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, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 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
}
B grade: { 32, 34 }
C grade: { 17, 19, 21, 23, 65, 66, 67, 68, 69, 70 }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 }
B grade: { 24, 26, 27, 28, 35, 36 }
C grade: { 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70
}
F grade: { }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43,
44, 45, 46, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64
B grade: { 14, 15, 25, 26, 41 }
C grade: { }
F grade: { 16, 17, 18, 19, 20, 21, 22, 23, 24, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70
}
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 42, 43, 44, 45, 46, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64 }
B grade: { 7, 40, 41, 58 }
C grade: { }
F grade: { 16, 17, 18, 19, 20, 21, 22, 23, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70 }
2.1.6 Sympy
A grade: { 9, 32, 33 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 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 }
2.1.7 Giac
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 46
}
B grade: { 24, 37, 38, 39, 40, 41, 42, 43, 44, 45, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 }
C grade: { }
F grade: { 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 47, 48, 49, 50, 51, 52, 53, 65, 66, 67, 68, 69, 70
}
2.2 Detailed conclusion table per each integral for all CAS systems
Detailed conclusion table per each integral is given by table below. The elapsed time is in
seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2)
if the failure was due to an exception being raised, which could indicate a bug in the system. If
the failure was due to integral not being evaluated within the time limit, then it is given just an
F.
In this table,the column normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)
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Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 81 | 78 | 81 | 188 | 0 | 107 |
normalized size | 1 | 1. | 0.93 | 0.9 | 0.93 | 2.16 | 0. | 1.23 |
time (sec) |
N/A |
0.056 |
0.302 |
0.031 |
0.937 |
0.908 |
0. |
1.18 |
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Problem 2 |
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 | 98 | 98 | 75 | 138 | 170 | 293 | 0 | 163 |
normalized size | 1 | 1. | 0.77 | 1.41 | 1.73 | 2.99 | 0. | 1.66 |
time (sec) |
N/A |
0.063 |
0.337 |
0.029 |
0.935 |
0.524 |
0. |
1.256 |
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Problem 3 |
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 | 65 | 65 | 61 | 58 | 58 | 140 | 0 | 77 |
normalized size | 1 | 1. | 0.94 | 0.89 | 0.89 | 2.15 | 0. | 1.18 |
time (sec) |
N/A |
0.044 |
0.194 |
0.032 |
0.928 |
0.468 |
0. |
1.184 |
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Problem 4 |
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 | 70 | 70 | 54 | 98 | 131 | 243 | 0 | 132 |
normalized size | 1 | 1. | 0.77 | 1.4 | 1.87 | 3.47 | 0. | 1.89 |
time (sec) |
N/A |
0.047 |
0.116 |
0.028 |
0.93 |
0.505 |
0. |
1.251 |
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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 |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 36 | 35 | 46 | 95 | 0 | 46 |
normalized size | 1 | 1. | 0.84 | 0.81 | 1.07 | 2.21 | 0. | 1.07 |
time (sec) |
N/A |
0.04 |
0.093 |
0.025 |
0.927 |
0.457 |
0. |
1.224 |
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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 | 40 | 40 | 48 | 59 | 78 | 192 | 0 | 81 |
normalized size | 1 | 1. | 1.2 | 1.48 | 1.95 | 4.8 | 0. | 2.02 |
time (sec) |
N/A |
0.025 |
0.014 |
0.026 |
0.928 |
0.494 |
0. |
1.185 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 15 | 16 | 20 | 76 | 0 | 20 |
normalized size | 1 | 1. | 1. | 1.07 | 1.33 | 5.07 | 0. | 1.33 |
time (sec) |
N/A |
0.013 |
0.003 |
0.015 |
0.921 |
0.466 |
0. |
1.145 |
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Problem 8 |
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 | 24 | 24 | 35 | 32 | 51 | 107 | 0 | 54 |
normalized size | 1 | 1. | 1.46 | 1.33 | 2.12 | 4.46 | 0. | 2.25 |
time (sec) |
N/A |
0.027 |
0.015 |
0.043 |
0.927 |
0.502 |
0. |
1.216 |
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Problem 9 |
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 | 31 | 31 | 33 | 37 | 50 | 72 | 51 | 50 |
normalized size | 1 | 1. | 1.06 | 1.19 | 1.61 | 2.32 | 1.65 | 1.61 |
time (sec) |
N/A |
0.03 |
0.034 |
0.044 |
1.411 |
0.471 |
21.15 |
1.168 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 50 | 33 | 36 | 69 | 0 | 46 |
normalized size |
1 |
1. |
1.67 |
1.1 |
1.2 |
2.3 |
0. |
1.53 |
time (sec) |
N/A |
0.047 |
0.018 |
0.05 |
0.921 |
0.471 |
0. |
1.128 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 45 | 65 | 99 | 119 | 0 | 99 |
normalized size |
1 |
1. |
0.74 |
1.07 |
1.62 |
1.95 |
0. |
1.62 |
time (sec) |
N/A |
0.044 |
0.087 |
0.05 |
1.408 |
0.486 |
0. |
1.164 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 50 | 50 | 71 | 54 | 58 | 113 | 0 | 77 |
normalized size |
1 |
1. |
1.42 |
1.08 |
1.16 |
2.26 |
0. |
1.54 |
time (sec) |
N/A |
0.068 |
0.022 |
0.051 |
0.922 |
0.476 |
0. |
1.211 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 68 | 86 | 139 | 167 | 0 | 130 |
normalized size |
1 |
1. |
0.76 |
0.97 |
1.56 |
1.88 |
0. |
1.46 |
time (sec) |
N/A |
0.059 |
0.103 |
0.056 |
1.407 |
0.495 |
0. |
1.205 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 512 | 410 | 81 | 0 | 0 |
normalized size | 1 | 1. | 1. | 19.69 | 15.77 | 3.12 | 0. | 0. |
time (sec) |
N/A |
0.04 |
0.314 |
0.385 |
1.996 |
0.495 |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 510 | 400 | 74 | 0 | 0 |
normalized size | 1 | 1. | 1. | 20.4 | 16. | 2.96 | 0. | 0. |
time (sec) |
N/A |
0.041 |
0.284 |
0.222 |
1.98 |
0.496 |
0. |
0. |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 84 | 251 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.76 |
2.28 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.081 |
1.06 |
0.352 |
0. |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 184 | 670 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.67 |
6.09 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.083 |
1.242 |
0.273 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 58 | 201 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.81 | 2.79 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.054 |
0.374 |
0.251 |
0. |
0. |
0. |
0. |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 126 | 588 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.85 | 8.65 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.057 |
0.709 |
0.279 |
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 |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 66 | 198 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.88 | 2.64 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.064 |
0.281 |
0.237 |
0. |
0. |
0. |
0. |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 133 | 613 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.73 |
7.96 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.064 |
1.152 |
0.217 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
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 | 112 | 112 | 79 | 241 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.71 |
2.15 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.09 |
0.692 |
0.265 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 112 | 112 | 143 | 636 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.28 |
5.68 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.086 |
1.417 |
0.261 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 41 | 0 | 53 | 0 | 63 |
normalized size | 1 | 1. | 1. | 1.95 | 0. | 2.52 | 0. | 3. |
time (sec) |
N/A |
0.023 |
0.133 |
0.684 |
0. |
0.481 |
0. |
1.393 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 506 | 382 | 69 | 0 | 0 |
normalized size | 1 | 1. | 1. | 24.1 | 18.19 | 3.29 | 0. | 0. |
time (sec) |
N/A |
0.029 |
0.253 |
0.398 |
1.973 |
0.493 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 70 | 57 | 46 | 0 | 32 |
normalized size | 1 | 1. | 1. | 3.68 | 3. | 2.42 | 0. | 1.68 |
time (sec) |
N/A |
0.024 |
0.06 |
0.03 |
0.932 |
0.465 |
0. |
1.176 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 56 | 41 | 46 | 0 | 45 |
normalized size | 1 | 1. | 1. | 2.95 | 2.16 | 2.42 | 0. | 2.37 |
time (sec) |
N/A |
0.023 |
0.03 |
0.028 |
0.923 |
0.461 |
0. |
1.175 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 47 | 45 | 46 | 0 | 34 |
normalized size | 1 | 1. | 1. | 2.47 | 2.37 | 2.42 | 0. | 1.79 |
time (sec) |
N/A |
0.023 |
0.031 |
0.028 |
0.922 |
0.455 |
0. |
1.197 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 19 | 19 | 19 | 34 | 27 | 46 | 0 | 30 |
normalized size | 1 | 1. | 1. | 1.79 | 1.42 | 2.42 | 0. | 1.58 |
time (sec) |
N/A |
0.023 |
0.056 |
0.021 |
0.926 |
0.448 |
0. |
1.141 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 17 | 17 | 17 | 18 | 30 | 46 | 0 | 35 |
normalized size | 1 | 1. | 1. | 1.06 | 1.76 | 2.71 | 0. | 2.06 |
time (sec) |
N/A |
0.015 |
0.013 |
0.025 |
0.923 |
0.446 |
0. |
1.154 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 11 | 11 | 11 | 12 | 15 | 43 | 0 | 16 |
normalized size | 1 | 1. | 1. | 1.09 | 1.36 | 3.91 | 0. | 1.45 |
time (sec) |
N/A |
0.013 |
0.004 |
0.011 |
0.918 |
0.446 |
0. |
1.194 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 23 | 12 | 15 | 23 | 14 | 16 |
normalized size | 1 | 1. | 2.09 | 1.09 | 1.36 | 2.09 | 1.27 | 1.45 |
time (sec) |
N/A |
0.005 |
0.01 |
0.011 |
0.916 |
0.459 |
0.194 |
1.161 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 17 | 17 | 33 | 18 | 31 | 41 | 49 | 20 |
normalized size | 1 | 1. | 1.94 | 1.06 | 1.82 | 2.41 | 2.88 | 1.18 |
time (sec) |
N/A |
0.023 |
0.013 |
0.046 |
0.928 |
0.466 |
52.326 |
1.139 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 46 | 32 | 28 | 43 | 0 | 31 |
normalized size |
1 |
1. |
2.42 |
1.68 |
1.47 |
2.26 |
0. |
1.63 |
time (sec) |
N/A |
0.024 |
0.02 |
0.049 |
0.926 |
0.465 |
0. |
1.157 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 31 | 45 | 45 | 43 | 0 | 34 |
normalized size |
1 |
1. |
1.63 |
2.37 |
2.37 |
2.26 |
0. |
1.79 |
time (sec) |
N/A |
0.024 |
0.029 |
0.047 |
0.926 |
0.469 |
0. |
1.158 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 38 | 52 | 41 | 43 | 0 | 45 |
normalized size |
1 |
1. |
2. |
2.74 |
2.16 |
2.26 |
0. |
2.37 |
time (sec) |
N/A |
0.025 |
0.029 |
0.068 |
0.922 |
0.473 |
0. |
1.126 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 76 | 92 | 128 | 266 | 0 | 221 |
normalized size | 1 | 1. | 0.89 | 1.08 | 1.51 | 3.13 | 0. | 2.6 |
time (sec) |
N/A |
0.072 |
0.229 |
0.03 |
0.935 |
0.505 |
0. |
1.221 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 60 | 72 | 95 | 236 | 0 | 165 |
normalized size | 1 | 1. | 0.95 | 1.14 | 1.51 | 3.75 | 0. | 2.62 |
time (sec) |
N/A |
0.058 |
0.148 |
0.028 |
0.929 |
0.498 |
0. |
1.289 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 47 | 51 | 78 | 198 | 0 | 144 |
normalized size | 1 | 1. | 1. | 1.09 | 1.66 | 4.21 | 0. | 3.06 |
time (sec) |
N/A |
0.043 |
0.017 |
0.023 |
0.938 |
0.495 |
0. |
1.229 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 24 | 32 | 42 | 162 | 0 | 77 |
normalized size | 1 | 1. | 1. | 1.33 | 1.75 | 6.75 | 0. | 3.21 |
time (sec) |
N/A |
0.018 |
0.012 |
0.014 |
0.922 |
0.488 |
0. |
1.219 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 16 | 30 | 50 | 95 | 0 | 58 |
normalized size | 1 | 1. | 1. | 1.88 | 3.12 | 5.94 | 0. | 3.62 |
time (sec) |
N/A |
0.029 |
0.004 |
0.04 |
0.933 |
0.499 |
0. |
1.208 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 26 | 21 | 27 | 38 | 0 | 53 |
normalized size | 1 | 1. | 1.73 | 1.4 | 1.8 | 2.53 | 0. | 3.53 |
time (sec) |
N/A |
0.032 |
0.009 |
0.044 |
0.936 |
0.468 |
0. |
1.171 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 35 | 38 | 46 | 72 | 0 | 111 |
normalized size |
1 |
1. |
0.92 |
1. |
1.21 |
1.89 |
0. |
2.92 |
time (sec) |
N/A |
0.042 |
0.062 |
0.046 |
0.927 |
0.475 |
0. |
1.182 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 57 | 49 | 62 | 105 | 0 | 132 |
normalized size |
1 |
1. |
1.06 |
0.91 |
1.15 |
1.94 |
0. |
2.44 |
time (sec) |
N/A |
0.051 |
0.062 |
0.046 |
0.937 |
0.482 |
0. |
1.225 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 76 | 76 | 73 | 60 | 77 | 136 | 0 | 189 |
normalized size |
1 |
1. |
0.96 |
0.79 |
1.01 |
1.79 |
0. |
2.49 |
time (sec) |
N/A |
0.065 |
0.118 |
0.049 |
0.925 |
0.485 |
0. |
1.185 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 89 | 70 | 93 | 173 | 0 | 208 |
normalized size |
1 |
1. |
0.97 |
0.76 |
1.01 |
1.88 |
0. |
2.26 |
time (sec) |
N/A |
0.076 |
0.104 |
0.108 |
0.937 |
0.495 |
0. |
1.161 |
|
|
|
|
|
|
|
|
|
|
|
|
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Problem 47 |
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 | 169 | 169 | 102 | 526 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.6 |
3.11 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.144 |
0.449 |
0.289 |
0. |
0. |
0. |
0. |
|
|
|
|
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|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 135 | 135 | 90 | 508 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.67 | 3.76 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.122 |
0.266 |
0.289 |
0. |
0. |
0. |
0. |
|
|
|
|
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 109 | 109 | 73 | 456 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.67 | 4.18 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.101 |
0.183 |
0.408 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 57 | 450 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.67 | 5.29 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.092 |
0.082 |
0.26 |
0. |
0. |
0. |
0. |
|
|
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 81 | 470 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.7 | 4.05 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.107 |
0.171 |
0.204 |
0. |
0. |
0. |
0. |
|
|
|
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|
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Problem 52 |
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 | 147 | 147 | 91 | 482 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.62 |
3.28 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.135 |
0.531 |
0.25 |
0. |
0. |
0. |
0. |
|
|
|
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|
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|
Problem 53 |
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 | 176 | 176 | 104 | 493 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.59 |
2.8 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.144 |
0.644 |
0.366 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 80 | 144 | 171 | 329 | 0 | 332 |
normalized size | 1 | 1. | 0.66 | 1.18 | 1.4 | 2.7 | 0. | 2.72 |
time (sec) |
N/A |
0.099 |
0.579 |
0.032 |
0.935 |
0.517 |
0. |
1.217 |
|
|
|
|
|
|
|
|
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|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 71 | 130 | 188 | 306 | 0 | 311 |
normalized size | 1 | 1. | 0.73 | 1.34 | 1.94 | 3.15 | 0. | 3.21 |
time (sec) |
N/A |
0.084 |
0.238 |
0.032 |
0.937 |
0.511 |
0. |
1.224 |
|
|
|
|
|
|
|
|
|
|
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|
|
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|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 51 | 83 | 107 | 250 | 0 | 219 |
normalized size | 1 | 1. | 0.65 | 1.06 | 1.37 | 3.21 | 0. | 2.81 |
time (sec) |
N/A |
0.081 |
0.2 |
0.028 |
0.932 |
0.507 |
0. |
1.21 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 59 | 70 | 101 | 220 | 0 | 155 |
normalized size | 1 | 1. | 1.16 | 1.37 | 1.98 | 4.31 | 0. | 3.04 |
time (sec) |
N/A |
0.051 |
0.015 |
0.028 |
0.935 |
0.501 |
0. |
1.196 |
|
|
|
|
|
|
|
|
|
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|
|
|
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|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 27 | 35 | 46 | 193 | 0 | 81 |
normalized size | 1 | 1. | 1. | 1.3 | 1.7 | 7.15 | 0. | 3. |
time (sec) |
N/A |
0.019 |
0.004 |
0.013 |
0.927 |
0.507 |
0. |
1.202 |
|
|
|
|
|
|
|
|
|
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|
|
|
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|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 38 | 41 | 62 | 120 | 0 | 95 |
normalized size | 1 | 1. | 1.41 | 1.52 | 2.3 | 4.44 | 0. | 3.52 |
time (sec) |
N/A |
0.05 |
0.017 |
0.044 |
0.934 |
0.507 |
0. |
1.164 |
|
|
|
|
|
|
|
|
|
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|
|
|
|
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|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 55 | 45 | 57 | 82 | 0 | 116 |
normalized size | 1 | 1. | 1.31 | 1.07 | 1.36 | 1.95 | 0. | 2.76 |
time (sec) |
N/A |
0.059 |
0.054 |
0.047 |
0.929 |
0.477 |
0. |
1.158 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
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|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 53 | 57 | 74 | 113 | 0 | 186 |
normalized size |
1 |
1. |
0.95 |
1.02 |
1.32 |
2.02 |
0. |
3.32 |
time (sec) |
N/A |
0.084 |
0.098 |
0.051 |
0.93 |
0.48 |
0. |
1.188 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 70 | 84 | 104 | 163 | 0 | 270 |
normalized size |
1 |
1. |
0.8 |
0.95 |
1.18 |
1.85 |
0. |
3.07 |
time (sec) |
N/A |
0.083 |
0.168 |
0.053 |
0.93 |
0.488 |
0. |
1.191 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 87 | 89 | 120 | 194 | 0 | 300 |
normalized size |
1 |
1. |
0.89 |
0.91 |
1.22 |
1.98 |
0. |
3.06 |
time (sec) |
N/A |
0.115 |
0.182 |
0.054 |
0.933 |
0.499 |
0. |
1.165 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 132 | 132 | 102 | 115 | 155 | 244 | 0 | 383 |
normalized size |
1 |
1. |
0.77 |
0.87 |
1.17 |
1.85 |
0. |
2.9 |
time (sec) |
N/A |
0.099 |
0.269 |
0.058 |
0.935 |
0.509 |
0. |
1.19 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 178 | 178 | 640 | 832 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
3.6 |
4.67 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.161 |
6.505 |
0.3 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 136 | 136 | 249 | 647 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.83 | 4.76 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.128 |
1.834 |
0.279 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 110 | 110 | 156 | 719 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.42 | 6.54 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.111 |
1.386 |
0.292 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 179 | 602 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.53 | 5.15 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.134 |
1.324 |
0.262 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 150 | 150 | 169 | 766 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.13 |
5.11 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.154 |
1.978 |
0.229 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 185 | 185 | 313 | 645 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.69 |
3.49 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.159 |
3.53 |
0.25 |
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 [ 0.2308
]
|
|
|
|
|
|
|
| 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 |
3 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
2 |
A |
4 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
3 |
A |
3 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
4 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
6 |
A |
2 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
7 |
A |
3 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
8 |
A |
2 |
2 |
1. |
19 |
0.105 |
|
|
|
|
|
|
|
9 |
A |
2 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
10 |
A |
3 |
2 |
1. |
21 |
0.095 |
|
|
|
|
|
|
|
11 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
12 |
A |
4 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
13 |
A |
4 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
14 |
A |
1 |
1 |
1. |
29 |
0.034 |
|
|
|
|
|
|
|
15 |
A |
1 |
1 |
1. |
28 |
0.036 |
|
|
|
|
|
|
|
16 |
A |
4 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
17 |
A |
4 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
19 |
A |
3 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
20 |
A |
3 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
|
21 |
A |
3 |
3 |
1. |
25 |
0.12 |
|
|
|
|
|
|
|
22 |
A |
4 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
23 |
A |
4 |
4 |
1. |
25 |
0.16 |
|
|
|
|
|
|
|
24 |
A |
1 |
1 |
1. |
23 |
0.043 |
|
|
|
|
|
|
|
25 |
A |
1 |
1 |
1. |
24 |
0.042 |
|
|
|
|
|
|
|
26 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
27 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
28 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
29 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
30 |
A |
1 |
1 |
1. |
19 |
0.053 |
|
|
|
|
|
|
|
31 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
32 |
A |
1 |
1 |
1. |
8 |
0.125 |
|
|
|
|
|
|
|
33 |
A |
1 |
1 |
1. |
19 |
0.053 |
|
|
|
|
|
|
|
34 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
35 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
36 |
A |
1 |
1 |
1. |
21 |
0.048 |
|
|
|
|
|
|
|
37 |
A |
7 |
5 |
1. |
28 |
0.179 |
|
|
|
|
|
|
|
38 |
A |
6 |
5 |
1. |
28 |
0.179 |
|
|
|
|
|
|
|
39 |
A |
6 |
6 |
1. |
26 |
0.231 |
|
|
|
|
|
|
|
40 |
A |
4 |
3 |
1. |
19 |
0.158 |
|
|
|
|
|
|
|
|
41 |
A |
4 |
4 |
1. |
26 |
0.154 |
|
|
|
|
|
|
|
42 |
A |
3 |
3 |
1. |
28 |
0.107 |
|
|
|
|
|
|
|
43 |
A |
5 |
5 |
1. |
28 |
0.179 |
|
|
|
|
|
|
|
44 |
A |
6 |
5 |
1. |
28 |
0.179 |
|
|
|
|
|
|
|
45 |
A |
7 |
5 |
1. |
28 |
0.179 |
|
|
|
|
|
|
|
46 |
A |
7 |
5 |
1. |
28 |
0.179 |
|
|
|
|
|
|
|
47 |
A |
10 |
7 |
1. |
32 |
0.219 |
|
|
|
|
|
|
|
48 |
A |
9 |
7 |
1. |
32 |
0.219 |
|
|
|
|
|
|
|
49 |
A |
8 |
7 |
1. |
32 |
0.219 |
|
|
|
|
|
|
|
50 |
A |
7 |
6 |
1. |
32 |
0.188 |
|
|
|
|
|
|
|
51 |
A |
8 |
7 |
1. |
32 |
0.219 |
|
|
|
|
|
|
|
52 |
A |
9 |
7 |
1. |
32 |
0.219 |
|
|
|
|
|
|
|
53 |
A |
10 |
7 |
1. |
32 |
0.219 |
|
|
|
|
|
|
|
54 |
A |
7 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
55 |
A |
6 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
56 |
A |
6 |
6 |
1. |
29 |
0.207 |
|
|
|
|
|
|
|
57 |
A |
5 |
5 |
1. |
27 |
0.185 |
|
|
|
|
|
|
|
58 |
A |
4 |
3 |
1. |
20 |
0.15 |
|
|
|
|
|
|
|
59 |
A |
4 |
4 |
1. |
27 |
0.148 |
|
|
|
|
|
|
|
60 |
A |
4 |
4 |
1. |
29 |
0.138 |
|
|
|
|
|
|
|
|
61 |
A |
6 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
62 |
A |
6 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
63 |
A |
8 |
6 |
1. |
29 |
0.207 |
|
|
|
|
|
|
|
64 |
A |
7 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
65 |
A |
8 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
66 |
A |
7 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
67 |
A |
6 |
5 |
1. |
33 |
0.152 |
|
|
|
|
|
|
|
68 |
A |
6 |
5 |
1. |
33 |
0.152 |
|
|
|
|
|
|
|
69 |
A |
7 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
70 |
A |
8 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|