Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 37, 38, 39, 40, 41, 42 }
B grade: { }
C grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32,
34, 35, 36 }
F grade: { 15, 26, 27, 33}
2.1.3 Maple
A grade: { 20, 21, 30, 31, 37, 38, 39, 40, 41, 42 }
B grade: { 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36 }
C grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 32 }
F grade: { 15, 33 }
2.1.4 Maxima
A grade: {
B grade: { }
C grade: { }
F grade: { 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 }
2.1.5 FriCAS
A grade: { 37, 38, 39, 40, 41, 42 }
B grade: { 34, 35, 36 }
C grade: { }
F grade: { 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 }
2.1.6 Sympy
A grade: { }
B grade: { }
C grade: { 1, 2, 3, 4, 10, 11 }
F grade: { 5, 6, 7, 8, 9, 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 }
2.1.7 Giac
A grade: { 37, 38, 41 }
B grade: { 42 }
C grade: { }
F grade: { 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, 39, 40 }
2.2 Detailed conclusion table per each integral for all CAS systems
Detailed conclusion table per each integral is given by table below. The elapsed time is in
seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2)
if the failure was due to an exception being raised, which could indicate a bug in the system. If
the failure was due to integral not being evaluated within the time limit, then it is given just an
F.
In this table,the column normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)
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Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 453 | 878 | 217 | 533 | 0 | 0 | 364 | 0 |
normalized size | 1 | 1.94 | 0.48 | 1.18 | 0. | 0. | 0.8 | 0. |
time (sec) |
N/A |
0.509 |
0.282 |
0.054 |
0. |
0. |
6.844 |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 367 | 706 | 159 | 403 | 0 | 0 | 262 | 0 |
normalized size | 1 | 1.92 | 0.43 | 1.1 | 0. | 0. | 0.71 | 0. |
time (sec) |
N/A |
0.396 |
0.171 |
0.008 |
0. |
0. |
4.888 |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 277 | 432 | 121 | 269 | 0 | 0 | 167 | 0 |
normalized size | 1 | 1.56 | 0.44 | 0.97 | 0. | 0. | 0.6 | 0. |
time (sec) |
N/A |
0.225 |
0.099 |
0.004 |
0. |
0. |
3.291 |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 226 | 226 | 77 | 169 | 0 | 0 | 78 | 0 |
normalized size | 1 | 1. | 0.34 | 0.75 | 0. | 0. | 0.35 | 0. |
time (sec) |
N/A |
0.066 |
0.034 |
0.005 |
0. |
0. |
1.591 |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 369 | 369 | 138 | 192 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.37 |
0.52 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.36 |
0.304 |
0.033 |
0. |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 641 | 641 | 297 | 679 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.46 |
1.06 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.189 |
1.238 |
0.025 |
0. |
0. |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 875 | 875 | 453 | 1591 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.52 |
1.82 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.784 |
2.292 |
0.027 |
0. |
0. |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 912 | 912 | 222 | 588 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.24 | 0.64 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.726 |
0.286 |
0.036 |
0. |
0. |
0. |
0. |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 694 | 694 | 166 | 458 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.24 | 0.66 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.532 |
0.178 |
0.006 |
0. |
0. |
0. |
0. |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 395 | 395 | 126 | 320 | 0 | 0 | 167 | 0 |
normalized size | 1 | 1. | 0.32 | 0.81 | 0. | 0. | 0.42 | 0. |
time (sec) |
N/A |
0.327 |
0.113 |
0.006 |
0. |
0. |
15.871 |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 262 | 262 | 99 | 217 | 0 | 0 | 78 | 0 |
normalized size | 1 | 1. | 0.38 | 0.83 | 0. | 0. | 0.3 | 0. |
time (sec) |
N/A |
0.096 |
0.042 |
0.004 |
0. |
0. |
8.421 |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 732 | 732 | 432 | 564 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.59 |
0.77 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.781 |
0.805 |
0.023 |
0. |
0. |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1494 | 1494 | 427 | 1384 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.29 |
0.93 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.007 |
1.626 |
0.033 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2452 | 2452 | 630 | 2326 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.26 |
0.95 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.971 |
3.033 |
0.035 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
F |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 169 | 169 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.219 |
0.396 |
0.066 |
0. |
0. |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 94 | 81 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.96 | 1.69 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.029 |
0.136 |
0.028 |
0. |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 755 | 755 | 4473 | 1708 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 5.92 | 2.26 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
1.357 |
6.506 |
0.021 |
0. |
0. |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 528 | 528 | 674 | 1201 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.28 | 2.27 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.726 |
4.58 |
0.009 |
0. |
0. |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 368 | 368 | 521 | 759 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.42 | 2.06 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.254 |
2.234 |
0.007 |
0. |
0. |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 283 | 283 | 302 | 362 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.07 | 1.28 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.081 |
0.265 |
0.006 |
0. |
0. |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 436 | 436 | 298 | 359 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.68 |
0.82 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.409 |
0.513 |
0.029 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 782 | 782 | 1853 | 1495 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.37 |
1.91 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.467 |
3.822 |
0.027 |
0. |
0. |
0. |
0. |
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Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1125 | 1125 | 781 | 4476 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.69 |
3.98 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.594 |
6.17 |
0.031 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 859 | 859 | 5432 | 2445 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
6.32 |
2.85 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.38 |
6.674 |
0.034 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 628 | 633 | 766 | 1891 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1.01 | 1.22 | 3.01 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.584 |
4.764 |
0.008 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 481 | 481 | 0 | 1390 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 2.89 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.387 |
0. |
0.006 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 398 | 398 | 0 | 931 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 2.34 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.195 |
0. |
0.006 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 867 | 1045 | 1736 | 3241 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1.21 |
2. |
3.74 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.424 |
5.243 |
0.026 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1301 | 2112 | 8031 | 8276 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1.62 |
6.17 |
6.36 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
3.386 |
8.007 |
0.041 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 273 | 273 | 310 | 365 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.14 |
1.34 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.136 |
0.537 |
0.056 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 271 | 271 | 312 | 369 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.15 |
1.36 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.187 |
0.443 |
0.042 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 74 | 93 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.7 | 0.88 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.105 |
0.183 |
0.028 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 218 | 218 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.463 |
0.242 |
0.042 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 125 | 296 | 0 | 744 | 0 | 0 |
normalized size | 1 | 1. | 1.18 | 2.79 | 0. | 7.02 | 0. | 0. |
time (sec) |
N/A |
0.129 |
0.201 |
0.053 |
0. |
2.419 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 145 | 145 | 441 | 238 | 0 | 1211 | 0 | 0 |
normalized size | 1 | 1. | 3.04 | 1.64 | 0. | 8.35 | 0. | 0. |
time (sec) |
N/A |
0.207 |
0.661 |
0.033 |
0. |
22.774 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 239 | 239 | 432 | 568 | 0 | 1364 | 0 | 0 |
normalized size | 1 | 1. | 1.81 | 2.38 | 0. | 5.71 | 0. | 0. |
time (sec) |
N/A |
0.184 |
0.694 |
0.072 |
0. |
22.672 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 309 | 309 | 214 | 442 | 0 | 1118 | 0 | 486 |
normalized size |
1 |
1. |
0.69 |
1.43 |
0. |
3.62 |
0. |
1.57 |
time (sec) |
N/A |
0.625 |
0.345 |
0.018 |
0. |
2.051 |
0. |
1.194 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 283 | 283 | 168 | 373 | 0 | 857 | 0 | 358 |
normalized size |
1 |
1. |
0.59 |
1.32 |
0. |
3.03 |
0. |
1.27 |
time (sec) |
N/A |
0.341 |
0.231 |
0.008 |
0. |
1.924 |
0. |
1.152 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 286 | 286 | 176 | 251 | 0 | 1798 | 0 | 0 |
normalized size |
1 |
1. |
0.62 |
0.88 |
0. |
6.29 |
0. |
0. |
time (sec) |
N/A |
0.754 |
0.322 |
0.007 |
0. |
7.154 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 171 | 289 | 0 | 1731 | 0 | 0 |
normalized size |
1 |
1. |
0.58 |
0.98 |
0. |
5.89 |
0. |
0. |
time (sec) |
N/A |
0.767 |
0.261 |
0.011 |
0. |
6.287 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 288 | 288 | 177 | 329 | 0 | 1766 | 0 | 500 |
normalized size |
1 |
1. |
0.61 |
1.14 |
0. |
6.13 |
0. |
1.74 |
time (sec) |
N/A |
0.784 |
0.359 |
0.01 |
0. |
5.942 |
0. |
1.236 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 189 | 412 | 0 | 1887 | 0 | 972 |
normalized size |
1 |
1. |
0.64 |
1.4 |
0. |
6.42 |
0. |
3.31 |
time (sec) |
N/A |
0.787 |
0.293 |
0.011 |
0. |
6.503 |
0. |
1.472 |
|
|
|
|
|
|
|
|
|
|
2.3 Detailed conclusion table specific for Rubi results
The following table is specific to Rubi. It gives additional statistics for each integral. the column
steps is the number of steps used by Rubi to obtain the antiderivative. The rules
column is the number of unique rules used. The integrand size column is the leaf size
of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the
integral was to solve. In this test, problem number [14] had the largest ratio of [ 0.3929
]
|
|
|
|
|
|
|
| Table 2.1Rubi specific breakdown of results for each
integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
15 |
5 |
1.94 |
28 |
0.179 |
|
|
|
|
|
|
|
2 |
A |
12 |
5 |
1.92 |
28 |
0.179 |
|
|
|
|
|
|
|
3 |
A |
8 |
5 |
1.56 |
26 |
0.192 |
|
|
|
|
|
|
|
4 |
A |
3 |
3 |
1. |
19 |
0.158 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1. |
28 |
0.107 |
|
|
|
|
|
|
|
6 |
A |
6 |
6 |
1. |
28 |
0.214 |
|
|
|
|
|
|
|
7 |
A |
7 |
6 |
1. |
28 |
0.214 |
|
|
|
|
|
|
|
8 |
A |
12 |
7 |
1. |
28 |
0.25 |
|
|
|
|
|
|
|
9 |
A |
10 |
6 |
1. |
28 |
0.214 |
|
|
|
|
|
|
|
10 |
A |
7 |
5 |
1. |
26 |
0.192 |
|
|
|
|
|
|
|
11 |
A |
4 |
4 |
1. |
19 |
0.21 |
|
|
|
|
|
|
|
12 |
A |
9 |
7 |
1. |
28 |
0.25 |
|
|
|
|
|
|
|
13 |
A |
15 |
10 |
1. |
28 |
0.357 |
|
|
|
|
|
|
|
14 |
A |
22 |
11 |
1. |
28 |
0.393 |
|
|
|
|
|
|
|
15 |
A |
6 |
3 |
1. |
26 |
0.115 |
|
|
|
|
|
|
|
16 |
A |
2 |
2 |
1. |
27 |
0.074 |
|
|
|
|
|
|
|
17 |
A |
6 |
4 |
1. |
33 |
0.121 |
|
|
|
|
|
|
|
18 |
A |
5 |
4 |
1. |
33 |
0.121 |
|
|
|
|
|
|
|
19 |
A |
4 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
20 |
A |
3 |
3 |
1. |
24 |
0.125 |
|
|
|
|
|
|
|
|
21 |
A |
3 |
3 |
1. |
33 |
0.091 |
|
|
|
|
|
|
|
22 |
A |
6 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
23 |
A |
7 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
24 |
A |
5 |
5 |
1. |
33 |
0.152 |
|
|
|
|
|
|
|
25 |
A |
4 |
4 |
1.01 |
33 |
0.121 |
|
|
|
|
|
|
|
26 |
A |
4 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
27 |
A |
4 |
4 |
1. |
24 |
0.167 |
|
|
|
|
|
|
|
28 |
A |
9 |
7 |
1.21 |
33 |
0.212 |
|
|
|
|
|
|
|
29 |
A |
15 |
10 |
1.62 |
33 |
0.303 |
|
|
|
|
|
|
|
30 |
A |
1 |
1 |
1. |
41 |
0.024 |
|
|
|
|
|
|
|
31 |
A |
1 |
1 |
1. |
41 |
0.024 |
|
|
|
|
|
|
|
32 |
A |
4 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
33 |
A |
6 |
3 |
1. |
31 |
0.097 |
|
|
|
|
|
|
|
34 |
A |
11 |
7 |
1. |
28 |
0.25 |
|
|
|
|
|
|
|
35 |
A |
4 |
3 |
1. |
30 |
0.1 |
|
|
|
|
|
|
|
36 |
A |
1 |
1 |
1. |
30 |
0.033 |
|
|
|
|
|
|
|
37 |
A |
6 |
6 |
1. |
38 |
0.158 |
|
|
|
|
|
|
|
38 |
A |
6 |
6 |
1. |
37 |
0.162 |
|
|
|
|
|
|
|
39 |
A |
8 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
40 |
A |
8 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
|
41 |
A |
8 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
42 |
A |
8 |
7 |
1. |
40 |
0.175 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|