????????????????? wrote:
So can you publish 8 failures for FriCAS?? please!
If you mean the 8 integrals FriCAS did not solve in
the 10,335 integrals test you are replying to, then
this is all given in the above link of the message you
are replying to?
Here it is again:
<https://12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/10_Hebisch/report.htm>
They are numbers { 833, 975, 2446, 3878, 5066, 5285, 5642, 8482 }
Here they are
integrate(((4*exp(x^2)-4*x)*log(exp(x)*exp(x^2)-exp(x)*x)*log(log(exp(x)*exp(x^2)-exp(x)*x))+(8*x+4)*exp(x^2)-
4*x-4)*(exp(x)*log(log(exp(x)*exp(x^2)-exp(x)*x)))^(2/9)/(9*exp(x^2)-9*x)/log(exp(x)*exp(x^2)-exp(x)*x)/log(lo
g(exp(x)*exp(x^2)-exp(x)*x)),x, algorithm="giac")
----------------
This one was calling GIAC, not FriCAS. Also no result or error message
is shown.
[...]
integrate((6*exp(4)*log(x)-2*exp(4)^2+(6-4*x)*exp(4)-2*x^2+6*x)*exp(1/3*log((-3*x*log(x)+x*exp(4)+x^2)/(x+exp(
4)))-1)*exp(2*exp(1/3*log((-3*x*log(x)+x*exp(4)+x^2)/(x+exp(4)))-1))/((9*x*exp(4)+9*x^2)*log(x)-3*x*exp(4)^2-6
*x^2*exp(4)-3*x^3),x, algorithm="fricas")
Exception raised: TypeError >> Error detected within library code: do_alg_rde: unimplemented kernel
----------------------------------
[...]
integrate(((-log(exp(x)*x)+x+9)*((-25*log(exp(x)*x)+200)/x)^(1/2)-2*log(exp(x)*x)+16)*exp(((-25*log(exp(x)*x)+
200)/x)^(1/2))/(2*x^2*log(exp(x)*x)-16*x^2),x, algorithm="fricas")
Exception raised: TypeError >> Error detected within library code: do_alg_rde: unimplemented kernel
-------------------------------
I am seeing those "do_alg_rde: unimplemented kernel" messages for the
first time, where I take "rde" to mean Risch Differential Equation.
Apparently, the rde-solver of FriCAS may fail for mixed transcendental- algebraic integrands.
Indeed, the first integrand involves:
(- x*(3*LN(x)/(x + #e^4) - 1))^(1/3)
while the second one contains:
SQRT((8 - LN(x))/x - 1)
Martin.
On Monday, December 27, 2021 at 2:48:09 AM UTC-8, Nasser M. Abbasi wrote:
On 12/27/2021 2:27 AM, ??????? ???????????? wrote:
Also please test on Integrate from Wolfram Math. 13.0.0,There should be a new build of CAS independent integration
since it has IntegrateAlgebraic indide.
tests which will have Mathematica V 13.0. But waiting for
Maple 2022 and sagemath 9.6 and most important for the next
version of Rubi to be released with its new test input files
with new integrals added.
This is because it takes about 2 months and lots of effort and
time to run all these tests, and do not want to do this now
and then have to do it again few months later.
Hopefully sometime next year.
--NasserThe long expression posted previously that Maxima could not integrate can be expanded and much of it is then integrated.
v:
(((-2)*x^3+34*x^2+392*x+800)*log((((-25)*x+(-100))*log(x)+(x^2+x))/(25*x+100))+((-6)*x^3+102*x^2+1176*x+2400))/((25*x^3+200*x^2+400*x)*log(x)+((-1)*x^4+(-5)*x^3+(-4)*x^2));
((-2*x^3+34*x^2+392*x+800)*log(((-25*x-100)*log(x)+x^2+x)/(25*x+100))-6*x^3+102*x^2+1176*x+2400)/((25*x^3+200*x^2+400*x)*log(x)-x^4-5*x^3-4*x^2)
There's one piece of the expansion that doesn't come out in the wash,
integrate((3*x^6-119*x^5-608*x^4+27320*x^3+339296*x^2+1411200*x+1920000)/((150*x^2+1200*x+2400)*log(x)-6*x^3-30*x^2-24*x),x)
So this after tossing out what I thought were extraneous to come up with a simple
"bug report" I came to the problem integrate( 1/(log(x)+x), x) which Maxima 5.45.1 apparently cannot do.
It seems to me that one can generate increasingly more challenging examples in a systematic fashion that would illustrate points of failure more effectively than trying out random algebraic tree generation.
For instance, irreducible polynomials of increasing degrees; one, two, ... more logarithmic extensions, exponential extensions, both, ..
Well, someone should train NN to solve those integrals,
that would be funny. BTW, I made a mistake in my Log[]
function for Mathematica, so
Integrate[Log (1 + x)/x/(1 + (1 + x)^(1/2))^(1/2), x]
should have been
Integrate[Log[1 + x]/x/(1 + (1 + x)^(1/2))^(1/2), x], still
works though even before 13.0, nonelementary, but simple.
Meanwhile
https://github.com/stblake/algebraic_integration/issues/2
is fixed so IntegrateAlgebraic cannot solve it, but Integrate
will be able due to another IntegralAlgebraic hack (in next
version of Mathematica).
To complete FriCAS Risch error branches see https://github.com/fricas/fricas/issues/78
"residue poly has multiple non-linear factors".
Who can say whether it is elementary? Sam Blake says
it is good enough that it is complex! So may it be that it is
only elementary with complex i inside? I find it improbable.
integrate((1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)), x) is
a very dangerous DoS case with fricas anyway. As for Math.
it cannot already solve slightly different
Integrate[(1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4))^(3/4), x]
Euler's elementary algebraic integral:
INT((1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)), x)
from paper E689 in the Euler Archive has the solution:
- ATANH(2*x*(x^4 - 6*x^2 + 1)^(1/4)/(SQRT(x^4 - 6*x^2 + 1) + x^2 + 1))
+ ATAN(x/((x^4 - 6*x^2 + 1)^(1/4) + 1))
- ATAN(((x^4 - 6*x^2 + 1)^(1/4) - 1)/x)
which is real and continuous where the integrand is real and
continuous. It appears that the awkward ATANH term cannot be decomposed
into simpler ones with real arguments.
Martin.
On Saturday, January 15, 2022 at 2:15:27 AM UTC-10, nob...@nowhere.invalid wrote:
Euler's elementary algebraic integral:
INT((1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)), x)
from paper E689 in the Euler Archive has the solution:
- ATANH(2*x*(x^4 - 6*x^2 + 1)^(1/4)/(SQRT(x^4 - 6*x^2 + 1) + x^2 + 1))
+ ATAN(x/((x^4 - 6*x^2 + 1)^(1/4) + 1))
- ATAN(((x^4 - 6*x^2 + 1)^(1/4) - 1)/x)
which is real and continuous where the integrand is real and
continuous. It appears that the awkward ATANH term cannot be
decomposed into simpler ones with real arguments.
What's wrong with combining the two arctan terms to get in
Mathematica notation:
ArcTan[(1 + x^2 - Sqrt[1 - 6*x^2 + x^4])/(2*x*(1 - 6*x^2 + x^4)^(1/4))] - ArcTanh[(2*x*(1 - 6*x^2 + x^4)^(1/4))/(1 + x^2 + Sqrt[1 - 6*x^2 + x^4])]
which is also real and continuous where the integrand is real and
continuous.
Albert Rich schrieb:
On Saturday, January 15, 2022 at 2:15:27 AM UTC-10, nob...@nowhere.invalid wrote:
Euler's elementary algebraic integral:
INT((1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)), x)
from paper E689 in the Euler Archive has the solution:
- ATANH(2*x*(x^4 - 6*x^2 + 1)^(1/4)/(SQRT(x^4 - 6*x^2 + 1) + x^2 + 1))
+ ATAN(x/((x^4 - 6*x^2 + 1)^(1/4) + 1))
- ATAN(((x^4 - 6*x^2 + 1)^(1/4) - 1)/x)
which is real and continuous where the integrand is real and
continuous. It appears that the awkward ATANH term cannot be
decomposed into simpler ones with real arguments.
What's wrong with combining the two arctan terms to get in
Mathematica notation:
ArcTan[(1 + x^2 - Sqrt[1 - 6*x^2 + x^4])/(2*x*(1 - 6*x^2 + x^4)^(1/4))] - ArcTanh[(2*x*(1 - 6*x^2 + x^4)^(1/4))/(1 + x^2 + Sqrt[1 - 6*x^2 + x^4])]
which is also real and continuous where the integrand is real and continuous.
Nothing really, only that ATANHs and ATANs broken down as far as
possible typically exhibit fewer unwarranted discontinuities; for the integral at hand, however, both formulations agree on the real axis.
In fact, Euler himself gives both at the end of §13 in his "Integratio Formulae Differentialis Maxime Irrationalis quam Tamen per Logarithmos
et Arcus Circulares Expedire Licet" of 1777 (paper E689).
But fusing two ATANHs or two ATANs is easier than recognizing how they
can perhaps be split. So, returning two terms instead of one may be considered a service to the reader or user.
Martin.
On Sunday, January 16, 2022 at 10:38:19 PM UTC-10, nob...@nowhere.invalid wrote:
Albert Rich schrieb:
On Saturday, January 15, 2022 at 2:15:27 AM UTC-10, nob...@nowhere.invalid wrote:
Euler's elementary algebraic integral:
INT((1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)), x)
from paper E689 in the Euler Archive has the solution:
- ATANH(2*x*(x^4 - 6*x^2 + 1)^(1/4)/(SQRT(x^4 - 6*x^2 + 1) + x^2 + 1)) + ATAN(x/((x^4 - 6*x^2 + 1)^(1/4) + 1))
- ATAN(((x^4 - 6*x^2 + 1)^(1/4) - 1)/x)
which is real and continuous where the integrand is real and continuous. It appears that the awkward ATANH term cannot be
decomposed into simpler ones with real arguments.
What's wrong with combining the two arctan terms to get in
Mathematica notation:
ArcTan[(1 + x^2 - Sqrt[1 - 6*x^2 + x^4])/(2*x*(1 - 6*x^2 + x^4)^(1/4))] - ArcTanh[(2*x*(1 - 6*x^2 + x^4)^(1/4))/(1 + x^2 + Sqrt[1 - 6*x^2 + x^4])]
which is also real and continuous where the integrand is real and continuous.
Nothing really, only that ATANHs and ATANs broken down as far as
possible typically exhibit fewer unwarranted discontinuities; for
the integral at hand, however, both formulations agree on the real
axis.
In fact, Euler himself gives both at the end of §13 in his
"Integratio Formulae Differentialis Maxime Irrationalis quam Tamen
per Logarithmos et Arcus Circulares Expedire Licet" of 1777 (paper
E689).
But fusing two ATANHs or two ATANs is easier than recognizing how
they can perhaps be split. So, returning two terms instead of one
may be considered a service to the reader or user.
Yes, fusing two arctangents into one composite arctangent is easier
than splitting one arctangent into two simpler arctangents. This
seems analogous to the fact that multiplying two expressions is
easier than factoring one expression into two simpler expressions.
There are numerous techniques for factoring expressions (integers, polynomials, etc). Are there techniques for “factoring” a
function f(x) into two simpler functions g(x) and h(x) such that
f(x) = (g(x)+h(x)) / (1-g(x)*h(x))
where neither g(x) nor h(x) are constant wrt x?
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