Computer Algebra Independent Integration Tests

These reports and the web pages themselves were written in LaTeX using TeXLive distribution and compiled to HTML using TeX4ht.

1 FULL CAS integration tests

These reports show the result of using full CAS test suite on diﬀerent CAS’s at same time. This compares how one CAS perform against another.

2 LITE CAS integration tests

These reports show the result of using same test suite on diﬀerent versions of same CAS. Only Maple, Mathematica and Rubi are used. This is for regression testing to ﬁnd how diﬀerent version of the same CAS improved over the years. The test suite used is the lite version of the full CAS intergation tests. It contains smaller number of integrals compared to the full test suite used above.

3 Note on build system

3.1 My PC during running the tests

This below shows example of CAS suddenly consuming all RAM in PC, and I had to terminate the process, since it did not time out as instructed, and just hanged.

4 my cheat sheet notes

4.1 Fricas

in case of error while installing the package, try sudo pacman -Rs sagemath and then repeat the previous command

If copy paste does not work between windows and Linux VBox, try VBoxClient --clipboard

\begin{axiom}
setSimplifyDenomsFlag(true)
integrate(sin(x),x)
\end{axiom}

To installl on Linux Manjaro, type yay -S fricas . Do not use the Manjaro GUI package manager, it always gives build error for some reason. Make sure to install yay ﬁrst using GUI manager.

Make sure when installing Linux on VBox, to add second CPU to conﬁguration of VBox. i.e. do not use ONE CPU (which is default). There is a bug in fricas installation which makes it fail due to use of multi-threading if only one CPU exist. This should be ﬁxed in future, but just in case.

#install sbcl
#sudo apt-get install sbcl #do not use this. bug
sudo apt-get install clisp #This lisp works
sudo apt-get install dvipng
sudo apt-get install auctex
bzip2 -dk fricas-1.3.2-full.tar.bz2
tar -xf fricas-1.3.2-full.tar

make clean
fricas-1.3.2>./configure --with-lisp=/usr/bin/clisp
checking build system type... x86_64-linux-gnu
checking host system type... x86_64-linux-gnu
checking target system type... x86_64-linux-gnu
checking for make... make
checking for gcc... gcc
checking whether the C compiler works... yes
checking for C compiler default output file name... a.out
checking for suffix of executables...
checking whether we are cross compiling... no
checking for suffix of object files... o
checking whether we are using the GNU C compiler... yes
checking whether gcc accepts -g... yes
checking for gcc option to accept ISO C89... none needed
checking for a BSD-compatible install... /usr/bin/install -c
checking for touch... touch
checking for mktemp... mktemp
checking for gawk... gawk
checking for gtar... no
checking for tar... tar
checking for ranlib... ranlib
checking for ar... ar
checking for latex... /usr/local/texlive/2017/bin/x86_64-linux/latex
checking for makeindex... makeindex
checking PREGENERATED... "/home/me/data/fricas/fricas-1.3.2/pre-generated"
checking for sbcl... /usr/bin/sbcl
checking Lisp implementation... This is SBCL 1.3.1.debian, an implementation of ANSI Common Lisp.
More information about SBCL is available at <http://www.sbcl.org/>.

SBCL is free software, provided as is, with absolutely no warranty.
It is mostly in the public domain; some portions are provided under
BSD-style licenses. See the CREDITS and COPYING files in the
distribution for more information.
* sbcl
checking how to run the C preprocessor... gcc -E
checking for grep that handles long lines and -e... /bin/grep
checking for egrep... /bin/grep -E
checking for ANSI C header files... yes
checking for sys/types.h... yes
checking for sys/stat.h... yes
checking for stdlib.h... yes
checking for string.h... yes
checking for memory.h... yes
checking for strings.h... yes
checking for inttypes.h... yes
checking for stdint.h... yes
checking for unistd.h... yes
fasl
checking dirent.h usability... yes
checking dirent.h presence... yes
checking for dirent.h... yes
checking whether closedir is declared... yes
checking whether opendir is declared... yes
checking whether readdir is declared... yes
checking whether dirfd is declared... yes
checking whether fchdir is declared... yes
checking signal.h usability... yes
checking signal.h presence... yes
checking for signal.h... yes
checking whether sigaction is declared... yes
checking for sys/stat.h... (cached) yes
checking for unistd.h... (cached) yes
checking whether getuid is declared... yes
checking whether geteuid is declared... yes
checking whether getgid is declared... yes
checking whether getegid is declared... yes
checking whether kill is declared... yes
checking sys/socket.h usability... yes
checking sys/socket.h presence... yes
checking for sys/socket.h... yes
checking util.h usability... no
checking util.h presence... no
checking for util.h... no
checking pty.h usability... yes
checking pty.h presence... yes
checking for pty.h... yes
checking whether openpty is declared... yes
checking for openpty in -lc... no
checking for openpty in -lutil... yes
checking sys/wait.h usability... yes
checking sys/wait.h presence... yes
checking for sys/wait.h... yes
checking whether wait is declared... yes
checking whether fork is declared... yes
checking for X... no
configure: The Graphics and HyperDoc components are disabled.
configure: WARNING: Aldor interface will not be built.
configure: creating ./config.status
config.status: creating src/clef/Makefile
config.status: creating src/sman/Makefile
config.status: creating src/hyper/Makefile
config.status: creating src/doc/Makefile
config.status: creating Makefile
config.status: creating src/Makefile
config.status: creating src/lib/Makefile
config.status: creating src/lisp/Makefile
config.status: creating src/boot/Makefile
config.status: creating src/interp/Makefile
config.status: creating src/algebra/Makefile
config.status: creating src/input/Makefile
config.status: creating src/etc/Makefile
config.status: creating src/aldor/Makefile
config.status: creating src/aldor/Makefile2
config.status: creating src/aldor/Makefile3
config.status: creating contrib/emacs/Makefile
config.status: creating config/fricas_c_macros.h
extracting list of SPAD type definitions
Type 'make' (without quotes) to build FriCAS

export PATH=/home/me/data/fricas/fricas-1.3.2/target/x86_64-linux-gnu/bin:$PATH
export AXIOM=/home/me/data/fricas/fricas-1.3.2/target/x86_64-linux-gnu
export DAASE=/home/me/data/fricas/fricas-1.3.2/target/x86_64-linux-gnu

To turn oﬀ, just do )set output tex off. Make sure to tex on ﬁrst. To send back to console, do )set output tex console

To read an input ﬁle into Axiom, use the )read system command. read For example, you can read a ﬁle in a particular directory by issuing

1. How to measure CPU time used from the int call?
(3) -> )set messages time on
(3) -> integrate(x^(3/2)/sqrt(1 + x^5), x)

Type: Union(Expression(Integer),...)
Time: 0.06 (EV) + 0.00 (OT) = 0.06 sec

That is command ')set messages time on' tell FriCAS to print
time needed to execute following command. The (EV) part means
actual computation (OT) printing and at the end there is total

2. How to measure leaf count/size of result? (Maple and Mathematica have
build in function to do this)

That is a bit tricky. FriCAS (like Axiom) has quite different
representation of expressions tham Maple or Mathematica.
In FriCAS expression is a quotient of two polnomials, with
variables beeing kernels:

(13) -> f := exp(x^3 + 1/x) + 1
(14) -> numer(f)

4
x + 1
------
x

(15) -> denom(f)

(16) -> variables(numer(f))

4
x + 1
------
x
(16) [%e ]
Type: List(Kernel(Expression(Integer)))
Time: 0.00 (OT) = 0.00 sec

From FriCAS point of view natural measure of size is number
of monomials in numerator and denominator:

(17) -> numberOfMonomials(numer(f))

(17) 2
(18) -> numberOfMonomials(denom(f))
(18) 1

but this may underestimate size because kernels may be big.
Also internally given kernel is stored only once, but in
printed output it may appear several times.

It is possible to traverse expression is somewhat tree like
manner, so it is possible to give better approximation
to say Maple result, but that would require investigating
what Maple is doing. I personally never used Maple node
count so I do not know it this is simple or if there are
some traps.

3. How to check if int passed or failed? Aborted? nil result? etc..

integrate may produce an error (in particular it will do so
if it can not decide if integral is elementary). Or may
produce unevaluated integral. Or normal result. At programistic
level it is possible to catch errors, but a bit tricky,
If you look at printed output, than errors are reasonably
easy to match via a regex:

(22) -> integrate(1/sqrt(exp(x) - x + 1), x)
integrate: implementation incomplete (constant residues)

the '>> Error' part indicates error. Unevaluated integrals
always have integral sign at top level. FriCAS never returns
partial results, so text match is relatively easy. Or you
may use code like:

test_int(f) ==
res : Union(Expression Integer, LE)
res := integrate(f, 'x)
res0 : List Expression Integer :=
res case Expression Integer =>
[res::(Expression Integer)]@(List Expression Integer)
res case List Expression Integer =>
res::(List Expression Integer)
error "test_int: impossible 1"
for ri in res0 repeat
#(kernels ri) > 0 and is?(operator first kernels ri, 'integral) =>
print("Unevaluated integral"::OutputForm)

Note1: this is part extracted from bigger test script, I did not
test it alone.

Note2: FriCAS may either return list of results or a single result.
Middle part converts this to have always a list (typically of
length 1). The last part is a loop so that all solutions can
be examined. It you only want to know if result is evaluated, than
loop is not needed: more than one result means that integral
is evaluated.

Note3: In FriCAS testsuite I use a bit different test for
evaluated integrals. Namely, FriCAS can do useful computations
on unevaluated integrals and in particular unevaluated integrals
may appear in the argument to integrate. Such unevaluated
integrals of course may propagate form input to the output.
The test above may misclasify such integral as unevaluated,
while better test checks that unevaluated integral came from
the input.

4. And most importantly, how to export the result (if it passed) to a
plain text file in _Latex_ format?

(24) -> )set output tex on
(24) -> integrate(exp(x)*exp(1/(exp(x)+1)-x), x)

Here one have to decide what to do with errors. Without error
trapping error will abort currently executing function
and propagate to top level (up to command line).

After doing:

)set break resume

after error FriCAS will continue executing file from next
command (but still abort current command). If you need
real loop then I can provide error catcher, but it is
a bit more complicated than snippets above.
Waldek Hebisch

)clear all

)clear completely

4.2 sympy

>sudo conda install sympy=1.4

## Package Plan ##
  environment location: /opt/anaconda

  added / updated specs:
    - sympy=1.4

The following packages will be downloaded:
    package                    |            build
    ---------------------------|-----------------
    sympy-1.4                  |           py37_0         9.7 MB
    ------------------------------------------------------------
                                           Total:         9.7 MB
The following packages will be REMOVED:

  anaconda-2019.03-py37_0

The following packages will be UPDATED:

  sympy                                          1.3-py37_0 --> 1.4-py37_0

>sudo conda install -c conda-forge matchpy
[sudo] password for me:
Collecting package metadata: done
Solving environment: done

## Package Plan ##

  environment location: /opt/anaconda

  added / updated specs:
    - matchpy

The following packages will be downloaded:

    package                    |            build
    ---------------------------|-----------------
    ca-certificates-2019.3.9   |       hecc5488_0         146 KB  conda-forge
    certifi-2019.3.9           |           py37_0         149 KB  conda-forge
    conda-4.6.14               |           py37_0         2.1 MB  conda-forge
    hopcroftkarp-1.2.4         |             py_0          20 KB  conda-forge
    matchpy-0.5.1              |             py_0          52 KB  conda-forge
    multiset-2.1.1             |             py_0          11 KB  conda-forge
    openssl-1.1.1b             |       h14c3975_1         4.0 MB  conda-forge
    ------------------------------------------------------------
                                           Total:         6.4 MB

The following NEW packages will be INSTALLED:

  hopcroftkarp       conda-forge/noarch::hopcroftkarp-1.2.4-py_0
  matchpy            conda-forge/noarch::matchpy-0.5.1-py_0
  multiset           conda-forge/noarch::multiset-2.1.1-py_0

The following packages will be UPDATED:

  ca-certificates    pkgs/main::ca-certificates-2019.1.23-0 --> conda-forge::ca-certificates-2019.3.9-hecc5488_0

The following packages will be SUPERSEDED by a higher-priority channel:

  certifi                                         pkgs/main --> conda-forge
  conda                                           pkgs/main --> conda-forge
  openssl              pkgs/main::openssl-1.1.1b-h7b6447c_1 --> conda-forge::openssl-1.1.1b-h14c3975_1

Proceed ([y]/n)? y

Downloading and Extracting Packages
multiset-2.1.1       | 11 KB     | ####################################################################### | 100%
openssl-1.1.1b       | 4.0 MB    | ####################################################################### | 100%
matchpy-0.5.1        | 52 KB     | ####################################################################### | 100%
conda-4.6.14         | 2.1 MB    | ####################################################################### | 100%
hopcroftkarp-1.2.4   | 20 KB     | ####################################################################### | 100%
ca-certificates-2019 | 146 KB    | ####################################################################### | 100%
certifi-2019.3.9     | 149 KB    | ####################################################################### | 100%
Preparing transaction: done
Verifying transaction: done
Executing transaction: done
>

In general, To install python package do conda install package-name to update do conda update package for example conda update spyder

>python
Python 3.7.3 (default, Mar 27 2019, 22:11:17)
[GCC 7.3.0] :: Anaconda, Inc. on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import *
>>> from sympy.integrals.rubi.rubi import rubi_integrate

from sympy import *

or

import sympy #but now have to add sympy. to each call

import os
os.getcwd()
os.chdir('X:\\data\\public_html\\my_notes\\CAS_integration_tests\\reports\\rubi_4_11\\code')
os.getcwd()
import math
init_printing()
init_printing(use_unicode=False, wrap_line=False, no_global=True)
x = symbols('x', real=True)
r=integrate(x,x)
r0=latex(r)
text_file = open("python.txt", "w")
text_file.write("\"%s\"" % r0)
text_file.close()

r=integrate(f*g, (x, L/2, L))
type(r) is integrals.Integral

Out[41]: True

or

isinstance(r, integrals.Integral)
Out[48]: True

r=integrate(x,x)
type(r) is integrals.Integral
Out[43]: False

if isinstance(r, integrals.Integral):
result = 0
else:
result=1

os.getcwd()

To get list of folder in cwd do

os.listdir(os.getcwd())

or

ls

Out[85]:
['1_Algebraic_functions',
'2_Exponentials',
'3_Logarithms',
'4_Trig_functions',
'5_Inverse_trig_functions',
'6_Hyperbolic_functions',
'7_Inverse_hyperbolic_functions',
'8_Special_functions',
'Independent_test_suites']

f = open('Hebisch_Problems.txt','r')
f.readline()
f.close()

def _timeout(self, function, timeout):
         def callback(x, y):
             signal.alarm(0)
             raise Skipped("Timeout")
         signal.signal(signal.SIGALRM, callback)
         signal.alarm(timeout)  # Set an alarm with a given timeout
         function()
         signal.alarm(0)  # Disable the alarm

AppellF1 --> #hypergeometric function of two variables
Ellipticpi-->
arctanh -->atanh
arctan --> atan
arccosh->acosh
Pi --> pi.
arcsin -> asin
arccos -> acos
hypergeom -> hyper,
arccoth -> acoth
GAMMA -> uppergamma()
arccsc -> acsc
arcsec -> asec
arccot -> acot,
EllipticF -> elliptic_f
EllipticE -> elliptic_e
Li -> Li
Si -> Si
Ci -> Ci
Ei -> Ei
FresnelS -> fresnels
FresnelC -> fresnelc

I also removed 4th ﬁeld that contains integrate(...) in it as it hangs the reading of the input ﬁle.

cd /media/data/public_html/my_notes/CAS_integration_tests/reports/rubi_4_11
python sympy_main.py

for n in range(0,1): #change the last number to the number of tests to do

conda update conda
conda update python

pip list | grep sympy
pip install --upgrade sympy

import sympy
sympy.__version__

To obtain a Rational in SymPy, one of these methods must be used:

>>> from sympy import Rational
>>> Rational(2, 7)
2/7

>>> for num,val in enumerate(sol,start=1):
... print("root number {} is {}".format(num,val))

from subprocess import STDOUT, check_output
output = check_output('dir', shell=True, stderr=STDOUT, timeout=5)
print(output)

sudo git clone git://github.com/sympy/sympy.git
sudo git clone https://github.com/HPAC/matchpy.git

>sudo conda install -c conda-forge matchpy
[sudo] password for me:
Solving environment: done

## Package Plan ##

  environment location: /opt/anaconda

  added / updated specs:
    - matchpy

The following packages will be downloaded:

    package                    |            build
    ---------------------------|-----------------
    hopcroftkarp-1.2.4         |             py_0          20 KB  conda-forge
    certifi-2018.4.16          |           py36_0         142 KB  conda-forge
    conda-4.5.5                |           py36_0         624 KB  conda-forge
    matchpy-0.4.4              |             py_0          49 KB  conda-forge
    multiset-2.1.1             |             py_0          11 KB  conda-forge
    ------------------------------------------------------------
                                           Total:         845 KB

The following NEW packages will be INSTALLED:

    hopcroftkarp: 1.2.4-py_0       conda-forge
    matchpy:      0.4.4-py_0       conda-forge
    multiset:     2.1.1-py_0       conda-forge

The following packages will be UPDATED:

    certifi:      2018.4.16-py36_0             --> 2018.4.16-py36_0 conda-forge
    conda:        4.5.4-py36_0                 --> 4.5.5-py36_0     conda-forge

Proceed ([y]/n)? y

Downloading and Extracting Packages
hopcroftkarp-1.2.4   |   20 KB | ########################################################################################### | 100%
certifi-2018.4.16    |  142 KB | ########################################################################################### | 100%
conda-4.5.5          |  624 KB | ########################################################################################### | 100%
matchpy-0.4.4        |   49 KB | ########################################################################################### | 100%
multiset-2.1.1       |   11 KB | ########################################################################################### | 100%
Preparing transaction: done
Verifying transaction: done
Executing transaction: done
>

To update sympy in conda do sudo conda update sympy I do not know now if I should always use sudo for this, but it works. To check conda version do

>conda list anaconda$
# packages in environment at /opt/anaconda:
#
# Name Version Build Channel
anaconda 5.2.0 py36_3

>sudo conda update conda
[sudo] password for me:
Solving environment: done

# All requested packages already installed.

>conda update anaconda
Solving environment: done

# All requested packages already installed.

>conda info

     active environment : None
       user config file : /home/me/.condarc
populated config files :
          conda version : 4.5.5
    conda-build version : 3.10.5
         python version : 3.6.5.final.0
       base environment : /opt/anaconda  (read only)
           channel URLs : https://repo.anaconda.com/pkgs/main/linux-64
                          https://repo.anaconda.com/pkgs/main/noarch
                          https://repo.anaconda.com/pkgs/free/linux-64
                          https://repo.anaconda.com/pkgs/free/noarch
                          https://repo.anaconda.com/pkgs/r/linux-64
                          https://repo.anaconda.com/pkgs/r/noarch
                          https://repo.anaconda.com/pkgs/pro/linux-64
                          https://repo.anaconda.com/pkgs/pro/noarch
          package cache : /opt/anaconda/pkgs
                          /home/me/.conda/pkgs
       envs directories : /home/me/.conda/envs
                          /opt/anaconda/envs
               platform : linux-64
             user-agent : conda/4.5.5 requests/2.18.4 CPython/3.6.5 Linux/4.14.53-1-MANJARO manjaro/17.1.11 glibc/2.27
                UID:GID : 1000:1000
             netrc file : None
           offline mode : False

4.3 sagemath hints

x = var('x')
y = function('y')

sage: desolve(diff(y(x),x,2) + y(x) ==0, y(x),algorithm='fricas')
_C0*cos(x) + _C1*sin(x)

sage: desolve(diff(y(x),x,2) + y(x) ==0, y(x),algorithm='maxima')
_K2*cos(x) + _K1*sin(x)

var('x')
sage: anti=integrate(sqrt(1+x^3),x)

sage: anti
integrate(sqrt(x^3 + 1), x)

sage: isinstance(anti.operator(), sage.symbolic.integration.integral.IndefiniteIntegral)

True

to ﬁnd what methods are there for some expressions to appy to, type expr.<TAB> i.e. hit TAB key after the dot.

4.4 Trying rubi in sympy

>sudo conda install -c conda-forge matchpy
Solving environment: done

## Package Plan ##

  environment location: /opt/anaconda

  added / updated specs:
    - matchpy

The following packages will be downloaded:

    package                    |            build
    ---------------------------|-----------------
    ca-certificates-2018.8.24  |       ha4d7672_0         136 KB  conda-forge
    certifi-2018.8.24          |           py36_1         139 KB  conda-forge
    openssl-1.0.2o             |       h470a237_1         3.5 MB  conda-forge
    conda-4.5.11               |           py36_0         625 KB  conda-forge
    hopcroftkarp-1.2.4         |             py_0          20 KB  conda-forge
    multiset-2.1.1             |             py_0          11 KB  conda-forge
    matchpy-0.4.4              |             py_0          49 KB  conda-forge
    ------------------------------------------------------------
                                           Total:         4.5 MB

The following NEW packages will be INSTALLED:

    hopcroftkarp:    1.2.4-py_0        conda-forge
    matchpy:         0.4.4-py_0        conda-forge
    multiset:        2.1.1-py_0        conda-forge

The following packages will be UPDATED:

    ca-certificates: 2018.03.07-0                  --> 2018.8.24-ha4d7672_0 conda-forge
    certifi:         2018.8.13-py36_0              --> 2018.8.24-py36_1     conda-forge
    conda:           4.5.10-py36_0                 --> 4.5.11-py36_0        conda-forge

The following packages will be DOWNGRADED:

    openssl:         1.0.2p-h14c3975_0             --> 1.0.2o-h470a237_1    conda-forge

Proceed ([y]/n)? y

Downloading and Extracting Packages
ca-certificates-2018 | 136 KB    | ####################################################### | 100%
certifi-2018.8.24    | 139 KB    | ####################################################### | 100%
openssl-1.0.2o       | 3.5 MB    | ####################################################### | 100%
conda-4.5.11         | 625 KB    | ####################################################### | 100%
hopcroftkarp-1.2.4   | 20 KB     | ####################################################### | 100%
multiset-2.1.1       | 11 KB     | ####################################################### | 100%
matchpy-0.4.4        | 49 KB     | ####################################################### | 100%
Preparing transaction: done
Verifying transaction: done
Executing transaction: done

>python
Python 3.6.5 |Anaconda, Inc.| (default, Apr 29 2018, 16:14:56)
[GCC 7.2.0] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import *
>>> from sympy.integrals.rubi.rubi import rubi_integrate
>>> x = symbols('x')
>>> rubi_integrate(exp(1-exp(x**2)*x+2*x**2)*(2*x**3+x)/(1-exp(x**2)*x)**2, x)
Integral((2*x**3 + x)*exp(2*x**2 - x*exp(x**2) + 1)/(-x*exp(x**2) + 1)**2, x)

>>> x,a,b = symbols('x a b')
>>> rubi_integrate(1/(a**2-b**2*x), x)
-log(a**2 - b**2*x)/b**2

>>> rubi_integrate(1/(a**2-b**2*x**2), x)
x*hyper((1, 1/2), (3/2,), b**2*x**2/a**2)/a**2

%the above seems to be wrong

>>> rubi_integrate(sec(2*a*x), x)
Integral(sec(2*a*x), x)

>>> rubi_integrate(1/(1+cos(x)), x)
Integral(1/(cos(x) + 1), x)

%why it can't do the above?

>>> integrate(1/(1+cos(x)), x)
tan(x/2)

%it looks like not all rules are there