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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
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 }
B grade: { }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 6, 7, 8, 9, 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 }
B grade: { }
C grade: { }
F grade: { 5, 10 }
2.1.4 Maxima
A grade: { 4, 9, 14, 15, 19, 23, 27, 28, 32, 36
B grade: { }
C grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 29, 30, 31, 33, 34, 35
}
F grade: { 5, 10 }
2.1.5 FriCAS
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 }
B grade: { }
C grade: { }
F grade: { }
2.1.6 Sympy
A grade: { 3, 4, 8, 9, 13, 14, 15, 18, 19, 22, 23, 26, 27, 28, 31, 32, 35, 36 }
B grade: { 12, 25 }
C grade: { }
F grade: { 1, 2, 5, 6, 7, 10, 11, 16, 17, 20, 21, 24, 29, 30, 33, 34 }
2.1.7 Giac
A grade: { 4, 9, 14, 15, 19, 23, 27, 28, 32, 36 }
B grade: { }
C grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 29, 30, 31, 33, 34, 35
}
F grade: { 5, 10 }
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 |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 249 | 249 | 157 | 204 | 4115 | 448 | 0 | 306 |
normalized size | 1 | 1. | 0.63 | 0.82 | 16.53 | 1.8 | 0. | 1.23 |
time (sec) |
N/A |
0.243 |
0.707 |
0.009 |
5.036 |
1.555 |
0. |
2.972 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 123 | 123 | 111 | 100 | 1305 | 325 | 0 | 247 |
normalized size | 1 | 1. | 0.9 | 0.81 | 10.61 | 2.64 | 0. | 2.01 |
time (sec) |
N/A |
0.037 |
0.371 |
0.007 |
2.729 |
1.562 |
0. |
1.221 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 84 | 82 | 393 | 279 | 88 | 182 |
normalized size | 1 | 1. | 0.87 | 0.85 | 4.05 | 2.88 | 0.91 | 1.88 |
time (sec) |
N/A |
0.021 |
0.173 |
0.007 |
2.316 |
1.35 |
0.547 |
1.215 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.008 |
4.173 |
0.047 |
0. |
0. |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 110 | 0 | 0 | 311 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 2.83 | 0. | 0. |
time (sec) |
N/A |
0.083 |
0.793 |
0.198 |
0. |
1.413 |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 251 | 251 | 165 | 202 | 4122 | 443 | 0 | 309 |
normalized size | 1 | 1. | 0.66 | 0.8 | 16.42 | 1.76 | 0. | 1.23 |
time (sec) |
N/A |
0.204 |
0.712 |
0.01 |
2.285 |
1.607 |
0. |
1.738 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 124 | 124 | 117 | 99 | 1310 | 323 | 0 | 250 |
normalized size | 1 | 1. | 0.94 | 0.8 | 10.56 | 2.6 | 0. | 2.02 |
time (sec) |
N/A |
0.044 |
0.318 |
0.008 |
1.708 |
1.529 |
0. |
2.446 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 89 | 80 | 393 | 278 | 94 | 185 |
normalized size | 1 | 1. | 0.91 | 0.82 | 4.01 | 2.84 | 0.96 | 1.89 |
time (sec) |
N/A |
0.025 |
0.185 |
0.009 |
2.049 |
1.585 |
0.545 |
2.795 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.008 |
4.792 |
0.046 |
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 |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 115 | 0 | 0 | 309 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 0. | 0. | 2.81 | 0. | 0. |
time (sec) |
N/A |
0.084 |
0.802 |
0.204 |
0. |
1.583 |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 66 | 59 | 211 | 228 | 0 | 101 |
normalized size | 1 | 1. | 0.8 | 0.72 | 2.57 | 2.78 | 0. | 1.23 |
time (sec) |
N/A |
0.033 |
0.117 |
0.009 |
3.979 |
1.483 |
0. |
2.418 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
B |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 39 | 30 | 166 | 124 | 160 | 88 |
normalized size | 1 | 1. | 0.95 | 0.73 | 4.05 | 3.02 | 3.9 | 2.15 |
time (sec) |
N/A |
0.013 |
0.05 |
0.009 |
4.143 |
1.39 |
1.148 |
1.297 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 24 | 20 | 95 | 89 | 29 | 53 |
normalized size | 1 | 1. | 1. | 0.83 | 3.96 | 3.71 | 1.21 | 2.21 |
time (sec) |
N/A |
0.006 |
0.027 |
0.007 |
3.77 |
1.354 |
0.5 |
1.175 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.008 |
14.481 |
0.063 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.025 |
11.554 |
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 |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 248 | 248 | 170 | 191 | 4236 | 440 | 0 | 289 |
normalized size | 1 | 1. | 0.69 | 0.77 | 17.08 | 1.77 | 0. | 1.17 |
time (sec) |
N/A |
0.227 |
0.655 |
0.018 |
3.616 |
1.609 |
0. |
1.243 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 118 | 95 | 1372 | 315 | 0 | 230 |
normalized size | 1 | 1. | 0.94 | 0.75 | 10.89 | 2.5 | 0. | 1.83 |
time (sec) |
N/A |
0.073 |
0.28 |
0.016 |
2.12 |
1.633 |
0. |
1.248 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 97 | 72 | 410 | 238 | 83 | 165 |
normalized size | 1 | 1. | 0.97 | 0.72 | 4.1 | 2.38 | 0.83 | 1.65 |
time (sec) |
N/A |
0.05 |
0.103 |
0.013 |
1.888 |
1.535 |
1.114 |
1.278 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.027 |
6.992 |
0.099 |
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 |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 248 | 248 | 175 | 199 | 4238 | 435 | 0 | 292 |
normalized size | 1 | 1. | 0.71 | 0.8 | 17.09 | 1.75 | 0. | 1.18 |
time (sec) |
N/A |
0.22 |
0.73 |
0.017 |
2.208 |
1.497 |
0. |
1.287 |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 122 | 99 | 1372 | 313 | 0 | 232 |
normalized size | 1 | 1. | 0.97 | 0.79 | 10.89 | 2.48 | 0. | 1.84 |
time (sec) |
N/A |
0.066 |
0.291 |
0.014 |
1.952 |
1.437 |
0. |
1.403 |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 102 | 76 | 410 | 235 | 88 | 167 |
normalized size | 1 | 1. | 1.02 | 0.76 | 4.1 | 2.35 | 0.88 | 1.67 |
time (sec) |
N/A |
0.048 |
0.11 |
0.015 |
2.073 |
1.536 |
1.145 |
1.346 |
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Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
7.299 |
0.097 |
0. |
0. |
0. |
0. |
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Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 77 | 64 | 231 | 215 | 0 | 86 |
normalized size | 1 | 1. | 0.91 | 0.75 | 2.72 | 2.53 | 0. | 1.01 |
time (sec) |
N/A |
0.062 |
0.145 |
0.017 |
2.33 |
1.653 |
0. |
1.352 |
|
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Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
B |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 42 | 35 | 182 | 135 | 121 | 73 |
normalized size | 1 | 1. | 0.91 | 0.76 | 3.96 | 2.93 | 2.63 | 1.59 |
time (sec) |
N/A |
0.03 |
0.061 |
0.014 |
2.042 |
1.66 |
1.648 |
1.289 |
|
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Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 27 | 20 | 46 | 74 | 22 | 35 |
normalized size | 1 | 1. | 1. | 0.74 | 1.7 | 2.74 | 0.81 | 1.3 |
time (sec) |
N/A |
0.015 |
0.013 |
0.013 |
1.737 |
1.685 |
0.887 |
1.281 |
|
|
|
|
|
|
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|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
10.992 |
0.106 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.049 |
11.036 |
0.162 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 285 | 285 | 186 | 399 | 5825 | 567 | 0 | 768 |
normalized size | 1 | 1. | 0.65 | 1.4 | 20.44 | 1.99 | 0. | 2.69 |
time (sec) |
N/A |
0.271 |
1.3 |
0.01 |
3.952 |
1.797 |
0. |
1.261 |
|
|
|
|
|
|
|
|
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|
|
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|
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|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 128 | 184 | 1702 | 359 | 0 | 441 |
normalized size | 1 | 1. | 0.91 | 1.31 | 12.16 | 2.56 | 0. | 3.15 |
time (sec) |
N/A |
0.056 |
0.668 |
0.01 |
2.714 |
1.741 |
0. |
1.282 |
|
|
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|
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|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 84 | 82 | 393 | 279 | 88 | 182 |
normalized size | 1 | 1. | 0.87 | 0.85 | 4.05 | 2.88 | 0.91 | 1.88 |
time (sec) |
N/A |
0.022 |
0.095 |
0.007 |
2.047 |
1.78 |
0.58 |
1.216 |
|
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Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.011 |
7.652 |
0.115 |
0. |
0. |
0. |
0. |
|
|
|
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Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 291 | 291 | 215 | 378 | 6030 | 608 | 0 | 726 |
normalized size | 1 | 1. | 0.74 | 1.3 | 20.72 | 2.09 | 0. | 2.49 |
time (sec) |
N/A |
0.337 |
1.19 |
0.017 |
3.79 |
1.71 |
0. |
1.349 |
|
|
|
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Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
F |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 150 | 150 | 140 | 170 | 1786 | 369 | 0 | 410 |
normalized size | 1 | 1. | 0.93 | 1.13 | 11.91 | 2.46 | 0. | 2.73 |
time (sec) |
N/A |
0.093 |
0.437 |
0.016 |
2.678 |
1.539 |
0. |
1.25 |
|
|
|
|
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|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
A |
C |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 97 | 72 | 410 | 238 | 83 | 165 |
normalized size | 1 | 1. | 0.97 | 0.72 | 4.1 | 2.38 | 0.83 | 1.65 |
time (sec) |
N/A |
0.048 |
0.073 |
0.013 |
2.167 |
1.467 |
1.2 |
1.322 |
|
|
|
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Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.037 |
7.409 |
0.23 |
0. |
0. |
0. |
0. |
|
|
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|
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 [24] had the largest ratio of [ 0.4667
]
|
|
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|
| 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 |
8 |
6 |
1. |
15 |
0.4 |
|
|
|
|
|
|
|
2 |
A |
4 |
4 |
1. |
13 |
0.308 |
|
|
|
|
|
|
|
3 |
A |
3 |
3 |
1. |
11 |
0.273 |
|
|
|
|
|
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|
4 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
5 |
A |
5 |
4 |
1. |
33 |
0.121 |
|
|
|
|
|
|
|
6 |
A |
8 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
7 |
A |
4 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
8 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
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|
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9 |
A |
0 |
0 |
0. |
0 |
0. |
|
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|
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|
|
10 |
A |
5 |
4 |
1. |
35 |
0.114 |
|
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|
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|
|
11 |
A |
6 |
6 |
1. |
13 |
0.462 |
|
|
|
|
|
|
|
12 |
A |
3 |
3 |
1. |
11 |
0.273 |
|
|
|
|
|
|
|
13 |
A |
2 |
2 |
1. |
9 |
0.222 |
|
|
|
|
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|
|
14 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
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|
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|
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15 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
16 |
A |
10 |
7 |
1. |
17 |
0.412 |
|
|
|
|
|
|
|
17 |
A |
6 |
5 |
1. |
15 |
0.333 |
|
|
|
|
|
|
|
18 |
A |
5 |
4 |
1. |
13 |
0.308 |
|
|
|
|
|
|
|
19 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
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|
|
20 |
A |
10 |
7 |
1. |
18 |
0.389 |
|
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|
|
|
|
|
21 |
A |
6 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
22 |
A |
5 |
4 |
1. |
14 |
0.286 |
|
|
|
|
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23 |
A |
0 |
0 |
0. |
0 |
0. |
|
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|
|
|
|
|
24 |
A |
8 |
7 |
1. |
15 |
0.467 |
|
|
|
|
|
|
|
25 |
A |
5 |
4 |
1. |
13 |
0.308 |
|
|
|
|
|
|
|
26 |
A |
4 |
3 |
1. |
11 |
0.273 |
|
|
|
|
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27 |
A |
0 |
0 |
0. |
0 |
0. |
|
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|
28 |
A |
0 |
0 |
0. |
0 |
0. |
|
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|
|
|
|
|
29 |
A |
8 |
6 |
1. |
19 |
0.316 |
|
|
|
|
|
|
|
30 |
A |
4 |
4 |
1. |
17 |
0.235 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1. |
11 |
0.273 |
|
|
|
|
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|
32 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
33 |
A |
10 |
7 |
1. |
21 |
0.333 |
|
|
|
|
|
|
|
34 |
A |
6 |
5 |
1. |
19 |
0.263 |
|
|
|
|
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|
|
35 |
A |
5 |
4 |
1. |
13 |
0.308 |
|
|
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|
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|
36 |
A |
0 |
0 |
0. |
0 |
0. |
|
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