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, 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 }
B grade: { }
C grade: { }
F grade: { 17}
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 6, 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, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63
}
B grade: { 7, 36, 44, 45, 49, 50, 61 }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 41, 42, 43, 46, 47, 48, 51, 52, 53, 57, 58, 62, 63 }
B grade: { 39, 40, 44, 45, 49, 50, 54, 55, 56, 59, 60, 61 }
C grade: { }
F grade: { 16, 17, 36, 37, 38 }
2.1.4 Maxima
A grade: { 4, 5, 9, 10, 14, 15, 21, 22, 23, 27, 28, 32, 33, 41, 42, 43, 47, 48, 52, 53
B grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 39, 40, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61 }
C grade: { }
F grade: { 16, 17, 18, 19, 20, 24, 25, 26, 29, 30, 31, 34, 35, 36, 37, 38, 57, 58, 62, 63 }
2.1.5 FriCAS
A grade: { 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, 34, 35,
36, 37, 38, 42, 43, 46, 47, 48, 51, 52, 53, 57, 58, 62, 63 }
B grade: { 3, 7, 13, 41, 56, 61 }
C grade: { 1, 2, 6, 11, 12, 39, 40, 44, 45, 49, 50, 54, 55, 59, 60 }
F grade: { }
2.1.6 Sympy
A grade: { 4, 5, 8, 9, 10, 14, 15, 18, 19, 20, 24, 25, 26, 29, 30, 31, 35, 42, 43, 47, 48, 52, 53, 57, 58, 62
}
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 21, 22, 23, 27, 28, 32, 33, 34, 36, 37, 38, 39, 40, 41, 44, 45, 46, 49,
50, 51, 54, 55, 56, 59, 60, 61, 63 }
2.1.7 Giac
A grade: { 4, 5, 9, 10, 14, 15, 18, 19, 20, 21, 24, 25, 26, 27, 29, 30, 31, 34, 35, 42, 43, 47, 48, 52, 53, 57, 58,
62, 63 }
B grade: { 8, 22, 23, 28, 32, 33 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 36, 37, 38, 39, 40, 41, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61
}
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 |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 106 | 125 | 325 | 802 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.18 | 3.07 | 7.57 | 0. | 0. |
time (sec) |
N/A |
0.155 |
0.064 |
0.087 |
1.694 |
1.611 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 77 | 103 | 219 | 562 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.34 | 2.84 | 7.3 | 0. | 0. |
time (sec) |
N/A |
0.134 |
0.008 |
0.035 |
1.672 |
1.621 |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 54 | 78 | 124 | 332 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.44 | 2.3 | 6.15 | 0. | 0. |
time (sec) |
N/A |
0.081 |
0.005 |
0.033 |
1.709 |
1.621 |
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 |
0.015 |
1.731 |
0.084 |
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 |
0.016 |
2.509 |
0.066 |
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 |
A |
A |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 115 | 133 | 864 | 614 | 0 | 0 |
normalized size | 1 | 1. | 1.17 | 1.36 | 8.82 | 6.27 | 0. | 0. |
time (sec) |
N/A |
0.167 |
0.619 |
0.054 |
1.783 |
1.566 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 73 | 73 | 160 | 108 | 348 | 387 | 0 | 0 |
normalized size | 1 | 1. | 2.19 | 1.48 | 4.77 | 5.3 | 0. | 0. |
time (sec) |
N/A |
0.112 |
5.41 |
0.046 |
1.782 |
1.629 |
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 |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 43 | 29 | 289 | 96 | 41 | 246 |
normalized size | 1 | 1. | 1.43 | 0.97 | 9.63 | 3.2 | 1.37 | 8.2 |
time (sec) |
N/A |
0.023 |
0.161 |
0.072 |
1.466 |
1.55 |
0.314 |
1.525 |
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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 |
2.083 |
0.069 |
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 |
0.03 |
2.417 |
0.099 |
0. |
0. |
0. |
0. |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 205 | 205 | 359 | 251 | 1624 | 938 | 0 | 0 |
normalized size | 1 | 1. | 1.75 | 1.22 | 7.92 | 4.58 | 0. | 0. |
time (sec) |
N/A |
0.292 |
6.513 |
0.069 |
2.192 |
1.695 |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 128 | 172 | 180 | 999 | 657 | 0 | 0 |
normalized size | 1 | 1. | 1.34 | 1.41 | 7.8 | 5.13 | 0. | 0. |
time (sec) |
N/A |
0.18 |
2.484 |
0.058 |
1.986 |
1.754 |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 171 | 122 | 524 | 396 | 0 | 0 |
normalized size | 1 | 1. | 1.9 | 1.36 | 5.82 | 4.4 | 0. | 0. |
time (sec) |
N/A |
0.103 |
4.31 |
0.051 |
1.876 |
1.607 |
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 |
0.027 |
4.573 |
0.552 |
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 |
0.028 |
3.164 |
0.722 |
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 |
F |
F |
A |
F |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 18 | 18 | 0 | 0 | 42 | 0 | 0 |
normalized size |
1 |
1. |
1. |
0. |
0. |
2.33 |
0. |
0. |
time (sec) |
N/A |
0.123 |
0.885 |
0.218 |
0. |
1.611 |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
F |
A |
F |
F |
A |
F |
F |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 0 | 17 | 0 | 0 | 35 | 0 | 0 |
normalized size | 1 | 0. | 1. | 0. | 0. | 2.06 | 0. | 0. |
time (sec) |
N/A |
0.035 |
0.56 |
0.253 |
0. |
1.558 |
0. |
0. |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 189 | 189 | 278 | 159 | 0 | 370 | 372 | 261 |
normalized size |
1 |
1. |
1.47 |
0.84 |
0. |
1.96 |
1.97 |
1.38 |
time (sec) |
N/A |
0.199 |
0.627 |
0.194 |
0. |
1.514 |
0.929 |
1.195 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 137 | 137 | 178 | 97 | 0 | 248 | 238 | 166 |
normalized size |
1 |
1. |
1.3 |
0.71 |
0. |
1.81 |
1.74 |
1.21 |
time (sec) |
N/A |
0.123 |
0.31 |
0.162 |
0. |
1.534 |
0.683 |
1.197 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 96 | 139 | 0 | 146 | 128 | 88 |
normalized size |
1 |
1. |
1.14 |
1.65 |
0. |
1.74 |
1.52 |
1.05 |
time (sec) |
N/A |
0.054 |
0.358 |
0.145 |
0. |
1.531 |
0.406 |
1.228 |
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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(-2) |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 161 | 161 | 166 | 65 | 149 | 119 | 0 | 192 |
normalized size |
1 |
1. |
1.03 |
0.4 |
0.93 |
0.74 |
0. |
1.19 |
time (sec) |
N/A |
0.277 |
0.311 |
0.15 |
1.191 |
1.596 |
0. |
1.178 |
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|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 168 | 168 | 224 | 96 | 162 | 212 | 0 | 797 |
normalized size |
1 |
1. |
1.33 |
0.57 |
0.96 |
1.26 |
0. |
4.74 |
time (sec) |
N/A |
0.24 |
0.767 |
0.171 |
1.313 |
1.801 |
0. |
1.442 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 285 | 216 | 211 | 300 | 0 | 732 |
normalized size |
1 |
1. |
1.26 |
0.95 |
0.93 |
1.32 |
0. |
3.22 |
time (sec) |
N/A |
0.322 |
1.046 |
0.217 |
1.569 |
1.638 |
0. |
1.207 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 270 | 270 | 473 | 272 | 0 | 639 | 666 | 517 |
normalized size |
1 |
1. |
1.75 |
1.01 |
0. |
2.37 |
2.47 |
1.91 |
time (sec) |
N/A |
0.293 |
1.246 |
0.298 |
0. |
1.624 |
1.861 |
1.193 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 202 | 202 | 282 | 162 | 0 | 406 | 420 | 306 |
normalized size |
1 |
1. |
1.4 |
0.8 |
0. |
2.01 |
2.08 |
1.51 |
time (sec) |
N/A |
0.208 |
0.668 |
0.252 |
0. |
1.588 |
1.254 |
1.185 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 151 | 151 | 130 | 82 | 0 | 220 | 228 | 144 |
normalized size |
1 |
1. |
0.86 |
0.54 |
0. |
1.46 |
1.51 |
0.95 |
time (sec) |
N/A |
0.142 |
0.463 |
0.22 |
0. |
1.589 |
0.765 |
1.161 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
A |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 305 | 305 | 211 | 114 | 257 | 203 | 0 | 567 |
normalized size |
1 |
1. |
0.69 |
0.37 |
0.84 |
0.67 |
0. |
1.86 |
time (sec) |
N/A |
0.772 |
0.422 |
0.234 |
1.424 |
1.564 |
0. |
1.19 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 436 | 436 | 467 | 175 | 284 | 362 | 0 | 1551 |
normalized size |
1 |
1. |
1.07 |
0.4 |
0.65 |
0.83 |
0. |
3.56 |
time (sec) |
N/A |
0.739 |
1.375 |
0.277 |
1.508 |
1.629 |
0. |
1.993 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 396 | 396 | 667 | 385 | 0 | 941 | 947 | 774 |
normalized size |
1 |
1. |
1.68 |
0.97 |
0. |
2.38 |
2.39 |
1.95 |
time (sec) |
N/A |
0.404 |
2.384 |
0.386 |
0. |
1.634 |
3.432 |
1.24 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 405 | 227 | 0 | 581 | 590 | 447 |
normalized size |
1 |
1. |
1.38 |
0.77 |
0. |
1.98 |
2.01 |
1.52 |
time (sec) |
N/A |
0.265 |
1.463 |
0.321 |
0. |
1.591 |
2.218 |
1.279 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 205 | 114 | 0 | 309 | 313 | 204 |
normalized size |
1 |
1. |
0.98 |
0.55 |
0. |
1.48 |
1.5 |
0.98 |
time (sec) |
N/A |
0.264 |
0.672 |
0.275 |
0. |
1.573 |
1.294 |
1.18 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 449 | 449 | 336 | 163 | 365 | 284 | 0 | 1142 |
normalized size |
1 |
1. |
0.75 |
0.36 |
0.81 |
0.63 |
0. |
2.54 |
time (sec) |
N/A |
1.783 |
0.654 |
0.305 |
1.377 |
1.662 |
0. |
1.258 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 712 | 712 | 833 | 787 | 397 | 509 | 0 | 2302 |
normalized size |
1 |
1. |
1.17 |
1.11 |
0.56 |
0.71 |
0. |
3.23 |
time (sec) |
N/A |
1.732 |
2.805 |
0.142 |
1.789 |
1.621 |
0. |
2.356 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.05 |
22.711 |
0.147 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
13.886 |
0.185 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
A |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 98 | 98 | 205 | 0 | 0 | 207 | 0 | 0 |
normalized size |
1 |
1. |
2.09 |
0. |
0. |
2.11 |
0. |
0. |
time (sec) |
N/A |
0.123 |
1.227 |
0.197 |
0. |
1.632 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 192 | 0 | 0 | 354 | 0 | 0 |
normalized size |
1 |
1. |
1.12 |
0. |
0. |
2.07 |
0. |
0. |
time (sec) |
N/A |
0.184 |
38.086 |
0.118 |
0. |
1.672 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 251 | 251 | 269 | 0 | 0 | 504 | 0 | 0 |
normalized size |
1 |
1. |
1.07 |
0. |
0. |
2.01 |
0. |
0. |
time (sec) |
N/A |
0.242 |
55.015 |
0.136 |
0. |
1.743 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
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 | 152 | 152 | 255 | 481 | 898 | 1257 | 0 | 0 |
normalized size | 1 | 1. | 1.68 | 3.16 | 5.91 | 8.27 | 0. | 0. |
time (sec) |
N/A |
0.251 |
0.327 |
0.143 |
1.803 |
1.737 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 115 | 115 | 191 | 295 | 498 | 814 | 0 | 0 |
normalized size | 1 | 1. | 1.66 | 2.57 | 4.33 | 7.08 | 0. | 0. |
time (sec) |
N/A |
0.208 |
0.115 |
0.074 |
1.682 |
1.683 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 87 | 143 | 176 | 420 | 0 | 0 |
normalized size | 1 | 1. | 1.04 | 1.7 | 2.1 | 5. | 0. | 0. |
time (sec) |
N/A |
0.119 |
0.014 |
0.061 |
1.64 |
1.63 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 |
0.028 |
1.922 |
0.184 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 |
0.027 |
5.718 |
0.201 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 300 | 300 | 1326 | 919 | 3401 | 1767 | 0 | 0 |
normalized size | 1 | 1. | 4.42 | 3.06 | 11.34 | 5.89 | 0. | 0. |
time (sec) |
N/A |
0.522 |
7.856 |
0.168 |
4.924 |
1.815 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 229 | 229 | 649 | 542 | 1713 | 1081 | 0 | 0 |
normalized size | 1 | 1. | 2.83 | 2.37 | 7.48 | 4.72 | 0. | 0. |
time (sec) |
N/A |
0.412 |
7.096 |
0.109 |
2.399 |
1.778 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 136 | 136 | 200 | 238 | 714 | 540 | 0 | 0 |
normalized size | 1 | 1. | 1.47 | 1.75 | 5.25 | 3.97 | 0. | 0. |
time (sec) |
N/A |
0.19 |
2.159 |
0.099 |
1.766 |
1.684 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
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 |
18.984 |
1.105 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
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 |
0.051 |
14.364 |
1.779 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 612 | 612 | 2572 | 1882 | 9189 | 2549 | 0 | 0 |
normalized size | 1 | 1. | 4.2 | 3.08 | 15.01 | 4.17 | 0. | 0. |
time (sec) |
N/A |
0.982 |
8.338 |
0.243 |
51.562 |
2.062 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 436 | 436 | 1846 | 1090 | 4578 | 1540 | 0 | 0 |
normalized size | 1 | 1. | 4.23 | 2.5 | 10.5 | 3.53 | 0. | 0. |
time (sec) |
N/A |
0.72 |
7.621 |
0.181 |
10.178 |
1.829 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 277 | 277 | 277 | 493 | 1782 | 752 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.78 | 6.43 | 2.71 | 0. | 0. |
time (sec) |
N/A |
0.338 |
3.533 |
0.136 |
3.144 |
1.744 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
Problem 52 |
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 |
13.592 |
1.775 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
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 |
0.05 |
18.698 |
3.512 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
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 | 243 | 243 | 297 | 1450 | 1319 | 2782 | 0 | 0 |
normalized size | 1 | 1. | 1.22 | 5.97 | 5.43 | 11.45 | 0. | 0. |
time (sec) |
N/A |
0.334 |
1.949 |
0.284 |
2.835 |
1.962 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
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 | 181 | 181 | 236 | 922 | 961 | 1993 | 0 | 0 |
normalized size | 1 | 1. | 1.3 | 5.09 | 5.31 | 11.01 | 0. | 0. |
time (sec) |
N/A |
0.272 |
1.412 |
0.179 |
2.356 |
1.852 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
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 | 125 | 125 | 177 | 462 | 539 | 1273 | 0 | 0 |
normalized size | 1 | 1. | 1.42 | 3.7 | 4.31 | 10.18 | 0. | 0. |
time (sec) |
N/A |
0.159 |
1.468 |
0.204 |
2.07 |
1.877 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
1.689 |
0.685 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.058 |
4.303 |
1.657 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 848 | 848 | 1673 | 3488 | 6329 | 5532 | 0 | 0 |
normalized size |
1 |
1. |
1.97 |
4.11 |
7.46 |
6.52 |
0. |
0. |
time (sec) |
N/A |
2.004 |
10.051 |
0.411 |
13.902 |
2.501 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 654 | 654 | 714 | 2160 | 3464 | 3513 | 0 | 0 |
normalized size |
1 |
1. |
1.09 |
3.3 |
5.3 |
5.37 |
0. |
0. |
time (sec) |
N/A |
1.537 |
7.912 |
0.32 |
5.494 |
2.136 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 214 | 214 | 745 | 999 | 1570 | 1952 | 0 | 0 |
normalized size |
1 |
1. |
3.48 |
4.67 |
7.34 |
9.12 |
0. |
0. |
time (sec) |
N/A |
0.275 |
6.87 |
0.279 |
3.376 |
2.018 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.061 |
15.847 |
5.47 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.059 |
16.207 |
10.503 |
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 |
2 |
1 |
1. |
45 |
0.022 |
|
|
|
|
|
|
|
17 |
F |
0 |
0 |
N/A |
0 |
N/A |
|
|
|
|
|
|
|
18 |
A |
5 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
19 |
A |
4 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
20 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
|
21 |
A |
7 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
22 |
A |
7 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
23 |
A |
8 |
5 |
1. |
23 |
0.217 |
|
|
|
|
|
|
|
24 |
A |
10 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
25 |
A |
8 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
26 |
A |
7 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
27 |
A |
21 |
5 |
1. |
23 |
0.217 |
|
|
|
|
|
|
|
28 |
A |
24 |
7 |
1. |
23 |
0.304 |
|
|
|
|
|
|
|
29 |
A |
14 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
30 |
A |
11 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
31 |
A |
11 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
32 |
A |
53 |
7 |
1. |
23 |
0.304 |
|
|
|
|
|
|
|
33 |
A |
60 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
34 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
35 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
36 |
A |
2 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
37 |
A |
4 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
38 |
A |
5 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
39 |
A |
8 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
40 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
|
41 |
A |
6 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
43 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
44 |
A |
15 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
45 |
A |
13 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
46 |
A |
9 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
47 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
48 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
49 |
A |
28 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
50 |
A |
22 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
51 |
A |
16 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
52 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
53 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
54 |
A |
6 |
6 |
1. |
20 |
0.3 |
|
|
|
|
|
|
|
55 |
A |
5 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
56 |
A |
4 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
57 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
58 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
59 |
A |
21 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
60 |
A |
18 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
|
61 |
A |
5 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
62 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
63 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|