1 Introduction
This report gives the result of running the computer algebra independent integration problems.
The listing of the problems are maintained by and can be downloaded from Albert Rich Rubi
web site.
1.1 Listing of CAS systems tested
The following systems were tested at this time.
-
Mathematica 11.3 (64 bit).
-
Rubi 4.15.2 in Mathematica 11.3.
-
Rubi in Sympy (Version 1.3) under Python 3.7.0 using Anaconda distribution.
-
Maple 2018.1 (64 bit).
-
Maxima 5.41 Using Lisp ECL 16.1.2.
-
Fricas 1.3.4.
-
Sympy 1.3 under Python 3.7.0 using Anaconda distribution.
-
Giac/Xcas 1.4.9.
Maxima, Fricas and Giac/Xcas were called from inside SageMath version 8.3. This was done
using SageMath integrate command by changing the name of the algorithm to use the different
CAS systems.
Sympy was called directly using Python. Rubi in Sympy was also called directly using sympy
1.3 in python.
1.2 Design of the test system
The following diagram gives a high level view of the current test build system.
1.3 Timing
The command AboluteTiming[] was used in Mathematica to obtain the elapsed time
for each integrate call. In Maple, the command Usage was used as in the following
example
cpu_time := Usage(assign ('result_of _int',int(expr,x)),output='realtime'
For all other CAS systems, the elapsed time to complete each integral was found by taking the
difference between the time after the call has completed from the time before the call was made.
This was done using Python’s time.time() call.
All elapsed times shown are in seconds. A time limit of 3 minutes was used for each integral. If
the integrate command did not complete within this time limit, the integral was aborted and
considered to have failed and assigned an F grade. The time used by failed integrals due to time
out is not counted in the final statistics.
1.4 Verification
A verification phase was applied on the result of integration for Rubi and Mathematica. Future
version of this report will implement verification for the other CAS systems. For the integrals
whose result was not run through a verification phase, it is assumed that the antiderivative
produced was correct.
Verification phase has 3 minutes time out. An integral whose result was not verified could still
be correct. Further investigation is needed on those integrals which failed verifications. Such
integrals are marked in the summary table below and also in each integral separate section so
they are easy to identify and locate.
1.5 Important notes about some of the results
Important note about Maxima results Since these integrals are run in a batch mode, using an
automated script, and by using sagemath (SageMath uses Maxima), then any integral where
Maxima needs an interactive response from the user to answer a question during
evaluation of the integral in order to complete the integration, will fail and is counted as
failed.
The exception raised is ValueError. Therefore Maxima result below is lower than what could
result if Maxima was run directly and each question Maxima asks was answered
correctly.
The percentage of such failures were not counted for each test file, but for an example, for the
Timofeev test file, there were about 30 such integrals out of total 705, or about 4
percent. This pecrentage can be higher or lower depending on the specific input test
file.
Such integrals can be indentified by looking at the output of the integration in each section for
Maxima. If the output was an exception ValueError then this is most likely due to this
reason.
Maxima integrate was run using SageMath with the following settings set by default
'besselexpand : true''display2d : false''domain : complex''keepfloat : true'
'load(to_poly_solve)'
'load(simplify_sum)'
'load(abs_integrate)' 'load(diag)'
SageMath loading of Maxima abs_integrate was found to cause some problem. So the
following code was added to disable this effect.
from sage.interfaces.maxima_lib import maxima_lib
maxima_lib.set('extra_definite_integration_methods', '[]')
maxima_lib.set('extra_integration_methods', '[]')
See https://ask.sagemath.org/question/43088/integrate-results-that-are-different-from-using-maxima/
for reference.
Important note about FriCAS and Giac/XCAS results There are Few integrals which failed due
to SageMath not able to translate the result back to SageMath syntax and not because these
CAS system were not able to do the integrations.
These will fail With error Exception raised: NotImplementedError
The number of such cases seems to be very small. About 1 or 2 percent of all integrals.
Hopefully the next version of SageMath will have complete translation of FriCAS and XCAS
syntax and I will re-run all the tests again when this happens.
Important note about finding leaf size of antiderivative For Mathematica, Rubi and
Maple, the buildin system function LeafSize is used to find the leaf size of each
antiderivative.
The other CAS systems (SageMath and Sympy) do not have special buildin function for this
purpose at this time. Therefore the leaf size is determined as follows.
For Fricas, Giac and Maxima (all called via sagemath) the following code is used
#see https://stackoverflow.com/questions/25202346/how-to-obtain-leaf-count-expression-size-in-sage
def tree(expr):
if expr.operator() is None:
return expr
else:
return [expr.operator()]+map(tree, expr.operands())
try:
# 1.35 is a fudge factor since this estimate of leaf count is bit lower than
#what it should be compared to Mathematica's
leafCount = round(1.35*len(flatten(tree(anti))))
except Exception as ee:
leafCount =1
For Sympy, called directly from Python, the following code is used
try: # 1.7 is a fudge factor since it is low side from actual leaf count
leafCount = round(1.7*count_ops(anti))
except Exception as ee:
leafCount =1
When these cas systems implement a buildin function to find the leaf size of expressions, it will
be used instead, and these tests run again.
1.6 Grading of results
The table below summarizes the grading of each CAS system.
Important note: A number of problems in this test suite have no antiderivative in closed form.
This means the antiderivative of these integrals can not be expressed in terms of elementary,
special functions or Hypergeometric2F1 functions. RootSum and RootOf are not
allowed.
If a CAS returns the above integral unevaluated within the time limit, then the result is
counted as passed and assigned an A grade.
However, if CAS times out, then it is assigned an F grade even if the integral is not integrable,
as this implies CAS could not determine that the integral is not integrable in the time
limit.
If a CAS returns an antiderivative to such an integral, it is assigned an A grade automatically
and this special result is listed in the introduction section of each individual test report to make
it easy to identify as this can be important result to investigate.
The results given in in the table below reflects the above.
|
|
|
System |
solved |
Failed |
|
|
|
|
|
|
Rubi |
% 99.21 ( 878 ) |
% 0.79 ( 7 ) |
|
|
|
Rubi in Sympy |
% 71.86 ( 636 ) |
% 28.14 ( 249 ) |
|
|
|
Mathematica |
% 91.86 ( 813 ) |
% 8.14 ( 72 ) |
|
|
|
Maple | % 81.92 ( 725 ) | % 18.08 ( 160 ) |
|
|
|
Maxima | % 36.38 ( 322 ) | % 63.62 ( 563 ) |
|
|
|
Fricas |
% 67.01 ( 593 ) |
% 32.99 ( 292 ) |
|
|
|
Sympy |
% 24.29 ( 215 ) |
% 75.71 ( 670 ) |
|
|
|
Giac |
% 45.88 ( 406 ) |
% 54.12 ( 479 ) |
|
|
|
|
The table below gives additional break down of the grading of quality of the antiderivatives
generated by each CAS. The grading is given using the letters A,B,C and F with A being the
best quality. The grading is accomplished by comparing the antiderivative generated with the
optimal antiderivatives included in the test suite. The following table describes the meaning of
these grades.
|
|
grade |
description |
|
|
|
|
A |
Integral was solved and antiderivative is optimal in quality and leaf
size. |
|
|
B |
Integral was solved and antiderivative is optimal in quality but leaf size
is larger than twice the optimal antiderivatives leaf size. |
|
|
C |
Integral was solved and antiderivative is non-optimal in quality. This can be
due to one or more of the following reasons
-
antiderivative contains a hypergeometric function and the
optimal antiderivative does not.
-
antiderivative contains a special function and the optimal
antiderivative does not.
-
antiderivative contains the imaginary unit and the optimal
antiderivative does not.
|
|
|
F |
Integral was not solved. Either the integral was returned unevaluated
within the time limit, or it timed out, or CAS hanged or crashed or an
exception was raised. |
|
|
|
Grading is currently implemented only for for Mathematica, Rubi and Maple results. For all
other CAS systems (Maxima, Fricas, Sympy, Giac, Rubi in sympy), the grading function is not
yet implemented. For these systems, a grade of A is assigned if the integrate command
completes successfully and a grade of F otherwise.
Based on the above, the following table summarizes the grading for this test suite.
|
|
|
|
|
System |
% A grade |
% B grade |
% C grade |
% F grade |
|
|
|
|
|
|
|
|
|
|
Rubi |
98.53 |
0.34 |
0.34 |
0.79 |
|
|
|
|
|
Rubi in Sympy |
71.86 |
0. |
0. |
28.14 |
|
|
|
|
|
Mathematica |
64.75 |
8.47 |
26.67 |
8.14 |
|
|
|
|
|
Maple |
49.83 | 20.45 | 11.64 | 18.08 |
|
|
|
|
|
Maxima | 36.38 | 0. | 0. | 63.62 |
|
|
|
|
|
Fricas |
67.01 |
0. |
0. |
32.99 |
|
|
|
|
|
Sympy |
24.29 |
0. |
0. |
75.71 |
|
|
|
|
|
Giac |
45.88 |
0. |
0. |
54.12 |
|
|
|
|
|
|
The following is a Bar chart illustration of the data in the above table.
The figure below compares the CAS systems for each grade level.
1.7 Performance
The table below summarizes the performance of each CAS system in terms of CPU time and
leaf size of results.
|
|
|
|
|
|
System |
Mean time (sec) |
Mean size |
Normalized mean |
Median size |
Normalized median |
|
|
|
|
|
|
|
|
|
|
|
|
Rubi |
0.39 |
128.99 |
1.01 |
70. |
1. |
|
|
|
|
|
|
Rubi in Sympy |
32.94 |
130.87 |
1.3 |
63. |
0.88 |
|
|
|
|
|
|
Mathematica |
0.79 |
359.78 |
2.66 |
86. |
1. |
|
|
|
|
|
|
Maple |
0.04 | 1612.26 | 13.38 | 76. | 1.15 |
|
|
|
|
|
|
Maxima |
0.74 |
73.95 |
1.3 |
36. |
1.1 |
|
|
|
|
|
|
Fricas |
0.61 |
154.69 |
1.93 |
45. |
1.2 |
|
|
|
|
|
|
Sympy |
12.54 |
219.47 |
2.65 |
49. |
0.95 |
|
|
|
|
|
|
Giac |
0.3 |
90.18 |
1.59 |
47. |
1.25 |
|
|
|
|
|
|
|
1.8 list of integrals that has no closed form antiderivative
{759, 760, 761, 762, 763, 764, 765, 766, 767, 768}
1.9 list of integrals not solved by each system
-
Not solved by Rubi
- {174, 455, 456, 857, 858, 879, 885}
-
Not solved by Rubi in Sympy
- {5, 6, 7, 8, 9, 18, 19, 24, 25, 26, 27, 28, 37, 38, 39, 40,
41, 46, 47, 48, 49, 50, 57, 58, 66, 67, 76, 86, 87, 88, 89, 98, 99, 100, 101, 123, 124,
125, 126, 141, 142, 143, 144, 149, 150, 173, 174, 175, 176, 177, 178, 179, 180, 181,
182, 187, 188, 195, 200, 201, 204, 246, 247, 258, 259, 260, 261, 262, 263, 264, 271,
272, 273, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 299,
300, 301, 302, 303, 304, 305, 306, 311, 312, 313, 314, 318, 319, 320, 321, 322, 323,
324, 325, 326, 328, 329, 330, 331, 333, 334, 335, 336, 338, 339, 340, 341, 342, 344,
345, 346, 347, 348, 350, 351, 352, 353, 354, 356, 358, 359, 360, 361, 383, 384, 386,
391, 392, 394, 406, 407, 408, 411, 412, 413, 415, 418, 435, 446, 447, 448, 449, 452,
453, 454, 455, 456, 457, 458, 459, 460, 461, 471, 472, 473, 474, 478, 479, 480, 511,
512, 513, 534, 551, 552, 556, 566, 568, 569, 571, 640, 641, 642, 643, 644, 645, 646,
647, 648, 649, 650, 651, 653, 654, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667,
669, 670, 675, 680, 693, 695, 697, 721, 722, 723, 724, 725, 726, 727, 728, 729, 735,
736, 740, 744, 745, 753, 755, 756, 757, 758, 774, 775, 793, 795, 797, 802, 806, 830,
857, 858, 859, 865, 879, 881, 884, 885}
-
Not solved by Mathematica
- {18, 19, 149, 150, 172, 174, 205, 206, 207, 213, 214, 292,
299, 300, 303, 304, 311, 318, 319, 322, 323, 327, 328, 332, 333, 338, 339, 344, 345,
346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362,
367, 368, 381, 382, 394, 397, 398, 426, 430, 431, 432, 433, 437, 661, 662, 663, 664,
665, 666, 769, 770, 771, 772, 857, 878, 879}
-
Not solved by Maple
- {5, 6, 7, 8, 18, 19, 24, 25, 26, 27, 37, 38, 39, 40, 46, 47, 48, 49,
55, 56, 57, 58, 64, 65, 66, 67, 73, 74, 75, 76, 82, 83, 84, 85, 86, 87, 88, 89, 94, 95, 96,
97, 98, 99, 100, 101, 108, 109, 110, 111, 117, 118, 119, 120, 149, 150, 154, 158, 162,
163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 174, 202, 203, 232, 292, 299, 300,
301, 302, 303, 304, 305, 306, 310, 311, 318, 319, 320, 321, 322, 323, 327, 328, 329,
330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346,
347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 401, 402,
403, 426, 427, 428, 429, 430, 431, 432, 433, 492, 493, 494, 495, 496, 530, 720, 769,
770, 771, 772, 800, 857, 858, 861, 862, 863, 870, 873, 874, 875, 876, 878, 879}
-
Not solved by Maxima
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88,
89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 153, 154, 157, 158, 161, 162, 163, 164, 165,
166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182,
183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199,
200, 201, 202, 203, 204, 205, 206, 207, 217, 219, 220, 222, 225, 226, 227, 228, 229,
230, 231, 232, 233, 234, 236, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250,
251, 252, 253, 254, 255, 256, 257, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274,
275, 276, 277, 278, 279, 280, 281, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299,
300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316,
317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333,
334, 335, 336, 338, 339, 340, 341, 342, 344, 345, 346, 347, 348, 349, 350, 351, 352,
353, 354, 356, 358, 360, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373,
374, 375, 376, 377, 378, 379, 380, 381, 382, 386, 387, 388, 389, 390, 394, 395, 396,
397, 398, 399, 400, 401, 402, 426, 427, 428, 429, 430, 431, 432, 433, 440, 461, 462,
463, 468, 469, 470, 475, 476, 477, 482, 483, 484, 489, 490, 491, 496, 497, 498, 499,
500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 525, 527, 529,
531, 532, 533, 534, 545, 546, 547, 548, 549, 550, 554, 564, 565, 567, 570, 571, 572,
584, 589, 597, 598, 599, 600, 601, 602, 603, 604, 605, 608, 609, 610, 611, 612, 613,
614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630,
631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647,
648, 650, 651, 655, 656, 657, 670, 671, 672, 674, 675, 676, 678, 720, 721, 722, 723,
725, 726, 729, 730, 732, 733, 734, 735, 736, 737, 739, 741, 745, 747, 750, 752, 753,
755, 756, 757, 758, 769, 770, 771, 772, 776, 777, 778, 779, 788, 796, 797, 798, 813,
814, 820, 823, 824, 825, 826, 827, 831, 843, 844, 845, 846, 847, 848, 849, 850, 851,
852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868,
870, 871, 872, 873, 874, 875, 876, 878, 879, 880, 881, 883, 884}
-
Not solved by Fricas
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24,
25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 82, 83, 84,
85, 86, 87, 88, 89, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 154, 158, 162, 163, 164, 165, 166, 167, 168,
169, 170, 171, 172, 173, 174, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 217, 219, 220, 222, 225, 226,
228, 230, 231, 232, 233, 234, 235, 236, 292, 311, 327, 328, 332, 333, 338, 339, 344,
345, 349, 350, 352, 356, 360, 362, 396, 397, 398, 399, 400, 401, 402, 413, 419, 426,
427, 428, 429, 430, 431, 432, 433, 440, 452, 453, 454, 455, 456, 496, 571, 572, 608,
609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625,
626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642,
643, 644, 645, 646, 647, 648, 720, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768,
771, 772, 788, 817, 842, 845, 846, 857, 858, 862, 863, 864, 865, 870, 873, 874, 875,
876, 879}
-
Not solved by Sympy
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 90, 91, 92, 93,
94, 95, 96, 97, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115,
116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 137, 138, 139, 140,
141, 142, 143, 144, 149, 150, 155, 156, 159, 160, 161, 162, 163, 164, 165, 166, 167,
168, 169, 170, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 187, 188, 193,
194, 195, 196, 197, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212,
213, 214, 215, 216, 217, 220, 221, 222, 223, 224, 225, 226, 228, 229, 230, 231, 232,
233, 234, 235, 236, 237, 238, 239, 240, 241, 243, 244, 245, 246, 248, 249, 250, 253,
254, 258, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276,
277, 278, 279, 280, 281, 283, 287, 288, 289, 290, 292, 296, 297, 298, 299, 300, 301,
302, 303, 304, 306, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322,
323, 324, 325, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 338, 339, 340, 341,
342, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359,
360, 361, 362, 367, 368, 373, 374, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388,
389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 407, 411,
412, 413, 415, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 430, 431, 432, 433,
434, 435, 436, 437, 438, 439, 440, 451, 452, 453, 454, 455, 456, 464, 465, 466, 467,
468, 469, 470, 471, 472, 473, 475, 476, 477, 478, 479, 480, 482, 483, 484, 485, 486,
487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 499, 500, 501, 502, 505, 506, 507,
508, 510, 511, 512, 513, 518, 519, 522, 526, 527, 529, 530, 531, 532, 533, 534, 535,
536, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 554, 556,
557, 559, 560, 561, 562, 563, 564, 565, 567, 568, 571, 572, 574, 576, 577, 578, 580,
581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597,
598, 599, 600, 601, 602, 603, 604, 605, 608, 609, 610, 611, 612, 613, 614, 615, 616,
617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633,
634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 651,
657, 660, 665, 666, 667, 668, 669, 670, 671, 672, 674, 675, 676, 677, 678, 682, 686,
688, 689, 691, 695, 697, 699, 701, 703, 705, 707, 708, 709, 710, 711, 713, 715, 717,
719, 720, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738,
739, 740, 741, 742, 743, 744, 745, 746, 747, 750, 751, 752, 753, 754, 755, 756, 757,
758, 769, 770, 771, 772, 773, 776, 777, 778, 779, 780, 781, 783, 784, 785, 787, 788,
789, 790, 792, 793, 795, 796, 798, 800, 801, 805, 813, 814, 818, 819, 820, 826, 827,
828, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 843, 844, 845, 846, 847,
848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864,
865, 866, 867, 868, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 883,
884, 885}
-
Not solved by Giac
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 90, 91, 92, 93, 102, 103, 104,
105, 106, 107, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 154, 158, 159, 162, 163, 164, 165, 166, 167,
168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184,
185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201,
202, 203, 204, 205, 207, 217, 219, 220, 222, 225, 226, 228, 230, 231, 232, 233, 234,
235, 236, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254,
262, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279,
280, 281, 287, 289, 290, 292, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306,
307, 308, 309, 310, 311, 315, 317, 318, 319, 320, 321, 322, 324, 325, 326, 327, 328,
329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345,
346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362,
365, 366, 367, 368, 371, 372, 373, 374, 375, 376, 379, 380, 381, 382, 396, 397, 398,
399, 400, 401, 402, 426, 427, 428, 429, 430, 431, 432, 433, 434, 436, 438, 440, 450,
451, 452, 453, 454, 455, 456, 468, 469, 470, 478, 479, 480, 481, 482, 483, 484, 489,
490, 491, 492, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 510,
511, 512, 513, 521, 529, 535, 536, 546, 550, 557, 558, 559, 560, 561, 562, 563, 570,
571, 572, 596, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621,
622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638,
639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 655, 656, 668, 673, 681, 682,
707, 708, 710, 713, 715, 717, 719, 720, 728, 729, 732, 733, 734, 735, 736, 737, 745,
769, 770, 771, 772, 778, 779, 798, 799, 800, 809, 828, 845, 846, 857, 858, 860, 861,
862, 863, 864, 865, 866, 867, 870, 871, 872, 873, 874, 875, 876, 878, 879, 880, 881}
1.10 list of integrals solved by CAS but has no known antiderivative
-
Rubi
- {}
-
Rubi in Sympy
- {}
-
Mathematica
- {}
-
Maple
- {}
-
Maxima
- {}
-
Fricas
- {}
-
Sympy
- {}
-
Giac
- {}
1.11 list of integrals solved by CAS but failed verification
The following are integrals solved by CAS but the verification phase failed to verify the
anti-derivative produced is correct. This does not mean necessarily that the anti-derivative is
wrong, as additional methods of verification might be needed, or more time is needed (3
minutes time limit was used). These integrals are listed here to make it easier to
do further investigation to determine why it was not possible to verify the result
produced.
-
Rubi
- {125, 128, 141, 142, 143, 144, 173, 187, 188, 194, 195, 200, 201, 204, 570, 618, 619,
621, 622, 624, 645, 648, 884}
-
Mathematica
- {1, 2, 3, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116,
117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 137, 138, 139, 140, 141,
142, 143, 144, 163, 164, 165, 166, 167, 168, 169, 170, 171, 173, 179, 180, 181, 182,
202, 203, 204, 226, 228, 231, 262, 287, 289, 290, 395, 396, 399, 400, 401, 402, 594,
597, 598, 599, 608, 609, 610, 611, 612, 613, 614, 616, 617, 619, 629, 630, 635, 637,
642, 643, 645, 646, 648, 708, 710, 729, 871, 872, 880, 883}
-
Maple
- Verification phase not implemented yet.
-
Maxima
- Verification phase not implemented yet.
-
Fricas
- Verification phase not implemented yet.
-
Sympy
- Verification phase not implemented yet.
-
Giac
- Verification phase not implemented yet.
-
Rubi in Sympy
- Verification phase not implemented yet.