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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 29, 30, 31, 32, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 53, 57, 58, 59 }
B grade: { 1, 3, 4, 5, 19, 20, 36, 37, 38, 52 }
C grade: { 21, 22, 23, 24, 25, 26, 27 }
F grade: { 33, 34, 35, 54, 55, 56}
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 11, 12, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 52, 53, 57, 58, 59
}
B grade: { 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51 }
C grade: { }
F grade: { 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 54, 55, 56 }
2.1.4 Maxima
A grade: { 4, 5, 6, 16, 36, 37, 38, 52, 53, 57, 58, 59
B grade: { 1, 2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17 }
C grade: { }
F grade: { 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 54, 55, 56 }
2.1.5 FriCAS
A grade: { 5, 6, 7, 8, 9, 10, 16, 43, 44, 45, 46, 47, 48, 52, 53, 57, 58, 59 }
B grade: { 1, 2, 3, 4, 11, 12, 13, 14, 15, 17, 18, 19, 20, 36, 37, 38, 39, 40, 41, 42, 49, 50, 51 }
C grade: { }
F grade: { 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 54, 55, 56 }
2.1.6 Sympy
A grade: { 57, 58 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 59
}
2.1.7 Giac
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 57, 58, 59
}
B grade: { 13, 14, 15, 17, 18, 36, 37, 38, 51 }
C grade: { }
F grade: { 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 54, 55, 56 }
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 |
B |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 113 | 89 | 162 | 537 | 0 | 130 |
normalized size | 1 | 1. | 2.05 | 1.62 | 2.95 | 9.76 | 0. | 2.36 |
time (sec) |
N/A |
0.071 |
0.765 |
0.033 |
0.994 |
0.497 |
0. |
1.463 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 83 | 67 | 131 | 433 | 0 | 99 |
normalized size | 1 | 1. | 1.89 | 1.52 | 2.98 | 9.84 | 0. | 2.25 |
time (sec) |
N/A |
0.066 |
0.325 |
0.033 |
0.991 |
0.488 |
0. |
1.395 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 63 | 45 | 92 | 301 | 0 | 72 |
normalized size | 1 | 1. | 2.33 | 1.67 | 3.41 | 11.15 | 0. | 2.67 |
time (sec) |
N/A |
0.087 |
0.164 |
0.028 |
1.031 |
0.487 |
0. |
1.348 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 44 | 24 | 42 | 204 | 0 | 32 |
normalized size | 1 | 1. | 2.2 | 1.2 | 2.1 | 10.2 | 0. | 1.6 |
time (sec) |
N/A |
0.056 |
0.053 |
0.027 |
0.97 |
0.481 |
0. |
1.369 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 26 | 14 | 22 | 68 | 0 | 18 |
normalized size | 1 | 1. | 2.17 | 1.17 | 1.83 | 5.67 | 0. | 1.5 |
time (sec) |
N/A |
0.02 |
0.024 |
0.024 |
1.02 |
0.444 |
0. |
1.337 |
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Problem 6 |
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 | 28 | 28 | 47 | 41 | 68 | 142 | 0 | 43 |
normalized size | 1 | 1. | 1.68 | 1.46 | 2.43 | 5.07 | 0. | 1.54 |
time (sec) |
N/A |
0.013 |
0.089 |
0.042 |
1.471 |
0.467 |
0. |
1.277 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 32 | 40 | 105 | 122 | 0 | 59 |
normalized size | 1 | 1. | 1.28 | 1.6 | 4.2 | 4.88 | 0. | 2.36 |
time (sec) |
N/A |
0.047 |
0.078 |
0.043 |
1.458 |
0.473 |
0. |
1.33 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 42 | 100 | 173 | 166 | 0 | 76 |
normalized size | 1 | 1. | 1.05 | 2.5 | 4.32 | 4.15 | 0. | 1.9 |
time (sec) |
N/A |
0.061 |
0.13 |
0.042 |
1.459 |
0.474 |
0. |
1.368 |
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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 |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 49 | 121 | 243 | 211 | 0 | 90 |
normalized size | 1 | 1. | 0.92 | 2.28 | 4.58 | 3.98 | 0. | 1.7 |
time (sec) |
N/A |
0.067 |
0.164 |
0.047 |
1.464 |
0.483 |
0. |
1.277 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 57 | 185 | 311 | 259 | 0 | 123 |
normalized size | 1 | 1. | 0.86 | 2.8 | 4.71 | 3.92 | 0. | 1.86 |
time (sec) |
N/A |
0.073 |
0.209 |
0.046 |
1.472 |
0.485 |
0. |
1.392 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 108 | 83 | 192 | 306 | 0 | 81 |
normalized size | 1 | 1. | 1.89 | 1.46 | 3.37 | 5.37 | 0. | 1.42 |
time (sec) |
N/A |
0.066 |
0.325 |
0.063 |
1.47 |
0.481 |
0. |
1.33 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 123 | 125 | 308 | 475 | 0 | 116 |
normalized size | 1 | 1. | 1.4 | 1.42 | 3.5 | 5.4 | 0. | 1.32 |
time (sec) |
N/A |
0.11 |
0.944 |
0.079 |
1.509 |
0.486 |
0. |
1.251 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 80 | 535 | 563 | 900 | 0 | 339 |
normalized size |
1 |
1. |
1.23 |
8.23 |
8.66 |
13.85 |
0. |
5.22 |
time (sec) |
N/A |
0.091 |
1.63 |
0.21 |
1.583 |
0.515 |
0. |
2.242 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 69 | 273 | 270 | 666 | 0 | 263 |
normalized size | 1 | 1. | 1.57 | 6.2 | 6.14 | 15.14 | 0. | 5.98 |
time (sec) |
N/A |
0.03 |
0.085 |
0.16 |
1.543 |
0.502 |
0. |
2.074 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 32 | 199 | 200 | 377 | 0 | 477 |
normalized size | 1 | 1. | 1.23 | 7.65 | 7.69 | 14.5 | 0. | 18.35 |
time (sec) |
N/A |
0.016 |
0.052 |
0.151 |
1.582 |
0.489 |
0. |
2.038 |
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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 |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 54 | 221 | 112 | 668 | 0 | 0 |
normalized size | 1 | 1. | 0.87 | 3.56 | 1.81 | 10.77 | 0. | 0. |
time (sec) |
N/A |
0.058 |
0.126 |
0.139 |
1.577 |
0.506 |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 129 | 1141 | 203 | 1299 | 0 | 410 |
normalized size | 1 | 1. | 1.59 | 14.09 | 2.51 | 16.04 | 0. | 5.06 |
time (sec) |
N/A |
0.102 |
0.411 |
0.184 |
1.544 |
0.532 |
0. |
2.973 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 100 | 100 | 139 | 1961 | 0 | 1677 | 0 | 470 |
normalized size | 1 | 1. | 1.39 | 19.61 | 0. | 16.77 | 0. | 4.7 |
time (sec) |
N/A |
0.148 |
0.503 |
0.148 |
0. |
0.546 |
0. |
2.529 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 108 | 114 | 0 | 755 | 0 | 0 |
normalized size | 1 | 1. | 2.92 | 3.08 | 0. | 20.41 | 0. | 0. |
time (sec) |
N/A |
0.059 |
0.408 |
0.359 |
0. |
0.535 |
0. |
0. |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 101 | 117 | 0 | 764 | 0 | 0 |
normalized size | 1 | 1. | 2.66 | 3.08 | 0. | 20.11 | 0. | 0. |
time (sec) |
N/A |
0.071 |
0.775 |
0.333 |
0. |
0.53 |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 102 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.4 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.281 |
0.382 |
0.384 |
0. |
0. |
0. |
0. |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 213 | 213 | 46 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.22 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.128 |
0.223 |
0.645 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 254 | 254 | 110 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.43 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.145 |
0.447 |
0.448 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 514 | 514 | 120 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.23 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.299 |
1.187 |
0.335 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 470 | 470 | 109 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.23 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.253 |
0.985 |
0.325 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 508 | 508 | 46 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.09 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.275 |
0.755 |
0.44 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 552 | 552 | 72 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.13 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.307 |
1.201 |
0.323 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 48 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.058 |
0.169 |
0.401 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 73 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.07 |
0.913 |
0.484 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 156 | 156 | 199 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.28 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.19 |
0.726 |
0.609 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 109 | 109 | 131 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.2 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.103 |
0.431 |
0.448 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 60 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.81 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.056 |
0.202 |
0.429 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.057 |
0.595 |
0.287 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.09 |
3.647 |
0.678 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.105 |
7.648 |
0.882 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 568 | 139 | 169 | 524 | 0 | 277 |
normalized size | 1 | 1. | 5.31 | 1.3 | 1.58 | 4.9 | 0. | 2.59 |
time (sec) |
N/A |
0.11 |
6.238 |
0.05 |
1.006 |
0.524 |
0. |
1.598 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 73 | 73 | 152 | 99 | 128 | 383 | 0 | 181 |
normalized size | 1 | 1. | 2.08 | 1.36 | 1.75 | 5.25 | 0. | 2.48 |
time (sec) |
N/A |
0.047 |
0.648 |
0.044 |
1.004 |
0.511 |
0. |
1.473 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 76 | 52 | 58 | 209 | 0 | 100 |
normalized size | 1 | 1. | 2.24 | 1.53 | 1.71 | 6.15 | 0. | 2.94 |
time (sec) |
N/A |
0.026 |
0.182 |
0.029 |
0.996 |
0.504 |
0. |
1.428 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 112 | 112 | 125 | 162 | 0 | 1378 | 0 | 262 |
normalized size |
1 |
1. |
1.12 |
1.45 |
0. |
12.3 |
0. |
2.34 |
time (sec) |
N/A |
0.407 |
1.631 |
0.05 |
0. |
0.898 |
0. |
1.387 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 144 | 112 | 0 | 1146 | 0 | 190 |
normalized size |
1 |
1. |
1.71 |
1.33 |
0. |
13.64 |
0. |
2.26 |
time (sec) |
N/A |
0.252 |
0.487 |
0.045 |
0. |
0.879 |
0. |
1.445 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 106 | 77 | 0 | 768 | 0 | 132 |
normalized size |
1 |
1. |
1.71 |
1.24 |
0. |
12.39 |
0. |
2.13 |
time (sec) |
N/A |
0.155 |
0.204 |
0.049 |
0. |
0.624 |
0. |
1.285 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 62 | 53 | 0 | 595 | 0 | 85 |
normalized size |
1 |
1. |
1.17 |
1. |
0. |
11.23 |
0. |
1.6 |
time (sec) |
N/A |
0.113 |
0.061 |
0.045 |
0. |
0.612 |
0. |
1.583 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 40 | 39 | 0 | 360 | 0 | 65 |
normalized size |
1 |
1. |
1. |
0.98 |
0. |
9. |
0. |
1.62 |
time (sec) |
N/A |
0.07 |
0.024 |
0.042 |
0. |
0.497 |
0. |
1.432 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 59 | 70 | 0 | 525 | 0 | 104 |
normalized size |
1 |
1. |
1.04 |
1.23 |
0. |
9.21 |
0. |
1.82 |
time (sec) |
N/A |
0.065 |
0.107 |
0.056 |
0. |
0.521 |
0. |
1.455 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 56 | 72 | 0 | 520 | 0 | 104 |
normalized size |
1 |
1. |
0.92 |
1.18 |
0. |
8.52 |
0. |
1.7 |
time (sec) |
N/A |
0.107 |
0.099 |
0.061 |
0. |
0.532 |
0. |
1.456 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 78 | 142 | 0 | 644 | 0 | 151 |
normalized size |
1 |
1. |
0.95 |
1.73 |
0. |
7.85 |
0. |
1.84 |
time (sec) |
N/A |
0.261 |
0.119 |
0.072 |
0. |
0.54 |
0. |
1.381 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 98 | 213 | 0 | 744 | 0 | 201 |
normalized size |
1 |
1. |
0.89 |
1.94 |
0. |
6.76 |
0. |
1.83 |
time (sec) |
N/A |
0.398 |
0.238 |
0.066 |
0. |
0.552 |
0. |
1.437 |
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 144 | 144 | 129 | 405 | 0 | 909 | 0 | 340 |
normalized size |
1 |
1. |
0.9 |
2.81 |
0. |
6.31 |
0. |
2.36 |
time (sec) |
N/A |
0.588 |
0.315 |
0.079 |
0. |
0.581 |
0. |
1.388 |
|
|
|
|
|
|
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|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 139 | 247 | 0 | 1076 | 0 | 213 |
normalized size |
1 |
1. |
1.29 |
2.29 |
0. |
9.96 |
0. |
1.97 |
time (sec) |
N/A |
0.171 |
0.456 |
0.093 |
0. |
0.563 |
0. |
1.384 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 170 | 170 | 216 | 796 | 0 | 1994 | 0 | 401 |
normalized size |
1 |
1. |
1.27 |
4.68 |
0. |
11.73 |
0. |
2.36 |
time (sec) |
N/A |
0.319 |
1.103 |
0.115 |
0. |
0.645 |
0. |
1.181 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 239 | 239 | 279 | 1912 | 0 | 3389 | 0 | 722 |
normalized size |
1 |
1. |
1.17 |
8. |
0. |
14.18 |
0. |
3.02 |
time (sec) |
N/A |
0.504 |
2.095 |
0.147 |
0. |
0.778 |
0. |
1.39 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 66 | 36 | 66 | 89 | 0 | 66 |
normalized size | 1 | 1. | 2.13 | 1.16 | 2.13 | 2.87 | 0. | 2.13 |
time (sec) |
N/A |
0.029 |
0.048 |
0.039 |
1.474 |
0.492 |
0. |
1.443 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
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 | 68 | 68 | 67 | 53 | 96 | 144 | 0 | 61 |
normalized size | 1 | 1. | 0.99 | 0.78 | 1.41 | 2.12 | 0. | 0.9 |
time (sec) |
N/A |
0.039 |
0.047 |
0.045 |
1.463 |
0.496 |
0. |
1.374 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 274 | 274 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.343 |
4.181 |
0.309 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 220 | 220 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.226 |
2.718 |
0.247 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 104 | 104 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.074 |
1.913 |
0.511 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
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 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.01 |
1.546 |
0.333 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
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 |
0.032 |
6.055 |
0.257 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.04 |
5.904 |
0.567 |
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 [39] had the largest ratio of [ 0.6923
]
|
|
|
|
|
|
|
| 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. |
13 |
0.385 |
|
|
|
|
|
|
|
2 |
A |
6 |
6 |
1. |
13 |
0.462 |
|
|
|
|
|
|
|
3 |
A |
4 |
4 |
1. |
13 |
0.308 |
|
|
|
|
|
|
|
4 |
A |
3 |
3 |
1. |
13 |
0.231 |
|
|
|
|
|
|
|
5 |
A |
1 |
1 |
1. |
11 |
0.091 |
|
|
|
|
|
|
|
6 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
7 |
A |
4 |
4 |
1. |
11 |
0.364 |
|
|
|
|
|
|
|
8 |
A |
5 |
5 |
1. |
13 |
0.385 |
|
|
|
|
|
|
|
9 |
A |
6 |
5 |
1. |
13 |
0.385 |
|
|
|
|
|
|
|
10 |
A |
7 |
5 |
1. |
13 |
0.385 |
|
|
|
|
|
|
|
11 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
12 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
13 |
A |
5 |
5 |
1. |
10 |
0.5 |
|
|
|
|
|
|
|
14 |
A |
4 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
15 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
16 |
A |
5 |
4 |
1. |
10 |
0.4 |
|
|
|
|
|
|
|
17 |
A |
6 |
5 |
1. |
10 |
0.5 |
|
|
|
|
|
|
|
18 |
A |
7 |
6 |
1. |
10 |
0.6 |
|
|
|
|
|
|
|
19 |
A |
2 |
2 |
1. |
25 |
0.08 |
|
|
|
|
|
|
|
20 |
A |
2 |
2 |
1. |
28 |
0.071 |
|
|
|
|
|
|
|
|
21 |
A |
4 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
22 |
A |
3 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
23 |
A |
4 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
24 |
A |
6 |
6 |
1. |
25 |
0.24 |
|
|
|
|
|
|
|
25 |
A |
5 |
5 |
1. |
25 |
0.2 |
|
|
|
|
|
|
|
26 |
A |
6 |
6 |
1. |
25 |
0.24 |
|
|
|
|
|
|
|
27 |
A |
7 |
6 |
1. |
25 |
0.24 |
|
|
|
|
|
|
|
28 |
A |
2 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
29 |
A |
3 |
3 |
1. |
24 |
0.125 |
|
|
|
|
|
|
|
30 |
A |
5 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
31 |
A |
4 |
4 |
1. |
21 |
0.19 |
|
|
|
|
|
|
|
32 |
A |
3 |
3 |
1. |
19 |
0.158 |
|
|
|
|
|
|
|
33 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1. |
19 |
0.158 |
|
|
|
|
|
|
|
35 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
36 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
37 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
38 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
39 |
A |
9 |
9 |
1. |
13 |
0.692 |
|
|
|
|
|
|
|
40 |
A |
8 |
8 |
1. |
13 |
0.615 |
|
|
|
|
|
|
|
|
41 |
A |
7 |
7 |
1. |
13 |
0.538 |
|
|
|
|
|
|
|
42 |
A |
6 |
6 |
1. |
13 |
0.462 |
|
|
|
|
|
|
|
43 |
A |
4 |
4 |
1. |
11 |
0.364 |
|
|
|
|
|
|
|
44 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
45 |
A |
6 |
6 |
1. |
11 |
0.546 |
|
|
|
|
|
|
|
46 |
A |
7 |
7 |
1. |
13 |
0.538 |
|
|
|
|
|
|
|
47 |
A |
8 |
7 |
1. |
13 |
0.538 |
|
|
|
|
|
|
|
48 |
A |
9 |
7 |
1. |
13 |
0.538 |
|
|
|
|
|
|
|
49 |
A |
6 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
50 |
A |
7 |
7 |
1. |
12 |
0.583 |
|
|
|
|
|
|
|
51 |
A |
8 |
7 |
1. |
12 |
0.583 |
|
|
|
|
|
|
|
52 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
53 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
54 |
A |
8 |
5 |
1. |
21 |
0.238 |
|
|
|
|
|
|
|
55 |
A |
7 |
4 |
1. |
21 |
0.19 |
|
|
|
|
|
|
|
56 |
A |
3 |
3 |
1. |
19 |
0.158 |
|
|
|
|
|
|
|
57 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
58 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
59 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|