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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
35, 37, 38, 39, 40, 41, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61 }
B grade: { 6, 11, 12, 34, 42, 43, 47 }
C grade: { 7, 13, 48, 59 }
F grade: { 36}
2.1.3 Maple
A grade: { 4, 5, 8, 9, 10, 14, 15, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 33, 40, 41, 45, 46, 50, 51, 55, 56, 60,
61 }
B grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 18, 22, 23, 28, 29, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 54, 57,
58, 59 }
C grade: { }
F grade: { 34, 35, 36 }
2.1.4 Maxima
A grade: { 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41,
44, 45, 46, 50, 51, 55, 56, 57, 58, 60, 61
B grade: { 1, 2, 3, 8, 12, 13, 37, 38, 42, 43, 47, 48, 49, 52, 53 }
C grade: { }
F grade: { 32, 33, 34, 35, 36, 39, 54, 59 }
2.1.5 FriCAS
A grade: { 4, 5, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 40, 41, 45, 46,
50, 51, 55, 56, 60, 61 }
B grade: { 3, 7, 8, 13, 22, 23, 27, 28, 39, 44, 49, 54, 59 }
C grade: { 1, 2, 6, 11, 12, 37, 38, 42, 43, 47, 48, 52, 53, 57, 58 }
F grade: { }
2.1.6 Sympy
A grade: { 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 22, 23, 24, 27, 28, 29, 32, 33, 40, 41, 45, 46, 50, 51, 55, 56, 60,
61 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 19, 20, 21, 25, 26, 30, 31, 34, 35, 36, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52,
53, 54, 57, 58, 59 }
2.1.7 Giac
A grade: { 4, 5, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 40, 41, 45,
46, 50, 51, 55, 56, 60, 61 }
B grade: { 8 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 34, 35, 36, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 54, 57, 58, 59
}
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 |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 91 | 200 | 230 | 644 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 2.3 | 2.64 | 7.4 | 0. | 0. |
time (sec) |
N/A |
0.159 |
0.01 |
0.101 |
1.048 |
2.541 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 66 | 166 | 186 | 489 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 2.63 | 2.95 | 7.76 | 0. | 0. |
time (sec) |
N/A |
0.135 |
0.01 |
0.03 |
1.105 |
2.194 |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 47 | 122 | 132 | 336 | 0 | 0 |
normalized size | 1 | 1. | 1.04 | 2.71 | 2.93 | 7.47 | 0. | 0. |
time (sec) |
N/A |
0.085 |
0.006 |
0.026 |
1.051 |
2.158 |
0. |
0. |
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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 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.016 |
12.211 |
0.085 |
0. |
0. |
0. |
0. |
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Problem 5 |
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 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.017 |
18.118 |
0.055 |
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 |
B |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 204 | 198 | 197 | 1643 | 0 | 0 |
normalized size | 1 | 1. | 2.34 | 2.28 | 2.26 | 18.89 | 0. | 0. |
time (sec) |
N/A |
0.178 |
3.921 |
0.028 |
1.379 |
2.244 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
A |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 163 | 156 | 146 | 1211 | 0 | 0 |
normalized size | 1 | 1. | 2.51 | 2.4 | 2.25 | 18.63 | 0. | 0. |
time (sec) |
N/A |
0.119 |
5.09 |
0.029 |
1.35 |
2.212 |
0. |
0. |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 46 | 54 | 155 | 477 | 80 | 132 |
normalized size | 1 | 1. | 1.48 | 1.74 | 5. | 15.39 | 2.58 | 4.26 |
time (sec) |
N/A |
0.027 |
0.153 |
0.029 |
1.184 |
2.147 |
4.74 |
1.178 |
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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 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
20.161 |
0.04 |
0. |
0. |
0. |
0. |
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Problem 10 |
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 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
19.436 |
0.063 |
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 |
B |
B |
A |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 179 | 179 | 390 | 375 | 408 | 5023 | 0 | 0 |
normalized size | 1 | 1. | 2.18 | 2.09 | 2.28 | 28.06 | 0. | 0. |
time (sec) |
N/A |
0.33 |
4.598 |
0.049 |
1.298 |
2.406 |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 295 | 246 | 305 | 3729 | 0 | 0 |
normalized size | 1 | 1. | 2.59 | 2.16 | 2.68 | 32.71 | 0. | 0. |
time (sec) |
N/A |
0.207 |
3.258 |
0.043 |
1.263 |
2.278 |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 232 | 164 | 201 | 2569 | 0 | 0 |
normalized size | 1 | 1. | 2.83 | 2. | 2.45 | 31.33 | 0. | 0. |
time (sec) |
N/A |
0.122 |
6.132 |
0.034 |
1.223 |
2.215 |
0. |
0. |
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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 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
14.899 |
0.29 |
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 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
11.14 |
0.227 |
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 |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 169 | 169 | 244 | 959 | 247 | 640 | 864 | 261 |
normalized size | 1 | 1. | 1.44 | 5.67 | 1.46 | 3.79 | 5.11 | 1.54 |
time (sec) |
N/A |
0.185 |
0.401 |
0.077 |
1.375 |
2.083 |
5.251 |
1.184 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 169 | 469 | 167 | 419 | 522 | 166 |
normalized size | 1 | 1. | 1.39 | 3.84 | 1.37 | 3.43 | 4.28 | 1.36 |
time (sec) |
N/A |
0.117 |
0.253 |
0.065 |
1.254 |
2.084 |
2.181 |
1.152 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 81 | 175 | 97 | 239 | 250 | 88 |
normalized size | 1 | 1. | 1.09 | 2.36 | 1.31 | 3.23 | 3.38 | 1.19 |
time (sec) |
N/A |
0.051 |
0.262 |
0.058 |
1.166 |
2.117 |
1.45 |
1.19 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 122 | 61 | 65 | 166 | 0 | 68 |
normalized size | 1 | 1. | 0.78 | 0.39 | 0.41 | 1.06 | 0. | 0.43 |
time (sec) |
N/A |
0.251 |
0.25 |
0.182 |
1.752 |
2.181 |
0. |
1.15 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 159 | 159 | 206 | 91 | 76 | 498 | 0 | 149 |
normalized size | 1 | 1. | 1.3 | 0.57 | 0.48 | 3.13 | 0. | 0.94 |
time (sec) |
N/A |
0.213 |
0.764 |
0.203 |
2.258 |
2.134 |
0. |
1.333 |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 211 | 211 | 265 | 210 | 92 | 748 | 0 | 242 |
normalized size | 1 | 1. | 1.26 | 1. | 0.44 | 3.55 | 0. | 1.15 |
time (sec) |
N/A |
0.301 |
1.073 |
0.219 |
3.059 |
2.17 |
0. |
1.149 |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 230 | 230 | 420 | 2200 | 401 | 1257 | 2200 | 517 |
normalized size | 1 | 1. | 1.83 | 9.57 | 1.74 | 5.47 | 9.57 | 2.25 |
time (sec) |
N/A |
0.268 |
1.023 |
0.108 |
2.493 |
2.171 |
10.887 |
1.167 |
|
|
|
|
|
|
|
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|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 170 | 170 | 207 | 1087 | 258 | 811 | 1358 | 306 |
normalized size | 1 | 1. | 1.22 | 6.39 | 1.52 | 4.77 | 7.99 | 1.8 |
time (sec) |
N/A |
0.192 |
0.946 |
0.097 |
2.028 |
2.266 |
3.463 |
1.143 |
|
|
|
|
|
|
|
|
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|
|
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|
|
Problem 24 |
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 | 133 | 133 | 114 | 74 | 144 | 454 | 700 | 147 |
normalized size | 1 | 1. | 0.86 | 0.56 | 1.08 | 3.41 | 5.26 | 1.11 |
time (sec) |
N/A |
0.129 |
0.522 |
0.122 |
1.51 |
2.037 |
2.846 |
1.145 |
|
|
|
|
|
|
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|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 297 | 297 | 199 | 106 | 109 | 304 | 0 | 105 |
normalized size | 1 | 1. | 0.67 | 0.36 | 0.37 | 1.02 | 0. | 0.35 |
time (sec) |
N/A |
0.704 |
0.439 |
0.27 |
3.381 |
2.193 |
0. |
1.191 |
|
|
|
|
|
|
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|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 420 | 420 | 442 | 164 | 135 | 1413 | 0 | 255 |
normalized size | 1 | 1. | 1.05 | 0.39 | 0.32 | 3.36 | 0. | 0.61 |
time (sec) |
N/A |
0.733 |
1.419 |
0.282 |
6.056 |
2.242 |
0. |
1.928 |
|
|
|
|
|
|
|
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|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 336 | 336 | 615 | 368 | 548 | 1877 | 3925 | 774 |
normalized size | 1 | 1. | 1.83 | 1.1 | 1.63 | 5.59 | 11.68 | 2.3 |
time (sec) |
N/A |
0.37 |
2.263 |
0.183 |
7.633 |
2.225 |
13.282 |
1.241 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 371 | 2098 | 343 | 1215 | 2448 | 447 |
normalized size | 1 | 1. | 1.51 | 8.53 | 1.39 | 4.94 | 9.95 | 1.82 |
time (sec) |
N/A |
0.258 |
1.372 |
0.125 |
5.602 |
2.132 |
5.252 |
1.259 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 185 | 791 | 186 | 709 | 1290 | 204 |
normalized size | 1 | 1. | 1.01 | 4.32 | 1.02 | 3.87 | 7.05 | 1.11 |
time (sec) |
N/A |
0.205 |
0.701 |
0.107 |
3.252 |
2.09 |
4.7 |
1.169 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 437 | 437 | 312 | 151 | 154 | 439 | 0 | 146 |
normalized size | 1 | 1. | 0.71 | 0.35 | 0.35 | 1. | 0. | 0.33 |
time (sec) |
N/A |
1.765 |
0.649 |
0.477 |
10.932 |
2.23 |
0. |
1.17 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 692 | 692 | 796 | 239 | 189 | 2685 | 0 | 358 |
normalized size | 1 | 1. | 1.15 | 0.35 | 0.27 | 3.88 | 0. | 0.52 |
time (sec) |
N/A |
1.762 |
3.331 |
0.408 |
23.914 |
2.339 |
0. |
2.382 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
A |
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 |
0.05 |
23.682 |
0.095 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
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 |
0.027 |
13.574 |
0.085 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 186 | 0 | 0 | 366 | 0 | 0 |
normalized size | 1 | 1. | 2.11 | 0. | 0. | 4.16 | 0. | 0. |
time (sec) |
N/A |
0.117 |
1.13 |
0.139 |
0. |
2.238 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 152 | 194 | 0 | 0 | 609 | 0 | 0 |
normalized size | 1 | 1. | 1.28 | 0. | 0. | 4.01 | 0. | 0. |
time (sec) |
N/A |
0.172 |
8.102 |
0.128 |
0. |
2.342 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
A |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 223 | 223 | 0 | 0 | 0 | 853 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 3.83 | 0. | 0. |
time (sec) |
N/A |
0.233 |
180.002 |
0.161 |
0. |
2.287 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 133 | 133 | 249 | 748 | 566 | 1207 | 0 | 0 |
normalized size | 1 | 1. | 1.87 | 5.62 | 4.26 | 9.08 | 0. | 0. |
time (sec) |
N/A |
0.26 |
0.358 |
0.153 |
1.414 |
2.359 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 101 | 101 | 149 | 447 | 324 | 782 | 0 | 0 |
normalized size | 1 | 1. | 1.48 | 4.43 | 3.21 | 7.74 | 0. | 0. |
time (sec) |
N/A |
0.217 |
0.217 |
0.055 |
1.382 |
2.243 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 87 | 201 | 0 | 435 | 0 | 0 |
normalized size | 1 | 1. | 1.16 | 2.68 | 0. | 5.8 | 0. | 0. |
time (sec) |
N/A |
0.126 |
0.056 |
0.043 |
0. |
2.119 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
4.994 |
0.14 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.029 |
20.337 |
0.134 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 271 | 271 | 857 | 1393 | 1054 | 7128 | 0 | 0 |
normalized size | 1 | 1. | 3.16 | 5.14 | 3.89 | 26.3 | 0. | 0. |
time (sec) |
N/A |
0.546 |
9.889 |
0.168 |
1.491 |
2.836 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 478 | 793 | 667 | 4251 | 0 | 0 |
normalized size | 1 | 1. | 2.29 | 3.79 | 3.19 | 20.34 | 0. | 0. |
time (sec) |
N/A |
0.404 |
10.212 |
0.088 |
1.438 |
2.665 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 192 | 318 | 329 | 2142 | 0 | 0 |
normalized size | 1 | 1. | 1.51 | 2.5 | 2.59 | 16.87 | 0. | 0. |
time (sec) |
N/A |
0.184 |
1.9 |
0.082 |
1.41 |
2.674 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
42.653 |
0.376 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.053 |
30.126 |
0.589 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 556 | 556 | 2043 | 2777 | 2067 | 23616 | 0 | 0 |
normalized size | 1 | 1. | 3.67 | 4.99 | 3.72 | 42.47 | 0. | 0. |
time (sec) |
N/A |
1.049 |
14.001 |
0.239 |
1.661 |
4.18 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 401 | 401 | 2031 | 1542 | 1346 | 13835 | 0 | 0 |
normalized size | 1 | 1. | 5.06 | 3.85 | 3.36 | 34.5 | 0. | 0. |
time (sec) |
N/A |
0.698 |
12.915 |
0.166 |
1.659 |
3.196 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 266 | 651 | 713 | 6711 | 0 | 0 |
normalized size | 1 | 1. | 1.02 | 2.49 | 2.73 | 25.71 | 0. | 0. |
time (sec) |
N/A |
0.35 |
3.457 |
0.121 |
1.504 |
2.63 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.055 |
62.508 |
0.599 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.053 |
59.527 |
0.924 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 239 | 1145 | 703 | 1808 | 0 | 0 |
normalized size | 1 | 1. | 1.14 | 5.45 | 3.35 | 8.61 | 0. | 0. |
time (sec) |
N/A |
0.345 |
3.366 |
0.203 |
1.803 |
2.373 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 156 | 156 | 191 | 722 | 448 | 1258 | 0 | 0 |
normalized size | 1 | 1. | 1.22 | 4.63 | 2.87 | 8.06 | 0. | 0. |
time (sec) |
N/A |
0.284 |
3.596 |
0.147 |
1.634 |
2.305 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 144 | 357 | 0 | 798 | 0 | 0 |
normalized size | 1 | 1. | 1.33 | 3.31 | 0. | 7.39 | 0. | 0. |
time (sec) |
N/A |
0.163 |
2.922 |
0.131 |
0. |
2.292 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.063 |
14.072 |
0.155 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.06 |
22.471 |
0.156 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 638 | 638 | 646 | 2039 | 1426 | 12840 | 0 | 0 |
normalized size |
1 |
1. |
1.01 |
3.2 |
2.24 |
20.13 |
0. |
0. |
time (sec) |
N/A |
2.274 |
12.299 |
0.301 |
2.775 |
3.555 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 475 | 475 | 509 | 1217 | 1014 | 7876 | 0 | 0 |
normalized size |
1 |
1. |
1.07 |
2.56 |
2.13 |
16.58 |
0. |
0. |
time (sec) |
N/A |
1.714 |
8.295 |
0.263 |
2.791 |
2.905 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 196 | 196 | 477 | 620 | 0 | 4058 | 0 | 0 |
normalized size |
1 |
1. |
2.43 |
3.16 |
0. |
20.7 |
0. |
0. |
time (sec) |
N/A |
0.296 |
6.004 |
0.234 |
0. |
2.559 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.062 |
144.035 |
0.752 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
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 | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.059 |
115.397 |
0.697 |
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 [11] had the largest ratio of [ 0.8333
]
|
|
|
|
|
|
|
| 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 |
6 |
1. |
10 |
0.6 |
|
|
|
|
|
|
|
2 |
A |
5 |
5 |
1. |
10 |
0.5 |
|
|
|
|
|
|
|
3 |
A |
4 |
4 |
1. |
8 |
0.5 |
|
|
|
|
|
|
|
4 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
5 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
6 |
A |
7 |
7 |
1. |
12 |
0.583 |
|
|
|
|
|
|
|
7 |
A |
6 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
8 |
A |
3 |
3 |
1. |
10 |
0.3 |
|
|
|
|
|
|
|
9 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
10 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
11 |
A |
13 |
10 |
1. |
12 |
0.833 |
|
|
|
|
|
|
|
12 |
A |
9 |
8 |
1. |
12 |
0.667 |
|
|
|
|
|
|
|
13 |
A |
7 |
7 |
1. |
10 |
0.7 |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
15 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
16 |
A |
5 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
17 |
A |
4 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1. |
18 |
0.167 |
|
|
|
|
|
|
|
19 |
A |
7 |
4 |
1. |
20 |
0.2 |
|
|
|
|
|
|
|
20 |
A |
7 |
4 |
1. |
20 |
0.2 |
|
|
|
|
|
|
|
|
21 |
A |
8 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
22 |
A |
10 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
23 |
A |
8 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
24 |
A |
7 |
3 |
1. |
18 |
0.167 |
|
|
|
|
|
|
|
25 |
A |
21 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
26 |
A |
24 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
27 |
A |
14 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
28 |
A |
11 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
29 |
A |
11 |
3 |
1. |
18 |
0.167 |
|
|
|
|
|
|
|
30 |
A |
53 |
7 |
1. |
20 |
0.35 |
|
|
|
|
|
|
|
31 |
A |
60 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
32 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
33 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
35 |
A |
4 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
36 |
A |
5 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
37 |
A |
8 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
38 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
39 |
A |
6 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
40 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
41 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
42 |
A |
15 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
43 |
A |
13 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
44 |
A |
9 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
47 |
A |
28 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
48 |
A |
22 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
49 |
A |
16 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
51 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
52 |
A |
6 |
6 |
1. |
20 |
0.3 |
|
|
|
|
|
|
|
53 |
A |
5 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
54 |
A |
4 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
55 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
56 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
57 |
A |
28 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
58 |
A |
24 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
59 |
A |
5 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
60 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
61 |
A |
0 |
0 |
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