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, 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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 27, 28, 32, 33, 35, 37, 38,
41, 42, 43, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 }
B grade: { 31, 36, 55, 56 }
C grade: { 6, 24, 25, 26, 29, 30, 44, 45, 46, 49, 50, 51, 57 }
F grade: { 34, 39, 40}
2.1.3 Maple
A grade: { 5, 6, 12, 14, 23, 25, 26, 27, 28, 29, 30, 33, 35, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 63,
64, 65, 66, 67, 68, 69, 70 }
B grade: { 1, 2, 3, 4, 7, 8, 9, 11, 13, 16, 19, 21, 22, 24, 31, 32, 38, 53, 60, 61, 62 }
C grade: { 10, 15, 17, 18, 20, 34, 36, 37, 39, 40, 52, 56, 57, 58, 59 }
F grade: { 41 }
2.1.4 Maxima
A grade: { 5, 24, 25, 26, 27, 29, 30, 44, 45, 46, 47, 48, 49, 50, 51, 55, 60, 65, 66, 69, 70
B grade: { 1, 2, 3, 6, 7, 9, 12, 14, 21, 22, 54 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 15, 16, 17, 18, 19, 20, 23, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 52,
53, 56, 57, 58, 59, 61, 62, 63, 64, 67, 68 }
2.1.5 FriCAS
A grade: { 3, 5, 6, 9, 12, 24, 25, 26, 27, 29, 42, 43, 44, 45, 46, 47, 49, 50, 51, 65, 66, 69, 70 }
B grade: { 1, 2, 7, 14, 30 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68 }
2.1.6 Sympy
A grade: { 1, 2, 3, 5, 6, 7, 9, 12, 23, 24, 25, 26, 27, 44, 45, 46, 47, 65, 66, 69 }
B grade: { 49, 50, 51 }
C grade: { }
F grade: { 4, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 48, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 70 }
2.1.7 Giac
A grade: { 5, 6, 23, 27, 29, 42, 43, 44, 45, 46, 47, 49, 50, 51, 65, 66, 69, 70 }
B grade: { 1, 2, 3, 7, 9, 14, 24, 25, 26, 30 }
C grade: { }
F grade: { 4, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68 }
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 |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 56 | 225 | 500 | 315 | 231 | 317 |
normalized size | 1 | 1. | 0.78 | 3.12 | 6.94 | 4.38 | 3.21 | 4.4 |
time (sec) |
N/A |
0.251 |
0.06 |
0.035 |
1.518 |
1.577 |
4.44 |
1.151 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 54 | 161 | 321 | 278 | 178 | 216 |
normalized size | 1 | 1. | 0.81 | 2.4 | 4.79 | 4.15 | 2.66 | 3.22 |
time (sec) |
N/A |
0.055 |
0.017 |
0.037 |
1.514 |
1.699 |
2.473 |
1.169 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 40 | 92 | 162 | 139 | 95 | 178 |
normalized size | 1 | 1. | 0.83 | 1.92 | 3.38 | 2.9 | 1.98 | 3.71 |
time (sec) |
N/A |
0.03 |
0.012 |
0.036 |
1.499 |
1.548 |
1.567 |
1.139 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 52 | 132 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.83 | 2.1 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.058 |
0.022 |
0.046 |
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 |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 50 | 73 | 124 | 184 | 2814 | 100 |
normalized size | 1 | 1. | 0.82 | 1.2 | 2.03 | 3.02 | 46.13 | 1.64 |
time (sec) |
N/A |
0.047 |
0.018 |
0.041 |
0.988 |
1.679 |
125.41 |
1.087 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 51 | 71 | 162 | 162 | 314 | 113 |
normalized size | 1 | 1. | 0.81 | 1.13 | 2.57 | 2.57 | 4.98 | 1.79 |
time (sec) |
N/A |
0.045 |
0.013 |
0.041 |
1.483 |
1.688 |
5.35 |
1.126 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 216 | 543 | 806 | 703 | 575 | 784 |
normalized size | 1 | 1. | 1.38 | 3.46 | 5.13 | 4.48 | 3.66 | 4.99 |
time (sec) |
N/A |
0.221 |
0.098 |
0.046 |
5.514 |
1.797 |
11.764 |
1.375 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 163 | 593 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.89 | 3.24 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.221 |
0.362 |
0.123 |
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 |
A |
B |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 107 | 220 | 294 | 335 | 240 | 294 |
normalized size | 1 | 1. | 1.13 | 2.32 | 3.09 | 3.53 | 2.53 | 3.09 |
time (sec) |
N/A |
0.119 |
0.058 |
0.046 |
5.202 |
1.723 |
3.045 |
1.238 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 170 | 1433 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.93 | 7.83 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.339 |
0.063 |
0.588 |
0. |
0. |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 135 | 471 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.13 |
3.96 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.187 |
0.191 |
0.125 |
0. |
0. |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 194 | 182 | 362 | 470 | 986 | 0 |
normalized size | 1 | 1. | 1.66 | 1.56 | 3.09 | 4.02 | 8.43 | 0. |
time (sec) |
N/A |
0.153 |
0.108 |
0.053 |
1.621 |
1.879 |
6.516 |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 194 | 194 | 163 | 547 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.84 |
2.82 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.257 |
0.606 |
0.125 |
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 |
A |
A |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 170 | 170 | 245 | 242 | 721 | 937 | 0 | 1902 |
normalized size |
1 |
1. |
1.44 |
1.42 |
4.24 |
5.51 |
0. |
11.19 |
time (sec) |
N/A |
0.227 |
0.271 |
0.054 |
1.879 |
2.094 |
0. |
1.494 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 271 | 271 | 349 | 3242 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.29 | 11.96 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.439 |
0.53 |
0.947 |
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 |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 164 | 164 | 196 | 567 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.2 | 3.46 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.243 |
0.267 |
0.132 |
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 |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 279 | 279 | 252 | 2894 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.9 | 10.37 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.458 |
0.092 |
0.377 |
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 |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 163 | 163 | 263 | 2696 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.61 | 16.54 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.3 |
0.449 |
0.4 |
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 |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 180 | 180 | 225 | 631 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.25 | 3.51 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.319 |
0.226 |
0.138 |
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 |
C |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 287 | 287 | 360 | 7083 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.25 |
24.68 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.5 |
0.824 |
0.864 |
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 |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 31 | 68 | 59 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 2.19 | 1.9 | 0. | 0. | 0. |
time (sec) |
N/A |
0.038 |
0.003 |
0.046 |
1.622 |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 34 | 98 | 166 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.83 | 2.39 | 4.05 | 0. | 0. | 0. |
time (sec) |
N/A |
0.045 |
0.007 |
0.046 |
1.682 |
0. |
0. |
0. |
|
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Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
F(-2) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.017 |
5.408 |
0.472 |
0. |
0. |
0. |
0. |
|
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Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 233 | 233 | 157 | 494 | 467 | 667 | 627 | 961 |
normalized size | 1 | 1. | 0.67 | 2.12 | 2. | 2.86 | 2.69 | 4.12 |
time (sec) |
N/A |
0.381 |
0.253 |
0.042 |
1.543 |
1.624 |
10.232 |
3.444 |
|
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Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 118 | 283 | 297 | 432 | 357 | 562 |
normalized size | 1 | 1. | 0.76 | 1.83 | 1.92 | 2.79 | 2.3 | 3.63 |
time (sec) |
N/A |
0.19 |
0.139 |
0.041 |
1.532 |
1.466 |
4.998 |
1.516 |
|
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Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 163 | 146 | 157 | 232 | 177 | 269 |
normalized size | 1 | 1. | 1.68 | 1.51 | 1.62 | 2.39 | 1.82 | 2.77 |
time (sec) |
N/A |
0.112 |
0.055 |
0.039 |
1.483 |
1.422 |
2.468 |
1.205 |
|
|
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Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 49 | 42 | 49 | 119 | 51 | 49 |
normalized size | 1 | 1. | 1.29 | 1.11 | 1.29 | 3.13 | 1.34 | 1.29 |
time (sec) |
N/A |
0.018 |
0.013 |
0.033 |
0.973 |
1.515 |
0.566 |
1.082 |
|
|
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Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 162 | 162 | 160 | 224 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 1.38 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.15 |
0.104 |
0.072 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 151 | 151 | 121 | 205 | 239 | 435 | 0 | 393 |
normalized size |
1 |
1. |
0.8 |
1.36 |
1.58 |
2.88 |
0. |
2.6 |
time (sec) |
N/A |
0.121 |
0.157 |
0.045 |
1.533 |
3.038 |
0. |
1.094 |
|
|
|
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Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 175 | 438 | 552 | 1415 | 0 | 1723 |
normalized size |
1 |
1. |
0.77 |
1.93 |
2.43 |
6.23 |
0. |
7.59 |
time (sec) |
N/A |
0.303 |
0.555 |
0.046 |
1.562 |
11.938 |
0. |
6.762 |
|
|
|
|
|
|
|
|
|
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|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 382 | 382 | 801 | 1622 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.1 | 4.25 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.574 |
3.655 |
0.133 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 222 | 222 | 264 | 748 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.19 | 3.37 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.373 |
0.379 |
0.119 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 102 | 102 | 109 | 180 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.07 | 1.76 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.107 |
0.073 |
0.105 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
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|
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|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 0 | 2149 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 8.23 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.165 |
5.445 |
1.339 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 568 | 568 | 419 | 1087 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.74 |
1.91 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.351 |
6.315 |
0.132 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 564 | 564 | 1844 | 6682 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 3.27 | 11.85 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.937 |
8.962 |
2.95 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 337 | 337 | 592 | 16362 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.76 | 48.55 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.632 |
0.679 |
1.261 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
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|
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|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 143 | 143 | 212 | 359 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.48 | 2.51 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.214 |
0.115 |
0.155 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 372 | 372 | 0 | 4389 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 11.8 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.203 |
71.529 |
0.765 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
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|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1233 | 1233 | 0 | 4764 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
3.86 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
2.311 |
70.268 |
1.013 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 177 | 177 | 162 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.92 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.246 |
0.288 |
1.284 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.058 |
4.445 |
1.105 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.056 |
0.405 |
1.097 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 95 | 132 | 140 | 203 | 155 | 139 |
normalized size | 1 | 1. | 0.9 | 1.25 | 1.32 | 1.92 | 1.46 | 1.31 |
time (sec) |
N/A |
0.111 |
0.07 |
0.039 |
1.504 |
1.685 |
2.368 |
1.108 |
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 114 | 95 | 115 | 161 | 117 | 111 |
normalized size | 1 | 1. | 1.44 | 1.2 | 1.46 | 2.04 | 1.48 | 1.41 |
time (sec) |
N/A |
0.092 |
0.048 |
0.036 |
1.503 |
1.749 |
1.432 |
1.093 |
|
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 90 | 66 | 92 | 123 | 78 | 81 |
normalized size | 1 | 1. | 1.5 | 1.1 | 1.53 | 2.05 | 1.3 | 1.35 |
time (sec) |
N/A |
0.055 |
0.031 |
0.035 |
1.514 |
1.693 |
0.965 |
1.1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 39 | 36 | 42 | 97 | 46 | 42 |
normalized size | 1 | 1. | 1.18 | 1.09 | 1.27 | 2.94 | 1.39 | 1.27 |
time (sec) |
N/A |
0.011 |
0.014 |
0.035 |
1.008 |
1.748 |
0.559 |
1.118 |
|
|
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 171 | 103 | 181 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.42 |
0.86 |
1.51 |
0. |
0. |
0. |
time (sec) |
N/A |
0.106 |
0.008 |
0.047 |
1.711 |
0. |
0. |
0. |
|
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 67 | 63 | 104 | 151 | 323 | 95 |
normalized size | 1 | 1. | 1.08 | 1.02 | 1.68 | 2.44 | 5.21 | 1.53 |
time (sec) |
N/A |
0.039 |
0.056 |
0.043 |
1.544 |
1.716 |
10.951 |
1.112 |
|
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 96 | 96 | 92 | 105 | 151 | 225 | 644 | 155 |
normalized size | 1 | 1. | 0.96 | 1.09 | 1.57 | 2.34 | 6.71 | 1.61 |
time (sec) |
N/A |
0.083 |
0.096 |
0.043 |
1.526 |
1.892 |
16.572 |
1.085 |
|
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
B |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 129 | 129 | 128 | 162 | 223 | 321 | 1127 | 239 |
normalized size | 1 | 1. | 0.99 | 1.26 | 1.73 | 2.49 | 8.74 | 1.85 |
time (sec) |
N/A |
0.114 |
0.136 |
0.043 |
1.543 |
1.653 |
26.956 |
1.101 |
|
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 863 | 863 | 701 | 631 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.81 |
0.73 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.207 |
0.805 |
0.636 |
0. |
0. |
0. |
0. |
|
|
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Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 543 | 543 | 409 | 2192 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.75 |
4.04 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.607 |
0.329 |
0.713 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 152 | 231 | 198 | 383 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.52 | 1.3 | 2.52 | 0. | 0. | 0. |
time (sec) |
N/A |
0.138 |
0.02 |
0.054 |
1.974 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
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|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 771 | 317 | 383 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.16 |
1.3 |
1.57 |
0. |
0. |
0. |
time (sec) |
N/A |
0.239 |
11.075 |
0.06 |
2.018 |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 668 | 668 | 1536 | 53434 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.3 |
79.99 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.855 |
21.489 |
2.177 |
0. |
0. |
0. |
0. |
|
|
|
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|
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|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 933 | 933 | 933 | 682 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1. |
0.73 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
1.367 |
7.074 |
0.658 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 673 | 673 | 604 | 344 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.9 |
0.51 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.898 |
0.518 |
0.338 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 770 | 770 | 666 | 377 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
0.49 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.982 |
0.719 |
0.318 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 274 | 274 | 283 | 833 | 443 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.03 | 3.04 | 1.62 | 0. | 0. | 0. |
time (sec) |
N/A |
0.271 |
0.053 |
0.444 |
1.835 |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 543 | 543 | 409 | 2192 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.75 |
4.04 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.6 |
0.354 |
0.702 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 367 | 367 | 443 | 4743 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.21 |
12.92 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.673 |
0.413 |
1.036 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 132 | 132 | 97 | 143 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.73 |
1.08 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.096 |
0.101 |
0.397 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 216 | 216 | 125 | 176 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.58 |
0.81 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.162 |
0.064 |
0.439 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
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|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 163 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
7.41 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.038 |
0.363 |
1.044 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 0 | 165 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
6.88 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.051 |
0.073 |
0.996 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
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|
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|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 187 | 187 | 145 | 187 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.78 |
1. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.216 |
0.657 |
0.619 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 281 | 281 | 189 | 222 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.67 |
0.79 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.331 |
0.133 |
1.125 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 0 | 181 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
6.24 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.131 |
1.433 |
1.329 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 0 | 225 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
7.26 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.179 |
0.312 |
1.3 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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 [35] had the largest ratio of [ 1.25
]
|
|
|
|
|
|
|
| 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 |
6 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
2 |
A |
6 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
3 |
A |
5 |
5 |
1. |
19 |
0.263 |
|
|
|
|
|
|
|
4 |
A |
5 |
4 |
1. |
21 |
0.19 |
|
|
|
|
|
|
|
5 |
A |
7 |
7 |
1. |
21 |
0.333 |
|
|
|
|
|
|
|
6 |
A |
5 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
7 |
A |
13 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
8 |
A |
11 |
10 |
1. |
23 |
0.435 |
|
|
|
|
|
|
|
9 |
A |
8 |
7 |
1. |
21 |
0.333 |
|
|
|
|
|
|
|
10 |
A |
8 |
7 |
1. |
23 |
0.304 |
|
|
|
|
|
|
|
11 |
A |
6 |
6 |
1. |
23 |
0.261 |
|
|
|
|
|
|
|
12 |
A |
10 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
13 |
A |
10 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
14 |
A |
15 |
10 |
1. |
23 |
0.435 |
|
|
|
|
|
|
|
15 |
A |
14 |
11 |
1. |
23 |
0.478 |
|
|
|
|
|
|
|
16 |
A |
10 |
10 |
1. |
21 |
0.476 |
|
|
|
|
|
|
|
17 |
A |
10 |
8 |
1. |
23 |
0.348 |
|
|
|
|
|
|
|
18 |
A |
7 |
8 |
1. |
23 |
0.348 |
|
|
|
|
|
|
|
19 |
A |
9 |
8 |
1. |
23 |
0.348 |
|
|
|
|
|
|
|
20 |
A |
16 |
13 |
1. |
23 |
0.565 |
|
|
|
|
|
|
|
|
21 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
22 |
A |
5 |
4 |
1. |
19 |
0.21 |
|
|
|
|
|
|
|
23 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
24 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
26 |
A |
7 |
6 |
1. |
16 |
0.375 |
|
|
|
|
|
|
|
27 |
A |
4 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
28 |
A |
5 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
29 |
A |
8 |
8 |
1. |
18 |
0.444 |
|
|
|
|
|
|
|
30 |
A |
9 |
8 |
1. |
18 |
0.444 |
|
|
|
|
|
|
|
31 |
A |
16 |
13 |
1. |
20 |
0.65 |
|
|
|
|
|
|
|
32 |
A |
13 |
10 |
1. |
18 |
0.556 |
|
|
|
|
|
|
|
33 |
A |
6 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
35 |
A |
25 |
25 |
1. |
20 |
1.25 |
|
|
|
|
|
|
|
36 |
A |
21 |
14 |
1. |
20 |
0.7 |
|
|
|
|
|
|
|
37 |
A |
15 |
11 |
1. |
18 |
0.611 |
|
|
|
|
|
|
|
38 |
A |
6 |
7 |
1. |
12 |
0.583 |
|
|
|
|
|
|
|
39 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
40 |
A |
35 |
22 |
1. |
20 |
1.1 |
|
|
|
|
|
|
|
|
41 |
A |
6 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
43 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
44 |
A |
7 |
6 |
1. |
10 |
0.6 |
|
|
|
|
|
|
|
45 |
A |
7 |
6 |
1. |
10 |
0.6 |
|
|
|
|
|
|
|
46 |
A |
7 |
6 |
1. |
8 |
0.75 |
|
|
|
|
|
|
|
47 |
A |
3 |
3 |
1. |
6 |
0.5 |
|
|
|
|
|
|
|
48 |
A |
5 |
5 |
1. |
10 |
0.5 |
|
|
|
|
|
|
|
49 |
A |
7 |
7 |
1. |
10 |
0.7 |
|
|
|
|
|
|
|
50 |
A |
8 |
7 |
1. |
10 |
0.7 |
|
|
|
|
|
|
|
51 |
A |
8 |
7 |
1. |
10 |
0.7 |
|
|
|
|
|
|
|
52 |
A |
23 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
53 |
A |
17 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
54 |
A |
5 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
55 |
A |
15 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
56 |
A |
25 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
57 |
A |
31 |
7 |
1. |
16 |
0.438 |
|
|
|
|
|
|
|
58 |
A |
31 |
13 |
1. |
18 |
0.722 |
|
|
|
|
|
|
|
59 |
A |
37 |
16 |
1. |
18 |
0.889 |
|
|
|
|
|
|
|
60 |
A |
17 |
5 |
1. |
14 |
0.357 |
|
|
|
|
|
|
|
|
61 |
A |
17 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
62 |
A |
12 |
8 |
1. |
19 |
0.421 |
|
|
|
|
|
|
|
63 |
A |
2 |
2 |
1. |
28 |
0.071 |
|
|
|
|
|
|
|
64 |
A |
3 |
3 |
1. |
33 |
0.091 |
|
|
|
|
|
|
|
65 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
66 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
67 |
A |
4 |
4 |
1. |
35 |
0.114 |
|
|
|
|
|
|
|
68 |
A |
5 |
5 |
1. |
40 |
0.125 |
|
|
|
|
|
|
|
69 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
70 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|