Fyi;

A new build of the independent CAS integration test suite is now
running.

Please see

https://www.12000.org/my_notes/CAS_integration_tests/index.htm

Under summer_2021 link

40 test files have been processed so far (24,710 integrals)
out of total 71,994 integrals.

It will take about another 2 months to complete the full test suite.

Will update the web page as more tests are completed. There are a
total of 208 test files.

I am posting this now, in case someone finds any problems
so I can fix them early on.

This build includes the following 6 changes and improvements ============================================================
1. New versions of Maple, Mathematica, Maxima, Sympy, Giac/XCAS
and sagemath.

2. Added new CAS: mupad included in current Matlab 2021a symbolic toolbox.

3. Failed integrals for those CAS run via sagemath now show the
exception error message. This makes it possible to determine if
the failed integral was due to sagemath interface error or due
to the CAS itself generating the exception internally.

4. A new table that shows percentage of failed integrals due to
exception vs. timeout vs. normal failure for each test file
and each CAS.

5. Two new Bar charts that compares antiderivatives mean size and
mean time used for all CAS systems for each test file.

6. For Maxima, any integral where it asks an interactive question to
continue, will fail. Each such failure throws an exception. So
Maxima result should be higher than shown in these tests but
it was not possible to bypass this or somehow tell Maxima not
to use the interactive feature.

To minimize the effect of false failure due to Maxima asking interactive questions on some integrals which sagemath interface does not support,
this version of the build program adds an assume(var>0) on all the variables other than the integration variable. This resulted in many more integral
now passing for Maxima compared to last year build. But it was not possible to eliminate all interactive questions this way.

Version of CAS changes and additional notes ===========================================
This build uses the same Rubi 4.16.1 test files used in the last
build on April of last year.

The versions of CAS’s used in this build vs. Last year's are

(Current)Summer 2021 April 2020
==================== ==================
Rubi 4.16.1 Rubi 4.16.1
Mathematica 12.3 Mathematica 12.1
Maple 2021.1 Maple 2020
Maxima 5.44/sagemath 9.3 Maxima 5.43/sagemath 8.9
Fricas 1.3.6/sagemath 9.3 Fricas 1.3.6/sagemath 9.0
Giac/Xcas 1.7/sagemath 9.3 Giac/Xcas 1.5/sagemath 8.9
Sympy 1.8/Python 3.7.9 Sympy 1.5/Python 3.7.3
Mupad/Matlab 2021a N/A

Three CAS systems are run using sagemath. These are Fricas,
Maxima and Giac.

There are still few integrals that fail due to interface issues
between sagemath and these CAS systems, but now with the
new sagemath 9.3, there are less of these.

The integrals that fail due to an exception are flagged with
error code F(-2) and the error message will now make it more clear
if the exception was due to an interface problem, or due to
the CAS itself generating it. For an example

integrate(x^(5/2)*(b*x+2)^(1/2),x, algorithm="giac")

The exception is

"Exception raised: NotImplementedError >> Unable to parse Giac output:
Warning, choosing root of ....."

Shows this integrals is flagged failed F(-2) due to interface error, since sagemath does not parse warning messages returned along with the antiderivative. Question on this posted at

<https://ask.sagemath.org/question/57312/result-that-comes-with-warning-from-giac-integrator/>

Another example of interface failure in sagemath with giac is

var('C B A e d x')
integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2)/(e*x+d),x, algorithm="giac") NotImplementedError: Unable to parse Giac output:

Even though the above integral can be solved using giac directly. This is due to using the letter `e`. Question on this posted at

<https://ask.sagemath.org/question/57442/why-giac-integrate-result-fail-to-be-parsed-when-using-letter-e/>

It might be a good idea for future Rubi test suite not to use letter `e`
as it is read as exp(1) by giac and that seems why sagemath had problems
with using it.

There are three type of Failed code: F, F(-1) and F(-2).

The first is normal failure, i.e. when integral returns unevaluated.
F(-1) means failed due to a timeout. F(-2) means failed due to an
exception.

For mupad, and since Mathworks in Matlab 2021a have removed the mupad
language interface, it was not possible to have the 3 minutes timeout
added to the call to integrate as Matlab itself does not have a
timelimit() function build in.

Hence for mupad, no timelimit is used. For those integrals that do
hang, they are assigned F(-2) manually. In addition, there is no
grading done for mupad now. Hence if the integral does not fail,
it is automatically given a B grade as a place holder for now.

Hopefully in the future I will add a grading function for mupad
by porting one of the current grading functions to Matlab's symbolic
toolbox.

Current stats as of now are (40 files, 24710 integrals) ======================================================

System solved Failed
======= ======= =======
Rubi % 99.86 % 0.14
Mathematica % 99.47 % 0.53
Maple % 86.85 % 13.15
Fricas % 75.24 % 24.76
Giac % 66.29 % 33.71
Mupad % 60.92 % 39.08
Maxima % 60.40 % 39.60
Sympy % 54.29 % 45.71


Percentage of A grade:

System
=======
Rubi 99.23 %
Mathematica 75.33 %
Maple 59.89 %
Fricas 56.82 %
Giac 50.53 %
Maxima 52.46 %
Sympy 38.02 %
Mupad N/A as Not Graded

The above numbers will of course change as more test files are processed and the web pages updated.

Some observations
================
Maple remains one of the fastest CAS to complete the integrals, but with
the largest leaf size (since no simplify or any other post-processing) is
done of the result. I am sure if simplify is used afterword, the leaf
size will be much less.

But this would have to be done for all CAS system. So the test are
always run without using simplify on the antiderivative for any CAS.

Sympy 1.8 took the longest time to complete, as was the case in last
year's build. It does not look like that fast C++ symEngine talked about
before is yet integrated in it.

https://github.com/symengine/symengine

There are still some sagemath interface issues with Fricas/giac and Maxima which makes the result of these CAS system a little lower than the actual result if these we called directly. May be about 1-2% for Fricas and Giac,
and more for Maxima due to the interactive questions issues which
can't really be fixed using sagemath interface.

But in this version of the build the full exception message is now displayed, so it is possible to determine the cause of the exception and see which
case it is.

Questions found about these sgemath inerface issues are reported to
asksagemath and some have tickets on them. May be in next version of
sagemath 9.4 there will less such issues.

If a new version of FriCAS comes out before the tests are completed, will re-run Fricas to update.

--Nasser

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

"Nasser M. Abbasi" schrieb:

Fyi;

A new build of the independent CAS integration test suite is now
running.

Please see

https://www.12000.org/my_notes/CAS_integration_tests/index.htm

Under summer_2021 link

[...]

Of the 705 Timofeev integrals from the independent CAS integration test
suites:

<https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rech1.htm#x2-10001>

the number solved varies as follows over the last three test runs:

Rubi 705 705 705

Mathematica 700 705 705

Maple 646 647 655

Fricas 647 650 652

Giac 564 564 587

Maxima 564 542 564

Mupad --- --- 542

Sympy 396 422 430

While Rubi and Mathematica have achieved saturation, Maple, FriCAS,
Giac, and SymPy continue to improve, and Maxima just recovers from a
decline. As Nasser has noted earlier, FriCAS 1.3.6 fails on the
(elementary) Timofeev integral 5.89 (#425) which earlier versions could
already handle:

integrate(cos(x)*cos(2*x)*sin(3*x)/(4*sin(x)^2-5)^(5/2),x) ?= 0

But even if this can be corrected, Maple will remain ahead for now!

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 6/4/2021 4:46 PM, clicliclic@freenet.de wrote:

Of the 705 Timofeev integrals from the independent CAS integration test suites:

<https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rech1.htm#x2-10001>

the number solved varies as follows over the last three test runs:

Rubi 705 705 705

Mathematica 700 705 705

Maple 646 647 655

Fricas 647 650 652

Giac 564 564 587

Maxima 564 542 564

Mupad --- --- 542

Sympy 396 422 430

While Rubi and Mathematica have achieved saturation, Maple, FriCAS,
Giac, and SymPy continue to improve, and Maxima just recovers from a
decline. As Nasser has noted earlier, FriCAS 1.3.6 fails on the
(elementary) Timofeev integral 5.89 (#425) which earlier versions could already handle:

integrate(cos(x)*cos(2*x)*sin(3*x)/(4*sin(x)^2-5)^(5/2),x) ?= 0

But even if this can be corrected, Maple will remain ahead for now!

Martin.

Fyi, Maple 2021 improved, going from solving 647 to 655 due to this
(under Advanced math)

https://www.maplesoft.com/products/maple/new_features/

"Integration has been enhanced with improved algorithms for
indefinite integration, and the ability to easily specify which
integration method should be used and to compare the results from different methods"

--Nasser

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

"Nasser M. Abbasi" schrieb:

On 6/4/2021 4:46 PM, clicliclic@freenet.de wrote:

Of the 705 Timofeev integrals from the independent CAS integration
test suites:

<https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rech1.htm#x2-10001>

the number solved varies as follows over the last three test runs:

Rubi 705 705 705

Mathematica 700 705 705

Maple 646 647 655

Fricas 647 650 652

Giac 564 564 587

Maxima 564 542 564

Mupad --- --- 542

Sympy 396 422 430

While Rubi and Mathematica have achieved saturation, Maple, FriCAS,
Giac, and SymPy continue to improve, and Maxima just recovers from a decline. As Nasser has noted earlier, FriCAS 1.3.6 fails on the (elementary) Timofeev integral 5.89 (#425) which earlier versions
could already handle:

integrate(cos(x)*cos(2*x)*sin(3*x)/(4*sin(x)^2-5)^(5/2),x) ?= 0

But even if this can be corrected, Maple will remain ahead for now!

Fyi, Maple 2021 improved, going from solving 647 to 655 due to this
(under Advanced math)

https://www.maplesoft.com/products/maple/new_features/

"Integration has been enhanced with improved algorithms for
indefinite integration, and the ability to easily specify which
integration method should be used and to compare the results from
different methods"

Wow, Maple users are now given documented access to its Risch
integrator by way of the revolutionary syntax int(f(x), x, method =
risch).

Just noticed that the FriCAS evaluation of Timofeev 5.89 (#425) is
counted as solved in the latest test run. So FriCAS would have to
handle four more of the Timofeev integrals in order to pull ahead of
the latest Maple.

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 6/4/2021 5:56 PM, clicliclic@freenet.de wrote:

Wow, Maple users are now given documented access to its Risch
integrator by way of the revolutionary syntax int(f(x), x, method =
risch).

Just noticed that the FriCAS evaluation of Timofeev 5.89 (#425) is
counted as solved in the latest test run. So FriCAS would have to
handle four more of the Timofeev integrals in order to pull ahead of
the latest Maple.

Martin.

That is correct. There is no verification done for Fricas.

Only Rubi and Mathematica have verification implemented for
them now, so their results are checked each time, and if it does
not verify it will flag the result as "not verified" (there are few
of these found). But only for Rubi and Mathematica.

For all other CAS systems, if they finish in the time limit, or
do not generate exception, and the result is not the same as the
input or do not has not integrate in them (and other checks),
(i.e. it "solved it"), it is automatically counted as "solved".

The grading functions do not check if the anti-derivative is
correct or not. It only checks for conditions given in the report on
what gives grade A, B, C. And actually result 0 gives A grade,
since leaf size gives it this grade and it passes all other conditions
for the grading. (no complex numbers, no special functions, etc...)

The grading function is listed here

https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/grade_sagemath.py

In the future, hopefully a verification will be added to all other CASes.

Integrals that passes but fail verification are also listed separately for
each file to make it easy to find and investigate.

--Nasser

--- SoupGate-Win32 v1.05
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On 6/4/2021 5:56 PM, clicliclic@freenet.de wrote:

Wow, Maple users are now given documented access to its Risch
integrator by way of the revolutionary syntax int(f(x), x, method =
risch).

...

Martin.

FYI;

I just run all 50 failed Maple integrals in the Timofeev_Problems using
risch, but still none of them passed. I wanted to see if any will now pass.

It is possible to have Maple try all methods it has one by one automatically also, like this

intergand:=exp(x)*arcsin(tanh(x));
int(intergand,x, method=_RETURNVERBOSE);

[FAILS = ("gosper", "lookup", "derivativedivides", "default",
"norman", "trager", "meijerg", "risch", "elliptic")]

The above says all of these methods failed on this integral including risch.


--Nasser

--- SoupGate-Win32 v1.05
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"Nasser M. Abbasi" schrieb:

On 6/4/2021 5:56 PM, clicliclic@freenet.de wrote:

Wow, Maple users are now given documented access to its Risch
integrator by way of the revolutionary syntax int(f(x), x, method =
risch).

...

FYI;

I just run all 50 failed Maple integrals in the Timofeev_Problems
using risch, but still none of them passed. I wanted to see if any
will now pass.

This is suggestive of Maple's updated integrator trying its Risch
method anyway when called in your test run.

A related question: Is Maple now able to evaluate purely algebraic
elementary integrals like int(1/(x*(x^2 - 1)^(1/4)), x) or int((3 +
x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x) without requiring users to
rewrite the integrand in terms of root objects? (I am guessing here
that older versions fail on these two.)

Martin.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 6/8/2021 2:12 PM, clicliclic@freenet.de wrote:

A related question: Is Maple now able to evaluate purely algebraic
elementary integrals like int(1/(x*(x^2 - 1)^(1/4)), x) or int((3 +
x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x) without requiring users to rewrite the integrand in terms of root objects?

Using Maple 2021.1, the first one gives result using "trager" and "meijerg" methods. The second integral you showed gives result using "trager" only.

Both outputs have RootOf when using "trager", and the output using
"trager" method have hypergeom and GAMMA but no RootOf.

------------------------
lprint(int(1/(x*(x^2 - 1)^(1/4)), x, method=_RETURNVERBOSE))

["trager" = -1/2*RootOf(_Z^4+1)*ln((2*(x^2-1)^(1/2)*RootOf(_Z^4+1)^3+2*(x^2-1)^ (1/4)*RootOf(_Z^4+1)^2-RootOf(_Z^4+1)*x^2-2*(x^2-1)^(3/4)+2*RootOf(_Z^4+1))/x^2 )-1/2*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^2-2*RootOf(_Z^4+1)^3+2*(x^2-1)^( 1/4)*RootOf(_Z^4+1)^2-2*(x^2-1)^(1/2)*RootOf(_Z^4+1)+2*(x^2-1)^(3/4))/x^2),

"meijerg" = 1/4/Pi*2^(1/2)*GAMMA(3/4)/signum(x^2-1)^(1/4)*(-signum(x^2-1))^(1/4 )*((-3*ln(2)-1/2*Pi+2*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4)+1/4*Pi*2^(1/2)/GAMMA(3/ 4)*x^2*hypergeom([1, 1, 5/4],[2, 2],x^2)), FAILS = ("gosper", "lookup", "derivativedivides", "default", "norman", "risch", "elliptic")] -------------------------


lprint(int((3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x,method=_RETURNVERBOSE))

["trager" = 1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(1/2)*x^4- RootOf(_Z^2+1)^3*x^6-RootOf(_Z^2+1)^2*(x^4+6*x^2+1)^(3/4)*x^3+RootOf(_Z^2+1)^3* (x^4+6*x^2+1)^(1/2)*x^2-5*RootOf(_Z^2+1)^3*x^4-(x^4+6*x^2+1)^(1/4)*x^5-RootOf( _Z^2+1)*(x^4+6*x^2+1)^(1/2)*x^2+RootOf(_Z^2+1)*x^4+(x^4+6*x^2+1)^(3/4)*x-4*(x^4 +6*x^2+1)^(1/4)*x^3-RootOf(_Z^2+1)*(x^4+6*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*( x^4+6*x^2+1)^(1/4)*x)/(RootOf(_Z^2+1)*x-1)^2/(RootOf(_Z^2+1)*x+1)^2)+1/2*ln(((x ^4+6*x^2+1)^(3/4)*x+(x^4+6*x^2+1)^(1/2)*x^2+(x^4+6*x^2+1)^(1/4)*x^3+x^4+(x^4+6* x^2+1)^(1/2)+3*(x^4+6*x^2+1)^(1/4)*x+5*x^2)/(x^2+1)), FAILS = ("gosper", "lookup", "derivativedivides", "default", "norman", "meijerg", "risch", "elliptic")]

--------------------------

(I am guessing here
that older versions fail on these two.)

Martin.

Actually, only your second integral could not be solved in Maple 2020.
The first one could be solved as is, giving the same output as shown
above using the "meijerg" (using hypergeom)

--Nasser

--- SoupGate-Win32 v1.05
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"Nasser M. Abbasi" schrieb:

On 6/8/2021 2:12 PM, clicliclic@freenet.de wrote:

A related question: Is Maple now able to evaluate purely algebraic elementary integrals like int(1/(x*(x^2 - 1)^(1/4)), x) or int((3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x) without requiring users to rewrite the integrand in terms of root objects?

Using Maple 2021.1, the first one gives result using "trager" and "meijerg" methods. The second integral you showed gives result using "trager" only.

Both outputs have RootOf when using "trager", and the output using
"trager" method have hypergeom and GAMMA but no RootOf.

------------------------
lprint(int(1/(x*(x^2 - 1)^(1/4)), x, method=_RETURNVERBOSE))

["trager" = -1/2*RootOf(_Z^4+1)*ln((2*(x^2-1)^(1/2)*RootOf(_Z^4+1)^3+2*(x^2-1)^
(1/4)*RootOf(_Z^4+1)^2-RootOf(_Z^4+1)*x^2-2*(x^2-1)^(3/4)+2*RootOf(_Z^4+1))/x^2
)-1/2*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^2-2*RootOf(_Z^4+1)^3+2*(x^2-1)^(
1/4)*RootOf(_Z^4+1)^2-2*(x^2-1)^(1/2)*RootOf(_Z^4+1)+2*(x^2-1)^(3/4))/x^2),

"meijerg" = 1/4/Pi*2^(1/2)*GAMMA(3/4)/signum(x^2-1)^(1/4)*(-signum(x^2-1))^(1/4
)*((-3*ln(2)-1/2*Pi+2*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4)+1/4*Pi*2^(1/2)/GAMMA(3/
4)*x^2*hypergeom([1, 1, 5/4],[2, 2],x^2)), FAILS = ("gosper", "lookup", "derivativedivides", "default", "norman", "risch", "elliptic")] -------------------------

lprint(int((3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x,method=_RETURNVERBOSE))

["trager" = 1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(1/2)*x^4- RootOf(_Z^2+1)^3*x^6-RootOf(_Z^2+1)^2*(x^4+6*x^2+1)^(3/4)*x^3+RootOf(_Z^2+1)^3*
(x^4+6*x^2+1)^(1/2)*x^2-5*RootOf(_Z^2+1)^3*x^4-(x^4+6*x^2+1)^(1/4)*x^5-RootOf(
_Z^2+1)*(x^4+6*x^2+1)^(1/2)*x^2+RootOf(_Z^2+1)*x^4+(x^4+6*x^2+1)^(3/4)*x-4*(x^4
+6*x^2+1)^(1/4)*x^3-RootOf(_Z^2+1)*(x^4+6*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*(
x^4+6*x^2+1)^(1/4)*x)/(RootOf(_Z^2+1)*x-1)^2/(RootOf(_Z^2+1)*x+1)^2)+1/2*ln(((x
^4+6*x^2+1)^(3/4)*x+(x^4+6*x^2+1)^(1/2)*x^2+(x^4+6*x^2+1)^(1/4)*x^3+x^4+(x^4+6*
x^2+1)^(1/2)+3*(x^4+6*x^2+1)^(1/4)*x+5*x^2)/(x^2+1)), FAILS = ("gosper", "lookup", "derivativedivides", "default", "norman", "meijerg", "risch", "elliptic")]

--------------------------

(I am guessing here
that older versions fail on these two.)

Actually, only your second integral could not be solved in Maple 2020.
The first one could be solved as is, giving the same output as shown
above using the "meijerg" (using hypergeom)

Do I understand this correctly?

When a user does not explicitly specify an integration method (nor does
convert the integrand to RootOf), the algebraic (3 + x^2)/((1 + x^2)*
(1 + 6*x^2 + x^4)^(1/4)) could not be integrated by Maple 2020, but can
be integrated by Maple 2021.1.

This algebraic would thus illustrate the way in which Maple has been
improved its performance on the test suites.

Martin.

--- SoupGate-Win32 v1.05
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On 6/8/2021 5:32 PM, clicliclic@freenet.de wrote:

"Nasser M. Abbasi" schrieb:

On 6/8/2021 2:12 PM, clicliclic@freenet.de wrote:

A related question: Is Maple now able to evaluate purely algebraic
elementary integrals like int(1/(x*(x^2 - 1)^(1/4)), x) or int((3 +
x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x) without requiring users to
rewrite the integrand in terms of root objects?

Using Maple 2021.1, the first one gives result using "trager" and "meijerg" >> methods. The second integral you showed gives result using "trager" only.

Both outputs have RootOf when using "trager", and the output using
"trager" method have hypergeom and GAMMA but no RootOf.

------------------------
lprint(int(1/(x*(x^2 - 1)^(1/4)), x, method=_RETURNVERBOSE))

["trager" = -1/2*RootOf(_Z^4+1)*ln((2*(x^2-1)^(1/2)*RootOf(_Z^4+1)^3+2*(x^2-1)^
(1/4)*RootOf(_Z^4+1)^2-RootOf(_Z^4+1)*x^2-2*(x^2-1)^(3/4)+2*RootOf(_Z^4+1))/x^2
)-1/2*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^2-2*RootOf(_Z^4+1)^3+2*(x^2-1)^(
1/4)*RootOf(_Z^4+1)^2-2*(x^2-1)^(1/2)*RootOf(_Z^4+1)+2*(x^2-1)^(3/4))/x^2), >>
"meijerg" = 1/4/Pi*2^(1/2)*GAMMA(3/4)/signum(x^2-1)^(1/4)*(-signum(x^2-1))^(1/4
)*((-3*ln(2)-1/2*Pi+2*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4)+1/4*Pi*2^(1/2)/GAMMA(3/
4)*x^2*hypergeom([1, 1, 5/4],[2, 2],x^2)), FAILS = ("gosper", "lookup",
"derivativedivides", "default", "norman", "risch", "elliptic")]
-------------------------

lprint(int((3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x,method=_RETURNVERBOSE))

["trager" = 1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(1/2)*x^4- >> RootOf(_Z^2+1)^3*x^6-RootOf(_Z^2+1)^2*(x^4+6*x^2+1)^(3/4)*x^3+RootOf(_Z^2+1)^3*
(x^4+6*x^2+1)^(1/2)*x^2-5*RootOf(_Z^2+1)^3*x^4-(x^4+6*x^2+1)^(1/4)*x^5-RootOf(
_Z^2+1)*(x^4+6*x^2+1)^(1/2)*x^2+RootOf(_Z^2+1)*x^4+(x^4+6*x^2+1)^(3/4)*x-4*(x^4
+6*x^2+1)^(1/4)*x^3-RootOf(_Z^2+1)*(x^4+6*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*(
x^4+6*x^2+1)^(1/4)*x)/(RootOf(_Z^2+1)*x-1)^2/(RootOf(_Z^2+1)*x+1)^2)+1/2*ln(((x
^4+6*x^2+1)^(3/4)*x+(x^4+6*x^2+1)^(1/2)*x^2+(x^4+6*x^2+1)^(1/4)*x^3+x^4+(x^4+6*
x^2+1)^(1/2)+3*(x^4+6*x^2+1)^(1/4)*x+5*x^2)/(x^2+1)), FAILS = ("gosper",
"lookup", "derivativedivides", "default", "norman", "meijerg", "risch",
"elliptic")]

--------------------------

(I am guessing here
that older versions fail on these two.)

Actually, only your second integral could not be solved in Maple 2020.
The first one could be solved as is, giving the same output as shown
above using the "meijerg" (using hypergeom)

Do I understand this correctly?

When a user does not explicitly specify an integration method (nor does convert the integrand to RootOf), the algebraic (3 + x^2)/((1 + x^2)*
(1 + 6*x^2 + x^4)^(1/4)) could not be integrated by Maple 2020, but can
be integrated by Maple 2021.1.

This algebraic would thus illustrate the way in which Maple has been
improved its performance on the test suites.

Martin.

Yes, that is correct. Your integral above could not be solved in
Maple 2020 but only in 2021:

----------------------------
interface(version)
Standard Worksheet Interface, Maple 2020.2, Windows 10,
November 11 2020 Build ID 1502365

lprint(int( (3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)),x)) int((x^2+3)/(x^2+1)/(x^4+6*x^2+1)^(1/4),x)
--------------------------

interface(version)

Standard Worksheet Interface, Maple 2021.1, Windows 10, May 19
2021 Build ID 1539851

lprint(int( (3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)),x)) 1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(1/2)*x^4-RootOf(_Z^2+1)^ 3*x^6-RootOf(_Z^2+1)^2*(x^4+6*x^2+1)^(3/4)*x^3+RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^( 1/2)*x^2-5*RootOf(_Z^2+1)^3*x^4-(x^4+6*x^2+1)^(1/4)*x^5-RootOf(_Z^2+1)*(x^4+6*x ^2+1)^(1/2)*x^2+RootOf(_Z^2+1)*x^4+(x^4+6*x^2+1)^(3/4)*x-4*(x^4+6*x^2+1)^(1/4)* x^3-RootOf(_Z^2+1)*(x^4+6*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*(x^4+6*x^2+1)^(1/ 4)*x)/(RootOf(_Z^2+1)*x-1)^2/(RootOf(_Z^2+1)*x+1)^2)+1/2*ln(((x^4+6*x^2+1)^(3/4 )*x+(x^4+6*x^2+1)^(1/2)*x^2+(x^4+6*x^2+1)^(1/4)*x^3+x^4+(x^4+6*x^2+1)^(1/2)+3*( x^4+6*x^2+1)^(1/4)*x+5*x^2)/(x^2+1))
------------------------------

it is part of that improvement they mentioned at Maplesoft website
for int in 2021 version.

--Nasser

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"Nasser M. Abbasi" schrieb:

On 6/8/2021 5:32 PM, clicliclic@freenet.de wrote:

"Nasser M. Abbasi" schrieb:

On 6/8/2021 2:12 PM, clicliclic@freenet.de wrote:

A related question: Is Maple now able to evaluate purely algebraic
elementary integrals like int(1/(x*(x^2 - 1)^(1/4)), x) or
int((3 > >>> + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x)
without requiring users to rewrite the integrand in terms of root
objects?

Using Maple 2021.1, the first one gives result using "trager" and "meijerg"
methods. The second integral you showed gives result using "trager" only. >>
Both outputs have RootOf when using "trager", and the output using
"trager" method have hypergeom and GAMMA but no RootOf.

------------------------
lprint(int(1/(x*(x^2 - 1)^(1/4)), x, method=_RETURNVERBOSE))

["trager" = -1/2*RootOf(_Z^4+1)*ln((2*(x^2-1)^(1/2)*RootOf(_Z^4+1)^3+2*(x^2-1)^
(1/4)*RootOf(_Z^4+1)^2-RootOf(_Z^4+1)*x^2-2*(x^2-1)^(3/4)+2*RootOf(_Z^4+1))/x^2
)-1/2*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^3*x^2-2*RootOf(_Z^4+1)^3+2*(x^2-1)^(
1/4)*RootOf(_Z^4+1)^2-2*(x^2-1)^(1/2)*RootOf(_Z^4+1)+2*(x^2-1)^(3/4))/x^2),

"meijerg" = 1/4/Pi*2^(1/2)*GAMMA(3/4)/signum(x^2-1)^(1/4)*(-signum(x^2-1))^(1/4
)*((-3*ln(2)-1/2*Pi+2*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4)+1/4*Pi*2^(1/2)/GAMMA(3/
4)*x^2*hypergeom([1, 1, 5/4],[2, 2],x^2)), FAILS = ("gosper", "lookup",
"derivativedivides", "default", "norman", "risch", "elliptic")]
-------------------------

lprint(int((3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)), x,method=_RETURNVERBOSE))

["trager" = 1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(1/2)*x^4-
RootOf(_Z^2+1)^3*x^6-RootOf(_Z^2+1)^2*(x^4+6*x^2+1)^(3/4)*x^3+RootOf(_Z^2+1)^3*
(x^4+6*x^2+1)^(1/2)*x^2-5*RootOf(_Z^2+1)^3*x^4-(x^4+6*x^2+1)^(1/4)*x^5-RootOf(
_Z^2+1)*(x^4+6*x^2+1)^(1/2)*x^2+RootOf(_Z^2+1)*x^4+(x^4+6*x^2+1)^(3/4)*x-4*(x^4
+6*x^2+1)^(1/4)*x^3-RootOf(_Z^2+1)*(x^4+6*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*(
x^4+6*x^2+1)^(1/4)*x)/(RootOf(_Z^2+1)*x-1)^2/(RootOf(_Z^2+1)*x+1)^2)+1/2*ln(((x
^4+6*x^2+1)^(3/4)*x+(x^4+6*x^2+1)^(1/2)*x^2+(x^4+6*x^2+1)^(1/4)*x^3+x^4+(x^4+6*
x^2+1)^(1/2)+3*(x^4+6*x^2+1)^(1/4)*x+5*x^2)/(x^2+1)), FAILS = ("gosper", >> "lookup", "derivativedivides", "default", "norman", "meijerg", "risch",
"elliptic")]

--------------------------

(I am guessing here
that older versions fail on these two.)

Actually, only your second integral could not be solved in Maple 2020.
The first one could be solved as is, giving the same output as shown
above using the "meijerg" (using hypergeom)

Do I understand this correctly?

When a user does not explicitly specify an integration method (nor does convert the integrand to RootOf), the algebraic (3 + x^2)/((1 + x^2)*
(1 + 6*x^2 + x^4)^(1/4)) could not be integrated by Maple 2020, but can
be integrated by Maple 2021.1.

This algebraic would thus illustrate the way in which Maple has been improved its performance on the test suites.

Yes, that is correct. Your integral above could not be solved in
Maple 2020 but only in 2021:

----------------------------
interface(version)
Standard Worksheet Interface, Maple 2020.2, Windows 10,
November 11 2020 Build ID 1502365

lprint(int( (3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)),x)) int((x^2+3)/(x^2+1)/(x^4+6*x^2+1)^(1/4),x)
--------------------------

interface(version)

Standard Worksheet Interface, Maple 2021.1, Windows 10, May 19
2021 Build ID 1539851

lprint(int( (3 + x^2)/((1 + x^2)*(1 + 6*x^2 + x^4)^(1/4)),x)) 1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(1/2)*x^4-RootOf(_Z^2+1)^
3*x^6-RootOf(_Z^2+1)^2*(x^4+6*x^2+1)^(3/4)*x^3+RootOf(_Z^2+1)^3*(x^4+6*x^2+1)^(
1/2)*x^2-5*RootOf(_Z^2+1)^3*x^4-(x^4+6*x^2+1)^(1/4)*x^5-RootOf(_Z^2+1)*(x^4+6*x
^2+1)^(1/2)*x^2+RootOf(_Z^2+1)*x^4+(x^4+6*x^2+1)^(3/4)*x-4*(x^4+6*x^2+1)^(1/4)*
x^3-RootOf(_Z^2+1)*(x^4+6*x^2+1)^(1/2)+5*RootOf(_Z^2+1)*x^2-3*(x^4+6*x^2+1)^(1/
4)*x)/(RootOf(_Z^2+1)*x-1)^2/(RootOf(_Z^2+1)*x+1)^2)+1/2*ln(((x^4+6*x^2+1)^(3/4
)*x+(x^4+6*x^2+1)^(1/2)*x^2+(x^4+6*x^2+1)^(1/4)*x^3+x^4+(x^4+6*x^2+1)^(1/2)+3*(
x^4+6*x^2+1)^(1/4)*x+5*x^2)/(x^2+1))
------------------------------

it is part of that improvement they mentioned at Maplesoft website
for int in 2021 version.

Thanks.

A user of Maple 2020 (or earlier) needed to convert the radicals in
his integrand to RootOf() first - a secret unearthed by Sam Blake last
year, who also helped cracking the Zodiac killer's 340-symbols cypher:

<https://en.wikipedia.org/wiki/Zodiac_Killer>

Martin.

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On 6/4/2021 12:00 AM, Nasser M. Abbasi wrote:

Fyi;

A new build of the independent CAS integration test suite is now
running.

Please see

https://www.12000.org/my_notes/CAS_integration_tests/index.htm

Under summer_2021 link

40 test files have been processed so far (24,710 integrals)
out of total 71,994 integrals.

FYI,

The following is the half way update of CAS integration tests build.

37,876 integrals (81 files) have now been processed out of total
71,994 (208 files).

This uses the same integrals used in last year's build of April 2020
which is Rubi's 4.16.1 test integrals maintained by Albert Rich.

The zip files containing the raw integrals can be downloaded from
Rubi's web site https://rulebasedintegration.org/testProblems.html

I rerun all of Mupad tests again, after finding a way to add a timeout
of 3 minutes using Matlab's parallel toolbox. So now mupad has
the same timeout as all the other CAS system.

The following is the current result as of now, showing percent solved,
and percent of A grade and the average time per integral in seconds
and mean normalized (relative to optimal) of result leaf size.

Fricas 1.3.7, Giac/Xcas 1.7 and Maxima 5.44 were all called
via sagemath 9.3 on Linux VBox running on top of windows 10.
Sympy was run using Ubuntu running under windows 10 Linux subsystem
and the rest were run directly on windows 10.

system % solved %A time(sec) leaf size ================== ========== ======== ========= ===========
Rubi 4.16.1 99.69 98.71 0.23 1
Mathematica 12.3 98.89 75.09 0.8 1.5
Maple 2021.1 83.27 54.99 0.36 8
Fricas 1.3.7 73.56 54.69 1.66 2.29
Giac/Xcas 1.7 61.54 46.48 0.97 1.94
Maxima 5.44 57.54 48.06 1.17 1.45
Mupad/Matlab 2021a 58.43 N/A 2.72 3.08
Sympy 1.8/Python 3.8.8 45.65 33.21 10.15 2.8

Mupad is not currently graded. B grade is given automatically
as a place holder for any integral that complete in time.

I will make a final update when the build completes in about another
month or may be more.

If you spot any problems in the results or have questions please feel
free to let me know.

--Nasser

--- SoupGate-Win32 v1.05
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"Nasser M. Abbasi" schrieb:

On 6/4/2021 12:00 AM, Nasser M. Abbasi wrote:

Fyi;

A new build of the independent CAS integration test suite is now
running.

Please see

https://www.12000.org/my_notes/CAS_integration_tests/index.htm

Under summer_2021 link

40 test files have been processed so far (24,710 integrals)
out of total 71,994 integrals.

FYI,

The following is the half way update of CAS integration tests build.

37,876 integrals (81 files) have now been processed out of total
71,994 (208 files).

This uses the same integrals used in last year's build of April 2020
which is Rubi's 4.16.1 test integrals maintained by Albert Rich.

The zip files containing the raw integrals can be downloaded from
Rubi's web site https://rulebasedintegration.org/testProblems.html

I rerun all of Mupad tests again, after finding a way to add a timeout
of 3 minutes using Matlab's parallel toolbox. So now mupad has
the same timeout as all the other CAS system.

The following is the current result as of now, showing percent solved,
and percent of A grade and the average time per integral in seconds
and mean normalized (relative to optimal) of result leaf size.

Fricas 1.3.7, Giac/Xcas 1.7 and Maxima 5.44 were all called
via sagemath 9.3 on Linux VBox running on top of windows 10.
Sympy was run using Ubuntu running under windows 10 Linux subsystem
and the rest were run directly on windows 10.

system % solved %A time(sec) leaf size ================== ========== ======== ========= ===========
Rubi 4.16.1 99.69 98.71 0.23 1
Mathematica 12.3 98.89 75.09 0.8 1.5
Maple 2021.1 83.27 54.99 0.36 8
Fricas 1.3.7 73.56 54.69 1.66 2.29
Giac/Xcas 1.7 61.54 46.48 0.97 1.94
Maxima 5.44 57.54 48.06 1.17 1.45
Mupad/Matlab 2021a 58.43 N/A 2.72 3.08
Sympy 1.8/Python 3.8.8 45.65 33.21 10.15 2.8

Mupad is not currently graded. B grade is given automatically
as a place holder for any integral that complete in time.

I will make a final update when the build completes in about another
month or may be more.

If you spot any problems in the results or have questions please feel
free to let me know.

FriCAS 1.3.7 manages to solve just one more (653 versus 652) of
Timofeev's integration problems compared to version 1.3.6, with both
versions called via sagemath 9.3. But which one is the newly solved
integral? And how is Timofeev's problem 5.89 (#425) handled now?

Martin.

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On 7/1/2021 7:53 AM, clicliclic@freenet.de wrote:

FriCAS 1.3.7 manages to solve just one more (653 versus 652) of
Timofeev's integration problems compared to version 1.3.6, with both
versions called via sagemath 9.3. But which one is the newly solved
integral? And how is Timofeev's problem 5.89 (#425) handled now?

Martin.

For the first question, the program now does not compare or
produce a regression report for each CAS to compare its current
result to the last build. That is something that can be added in
the future, but the problem is that once Albert Rich in the future
adds or removes integrals from the input raw files used, it will
break the regression/comparison since the integrals now for each build
will be different, that is why I did not want to do it.

But it is fairly easy to see this difference manually per each file, as
the build program produces a list of all failed integrals for test file.

Each test file, under "detailed summary tables of results"
has a list of integrals per grade.

Looking at the Failed list, and by opening two browser windows,
one for Fricas 1.3.6 and window for 1.3.7 and putting them side
by side one can quickly see the difference.

In this case, I see that it was #413 Here is screen shot

https://www.12000.org/tmp/07012021/diff_fricas.png

integrate((sin(x)^5/cos(x))^(1/2),x, algorithm="fricas")

Output too large to paste here. But here is the link. it got grade B

https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rese413.htm#x417-4410003.413

In Fricas 1.3.6, it timedout.

For your second question about #425, in 1.3.7 I see it still gives zero
as in 1.3.6:

https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rese425.htm#x429-4530003.425

integrate(cos(x)*cos(2*x)*sin(3*x)/(-5+4*sin(x)^2)^(5/2),x, algorithm="fricas")
0

The optimal should be

-1/4/(-5+4*sin(x)^2)^(3/2)-5/8/(-5+4*sin(x)^2)^(1/2)+1/8*(-5+4*sin(x)^2)^(1/2)

--Nasser

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"Nasser M. Abbasi" schrieb:

On 7/1/2021 7:53 AM, clicliclic@freenet.de wrote:

FriCAS 1.3.7 manages to solve just one more (653 versus 652) of
Timofeev's integration problems compared to version 1.3.6, with both versions called via sagemath 9.3. But which one is the newly solved integral? And how is Timofeev's problem 5.89 (#425) handled now?

For the first question, the program now does not compare or
produce a regression report for each CAS to compare its current
result to the last build. That is something that can be added in
the future, but the problem is that once Albert Rich in the future
adds or removes integrals from the input raw files used, it will
break the regression/comparison since the integrals now for each build
will be different, that is why I did not want to do it.

But it is fairly easy to see this difference manually per each file, as
the build program produces a list of all failed integrals for test file.

Each test file, under "detailed summary tables of results"
has a list of integrals per grade.

Looking at the Failed list, and by opening two browser windows,
one for Fricas 1.3.6 and window for 1.3.7 and putting them side
by side one can quickly see the difference.

In this case, I see that it was #413 Here is screen shot

https://www.12000.org/tmp/07012021/diff_fricas.png

integrate((sin(x)^5/cos(x))^(1/2),x, algorithm="fricas")

Output too large to paste here. But here is the link. it got grade B

https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rese413.htm#x417-4410003.413

In Fricas 1.3.6, it timedout.

For your second question about #425, in 1.3.7 I see it still gives zero
as in 1.3.6:

https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/test_cases/0_Independent_test_suites/Timofeev_Problems/rese425.htm#x429-4530003.425

integrate(cos(x)*cos(2*x)*sin(3*x)/(-5+4*sin(x)^2)^(5/2),x, algorithm="fricas")
0

The optimal should be

-1/4/(-5+4*sin(x)^2)^(3/2)-5/8/(-5+4*sin(x)^2)^(1/2)+1/8*(-5+4*sin(x)^2)^(1/2)

Thanks.

More improvements are needed for the FriCAS integrator to beat Maple
again in the future.

Martin.

--- SoupGate-Win32 v1.05
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On 6/4/2021 12:00 AM, Nasser M. Abbasi wrote:

Fyi;

A new build of the independent CAS integration test suite is now
running.

FYI;

Summer 2021 CAS integration tests is now complete.

https://www.12000.org/my_notes/CAS_integration_tests/index.htm https://www.12000.org/my_notes/CAS_integration_tests/reports/summer_2021/index.htm

Below is a summary that compares current build result with last years.

Versions used:
===============
April 2020 summer 2021
====================== ============================
Rubi 4.16.1 4.16.1
Mathematica 12.1 on windows 10 12.3.1 on windows 10
Maple 2020 on windows 10 2021.1 on windows 10
Fricas 1.3.6 (via sagemath 9.0) 1.3.7 (via sagemath 9.3)
Mupad N/A Matlab 2021a/windows 10
Giac/XCas 1.5 (via sagemath 8.9) 1.7 (via sagemath 9.3)
Maxima 5.43 (via sagemath 8.9) 5.44 (via sagemath 9.3)
Sympy 1.5/Python 3.7.3 1.8/Python 3.8.8


Percentage solved:
===================
April 2020 summer 2021 change
=========== ============= ========
Rubi 99.52 % 99.52 % 0.0 %
Mathematica 98.35 % 98.39 % +0.04 %
Maple 83.46 % 83.58 % +0.14 %
Fricas 68.07 % 68.7 % +0.63 %
Mupad N/A 52.55 % N/A
Giac/XCas 52.51 % 52.43 % -0.08 %
Maxima 43.03 % 52.72 % +9.69 %
Sympy 32.41 % 34.47 % +2.06 %


A grade Percentage:
===================
April 2020 summer 2021 change
=========== ============= ========
Rubi 98.89 % 98.89 % 0.0 %
Mathematica 74.67 % 74.75 % +0.08 %
Maple 52.8 % 52.77 % -0.03 %
Fricas 48.52 % 48.82 % +0.3 %
Giac/XCas 39 % 38.1 % -0.9 %
Maxima 33.24 % 41.43 % +8.19 %
Sympy 25.08 % 26.94 % +1.86 %
Mupad N/A N/A N/A

Mean time to solve one integral (sec)
======================================
April 2020 summer 2021
=========== =============
Rubi 0.28 0.28
Mathematica 1.77 1.84
Maple 0.46 0.79
Fricas 2.81 1.56
Giac/XCas 1.53 1.05
Maxima 1.34 0.92
Sympy 9.65 8.89
Mupad N/A 2.73


Normalized mean leaf size of result
======================================
April 2020 summer 2021
=========== =============
Rubi 1 1
Mathematica 2.8 2.8
Maple 743.6 746.4
Fricas 6.76 2.78
Giac/XCas 2.55 2
Maxima 2.46 1.76
Sympy 2.53 2.77
Mupad N/A 3.35

Some observations
==================
1. There is some regression in GIAC result in terms of percentage
solved. It was the only CAS whose percentage solved reduced from
last build.

This could be due to sagemath interface issue or due to Giac itself.
Hard to know now. One needs to go over the failed integrals and
find out why.

During running the tests, I could see many integrals
would crash the giac process and sagemmath will restart giac
automatically each time before going to the next integral to
process.

2. Most improved result came from Maxima. Part of this because in
summer 2021 build, assumptions were added that all symbols in
integrand (other than integration variable) are now assumed
positive, which made Maxima not ask as many questions as before
about if something is positive, negative or zero, which would have
caused the integration to fail before since interactive integration
does not work via sagemath interface.

There are still integrals that fail due to this interactive format,
but these are not possible to eliminate completely.

3. In terms of speed, Fricas improved the most among all CAS systems.
1.3.7 is more than twice as fast now as 1.3.7 (on same PC).
Also Maxima and Giac speed improved. This could be due to improvement
in the sagemath interface in 9.3.

Also Fricas leaf size has improved the most, going from
6.76 down to 2.78.

Any problems please let me know.

--Nasser

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