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, 71, 72 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 9, 11, 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, 59, 60, 61, 62, 64, 65,
66, 67, 68, 69, 70, 71, 72 }
B grade: { 4, 58, 63 }
C grade: { 5, 6, 7, 8, 10, 12 }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 5, 6, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 54, 56, 57 }
B grade: { 7, 20, 21, 24, 25, 28, 29, 32, 33, 50, 51, 52, 53, 55 }
C grade: { }
F grade: { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72
}
2.1.4 Maxima
A grade: { 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48
B grade: { 20, 21, 24, 25, 28, 29, 32, 33, 37, 41, 45, 49 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 }
2.1.5 FriCAS
A grade: { 1, 2, 3, 5, 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 }
B grade: { 4, 6, 7 }
C grade: { }
F grade: { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72 }
2.1.6 Sympy
A grade: { 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 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 }
2.1.7 Giac
A grade: { 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 }
B grade: { 6 }
C grade: { }
F grade: { 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72 }
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 |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 154 | 75 | 0 | 360 | 0 | 0 |
normalized size |
1 |
1. |
1.29 |
0.63 |
0. |
3.03 |
0. |
0. |
time (sec) |
N/A |
0.071 |
0.654 |
0.092 |
0. |
1.63 |
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 |
F |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 117 | 65 | 0 | 292 | 0 | 0 |
normalized size |
1 |
1. |
1.31 |
0.73 |
0. |
3.28 |
0. |
0. |
time (sec) |
N/A |
0.047 |
0.32 |
0.074 |
0. |
1.603 |
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 |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 89 | 53 | 0 | 204 | 0 | 0 |
normalized size | 1 | 1. | 1.51 | 0.9 | 0. | 3.46 | 0. | 0. |
time (sec) |
N/A |
0.029 |
0.131 |
0.082 |
0. |
1.642 |
0. |
0. |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
A |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 65 | 43 | 0 | 136 | 0 | 0 |
normalized size | 1 | 1. | 2.5 | 1.65 | 0. | 5.23 | 0. | 0. |
time (sec) |
N/A |
0.013 |
0.04 |
0.071 |
0. |
1.529 |
0. |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
A |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 73 | 75 | 0 | 463 | 0 | 0 |
normalized size | 1 | 1. | 1.55 | 1.6 | 0. | 9.85 | 0. | 0. |
time (sec) |
N/A |
0.025 |
0.069 |
0.072 |
0. |
1.646 |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
A |
F |
B |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 108 | 125 | 0 | 666 | 0 | 396 |
normalized size | 1 | 1. | 1.4 | 1.62 | 0. | 8.65 | 0. | 5.14 |
time (sec) |
N/A |
0.042 |
0.172 |
0.082 |
0. |
1.769 |
0. |
2.016 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
B |
F |
B |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 196 | 195 | 0 | 851 | 0 | 0 |
normalized size | 1 | 1. | 1.83 | 1.82 | 0. | 7.95 | 0. | 0. |
time (sec) |
N/A |
0.063 |
0.159 |
0.102 |
0. |
1.844 |
0. |
0. |
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Problem 8 |
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 | 67 | 67 | 341 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
5.09 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.036 |
2.494 |
0.046 |
0. |
0. |
0. |
0. |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 124 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.88 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.034 |
0.21 |
0.103 |
0. |
0. |
0. |
0. |
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Problem 10 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 270 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 4.09 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.031 |
2.443 |
0.069 |
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 |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 70 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.03 |
0.109 |
0.069 |
0. |
0. |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 604 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 9.15 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.033 |
6.098 |
0.066 |
0. |
0. |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 130 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.88 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.034 |
0.287 |
0.043 |
0. |
0. |
0. |
0. |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 90 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.22 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.038 |
0.188 |
0.31 |
0. |
0. |
0. |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 91 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.23 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.034 |
0.142 |
0.326 |
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 |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 90 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.5 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.018 |
0.115 |
0.207 |
0. |
0. |
0. |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 90 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.53 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.016 |
0.106 |
0.212 |
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 |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 56 | 20 | 35 | 72 | 46 | 66 |
normalized size | 1 | 1. | 1.81 | 0.65 | 1.13 | 2.32 | 1.48 | 2.13 |
time (sec) |
N/A |
0.013 |
0.026 |
0.017 |
1.421 |
1.872 |
0.868 |
1.173 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 91 | 92 | 126 | 162 | 388 | 128 |
normalized size | 1 | 1. | 1.62 | 1.64 | 2.25 | 2.89 | 6.93 | 2.29 |
time (sec) |
N/A |
0.034 |
0.126 |
0.03 |
1.456 |
1.997 |
2.942 |
1.098 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 113 | 184 | 234 | 271 | 918 | 163 |
normalized size | 1 | 1. | 1.4 | 2.27 | 2.89 | 3.35 | 11.33 | 2.01 |
time (sec) |
N/A |
0.061 |
0.302 |
0.038 |
1.458 |
2.103 |
9.11 |
1.12 |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 133 | 272 | 342 | 382 | 1693 | 198 |
normalized size | 1 | 1. | 1.25 | 2.57 | 3.23 | 3.6 | 15.97 | 1.87 |
time (sec) |
N/A |
0.098 |
0.421 |
0.037 |
1.48 |
2.064 |
73.773 |
1.12 |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 56 | 20 | 35 | 72 | 46 | 68 |
normalized size | 1 | 1. | 1.7 | 0.61 | 1.06 | 2.18 | 1.39 | 2.06 |
time (sec) |
N/A |
0.013 |
0.025 |
0.014 |
1.856 |
1.968 |
0.91 |
1.118 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
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 | 58 | 58 | 91 | 92 | 126 | 162 | 384 | 130 |
normalized size | 1 | 1. | 1.57 | 1.59 | 2.17 | 2.79 | 6.62 | 2.24 |
time (sec) |
N/A |
0.031 |
0.137 |
0.03 |
1.557 |
1.888 |
2.959 |
1.122 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 113 | 184 | 234 | 271 | 915 | 165 |
normalized size | 1 | 1. | 1.36 | 2.22 | 2.82 | 3.27 | 11.02 | 1.99 |
time (sec) |
N/A |
0.063 |
0.423 |
0.036 |
1.45 |
1.995 |
9.024 |
1.179 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 133 | 272 | 342 | 383 | 1690 | 200 |
normalized size | 1 | 1. | 1.23 | 2.52 | 3.17 | 3.55 | 15.65 | 1.85 |
time (sec) |
N/A |
0.096 |
0.435 |
0.037 |
1.467 |
2.013 |
73.772 |
1.17 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 56 | 20 | 35 | 73 | 48 | 68 |
normalized size | 1 | 1. | 1.7 | 0.61 | 1.06 | 2.21 | 1.45 | 2.06 |
time (sec) |
N/A |
0.012 |
0.022 |
0.013 |
1.433 |
1.87 |
0.907 |
1.138 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 91 | 92 | 126 | 162 | 384 | 130 |
normalized size | 1 | 1. | 1.57 | 1.59 | 2.17 | 2.79 | 6.62 | 2.24 |
time (sec) |
N/A |
0.032 |
0.023 |
0.029 |
1.456 |
2.03 |
2.93 |
1.225 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 113 | 184 | 234 | 273 | 915 | 165 |
normalized size | 1 | 1. | 1.36 | 2.22 | 2.82 | 3.29 | 11.02 | 1.99 |
time (sec) |
N/A |
0.063 |
0.279 |
0.036 |
1.461 |
2.003 |
8.915 |
1.224 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 133 | 272 | 342 | 383 | 1690 | 200 |
normalized size | 1 | 1. | 1.23 | 2.52 | 3.17 | 3.55 | 15.65 | 1.85 |
time (sec) |
N/A |
0.095 |
0.108 |
0.033 |
1.471 |
2.02 |
72.412 |
1.138 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 31 | 31 | 56 | 20 | 35 | 73 | 49 | 66 |
normalized size | 1 | 1. | 1.81 | 0.65 | 1.13 | 2.35 | 1.58 | 2.13 |
time (sec) |
N/A |
0.012 |
0.024 |
0.016 |
1.434 |
1.945 |
0.944 |
1.126 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 56 | 56 | 91 | 92 | 126 | 162 | 389 | 128 |
normalized size | 1 | 1. | 1.62 | 1.64 | 2.25 | 2.89 | 6.95 | 2.29 |
time (sec) |
N/A |
0.031 |
0.099 |
0.03 |
1.442 |
1.937 |
3.071 |
1.187 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 114 | 184 | 234 | 273 | 920 | 163 |
normalized size | 1 | 1. | 1.41 | 2.27 | 2.89 | 3.37 | 11.36 | 2.01 |
time (sec) |
N/A |
0.061 |
0.343 |
0.035 |
1.455 |
1.982 |
8.991 |
1.141 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 133 | 272 | 342 | 382 | 1695 | 198 |
normalized size | 1 | 1. | 1.25 | 2.57 | 3.23 | 3.6 | 15.99 | 1.87 |
time (sec) |
N/A |
0.09 |
0.412 |
0.037 |
1.472 |
2.184 |
73.175 |
1.132 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 63 | 63 | 63 | 38 | 66 | 128 | 42 | 49 |
normalized size | 1 | 1. | 1. | 0.6 | 1.05 | 2.03 | 0.67 | 0.78 |
time (sec) |
N/A |
0.023 |
0.023 |
0.017 |
0.962 |
1.962 |
0.729 |
1.187 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 88 | 88 | 126 | 76 | 155 | 247 | 466 | 109 |
normalized size | 1 | 1. | 1.43 | 0.86 | 1.76 | 2.81 | 5.3 | 1.24 |
time (sec) |
N/A |
0.048 |
0.186 |
0.035 |
0.97 |
1.865 |
2.179 |
1.17 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 113 | 113 | 180 | 114 | 263 | 390 | 1227 | 144 |
normalized size | 1 | 1. | 1.59 | 1.01 | 2.33 | 3.45 | 10.86 | 1.27 |
time (sec) |
N/A |
0.078 |
0.465 |
0.04 |
0.98 |
1.766 |
5.987 |
1.198 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 138 | 138 | 235 | 152 | 371 | 543 | 2356 | 180 |
normalized size | 1 | 1. | 1.7 | 1.1 | 2.69 | 3.93 | 17.07 | 1.3 |
time (sec) |
N/A |
0.111 |
0.872 |
0.04 |
0.979 |
1.713 |
31.861 |
1.13 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
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 | 65 | 65 | 65 | 38 | 66 | 127 | 42 | 49 |
normalized size | 1 | 1. | 1. | 0.58 | 1.02 | 1.95 | 0.65 | 0.75 |
time (sec) |
N/A |
0.023 |
0.023 |
0.017 |
0.955 |
1.701 |
0.76 |
1.19 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
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 | 90 | 90 | 130 | 76 | 155 | 248 | 462 | 109 |
normalized size | 1 | 1. | 1.44 | 0.84 | 1.72 | 2.76 | 5.13 | 1.21 |
time (sec) |
N/A |
0.046 |
0.185 |
0.039 |
0.96 |
1.774 |
2.202 |
1.168 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 | 115 | 115 | 184 | 114 | 263 | 389 | 1224 | 144 |
normalized size | 1 | 1. | 1.6 | 0.99 | 2.29 | 3.38 | 10.64 | 1.25 |
time (sec) |
N/A |
0.078 |
0.503 |
0.037 |
0.971 |
1.77 |
6.482 |
1.176 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 241 | 152 | 371 | 541 | 2353 | 180 |
normalized size | 1 | 1. | 1.72 | 1.09 | 2.65 | 3.86 | 16.81 | 1.29 |
time (sec) |
N/A |
0.115 |
0.926 |
0.041 |
0.984 |
1.61 |
31.101 |
1.175 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 | 65 | 65 | 65 | 38 | 66 | 128 | 42 | 49 |
normalized size | 1 | 1. | 1. | 0.58 | 1.02 | 1.97 | 0.65 | 0.75 |
time (sec) |
N/A |
0.022 |
0.021 |
0.014 |
0.975 |
1.582 |
0.696 |
1.156 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 | 90 | 90 | 130 | 76 | 155 | 248 | 462 | 109 |
normalized size | 1 | 1. | 1.44 | 0.84 | 1.72 | 2.76 | 5.13 | 1.21 |
time (sec) |
N/A |
0.048 |
0.019 |
0.034 |
0.966 |
1.712 |
2.176 |
1.174 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 115 | 115 | 183 | 114 | 263 | 390 | 1224 | 144 |
normalized size | 1 | 1. | 1.59 | 0.99 | 2.29 | 3.39 | 10.64 | 1.25 |
time (sec) |
N/A |
0.079 |
0.472 |
0.04 |
0.984 |
1.747 |
6.895 |
1.152 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 241 | 152 | 371 | 541 | 2353 | 180 |
normalized size | 1 | 1. | 1.72 | 1.09 | 2.65 | 3.86 | 16.81 | 1.29 |
time (sec) |
N/A |
0.116 |
0.024 |
0.038 |
0.988 |
2.073 |
34.166 |
1.16 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 | 63 | 63 | 63 | 38 | 66 | 127 | 44 | 49 |
normalized size | 1 | 1. | 1. | 0.6 | 1.05 | 2.02 | 0.7 | 0.78 |
time (sec) |
N/A |
0.022 |
0.023 |
0.017 |
0.966 |
2.015 |
1.032 |
1.181 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 126 | 76 | 155 | 247 | 468 | 109 |
normalized size | 1 | 1. | 1.43 | 0.86 | 1.76 | 2.81 | 5.32 | 1.24 |
time (sec) |
N/A |
0.043 |
0.151 |
0.033 |
0.965 |
2.12 |
2.289 |
1.175 |
|
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
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|
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|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 113 | 113 | 179 | 114 | 263 | 389 | 1229 | 144 |
normalized size | 1 | 1. | 1.58 | 1.01 | 2.33 | 3.44 | 10.88 | 1.27 |
time (sec) |
N/A |
0.077 |
0.462 |
0.036 |
0.978 |
1.95 |
6.38 |
1.164 |
|
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
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|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 138 | 138 | 235 | 152 | 371 | 543 | 2358 | 180 |
normalized size | 1 | 1. | 1.7 | 1.1 | 2.69 | 3.93 | 17.09 | 1.3 |
time (sec) |
N/A |
0.113 |
0.823 |
0.04 |
1.012 |
2.023 |
32.995 |
1.188 |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
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|
|
|
grade |
A |
A |
A |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 256 | 256 | 220 | 1040 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.86 |
4.06 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.382 |
0.969 |
0.199 |
0. |
0. |
0. |
0. |
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 207 | 207 | 185 | 890 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.89 |
4.3 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.26 |
0.939 |
0.137 |
0. |
0. |
0. |
0. |
|
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
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|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 167 | 167 | 142 | 656 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.85 | 3.93 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.164 |
0.786 |
0.126 |
0. |
0. |
0. |
0. |
|
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Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 61 | 239 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 3.85 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.035 |
2.173 |
0.135 |
0. |
0. |
0. |
0. |
|
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Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 61 | 126 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.98 | 2.03 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.042 |
0.058 |
0.1 |
0. |
0. |
0. |
0. |
|
|
|
|
|
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Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 111 | 111 | 87 | 443 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.78 | 3.99 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.065 |
0.289 |
0.158 |
0. |
0. |
0. |
0. |
|
|
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Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 231 | 231 | 166 | 497 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.72 | 2.15 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.228 |
0.905 |
0.339 |
0. |
0. |
0. |
0. |
|
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Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 198 | 584 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.68 |
2. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.35 |
1.291 |
0.412 |
0. |
0. |
0. |
0. |
|
|
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Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 109 | 109 | 244 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.24 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.079 |
1.57 |
0.054 |
0. |
0. |
0. |
0. |
|
|
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Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.11 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.068 |
0.207 |
0.204 |
0. |
0. |
0. |
0. |
|
|
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Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.11 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.066 |
0.194 |
0.161 |
0. |
0. |
0. |
0. |
|
|
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Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.11 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.064 |
0.187 |
0.097 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
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|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 116 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.09 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.067 |
0.18 |
0.078 |
0. |
0. |
0. |
0. |
|
|
|
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|
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|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 111 | 111 | 262 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.36 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.072 |
1.803 |
0.066 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
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|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 104 | 104 | 120 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.15 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.066 |
0.245 |
0.405 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
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|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 72 | 72 | 83 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.15 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.041 |
0.149 |
0.453 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 84 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.22 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.035 |
0.166 |
0.49 |
0. |
0. |
0. |
0. |
|
|
|
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|
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|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 99 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.55 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.035 |
0.125 |
0.346 |
0. |
0. |
0. |
0. |
|
|
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Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 96 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.39 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.034 |
0.129 |
0.369 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
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|
Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 83 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.24 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.038 |
0.148 |
0.372 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
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|
|
Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 84 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.31 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.035 |
0.162 |
0.374 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 95 | 95 | 95 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.051 |
0.119 |
0.268 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 72 |
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 | 110 | 110 | 100 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.91 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.067 |
0.133 |
0.264 |
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 [36] had the largest ratio of [ 0.5
]
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|
| Table 2.1Rubi specific breakdown of results for each
integral
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|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
4 |
2 |
1. |
14 |
0.143 |
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|
|
|
|
|
|
2 |
A |
3 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
4 |
A |
1 |
1 |
1. |
14 |
0.071 |
|
|
|
|
|
|
|
5 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
6 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
7 |
A |
4 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
8 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
9 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
10 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
11 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
12 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
13 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
14 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
15 |
A |
2 |
2 |
1. |
13 |
0.154 |
|
|
|
|
|
|
|
16 |
A |
1 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
17 |
A |
1 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
18 |
A |
1 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
19 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
20 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
|
21 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
22 |
A |
1 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
23 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
24 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
26 |
A |
1 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
27 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
28 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
29 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
30 |
A |
1 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
32 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
33 |
A |
5 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
34 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
35 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
36 |
A |
7 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
37 |
A |
8 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
38 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
39 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
40 |
A |
7 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
|
41 |
A |
8 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
42 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
43 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
44 |
A |
7 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
45 |
A |
8 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
46 |
A |
4 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
47 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
48 |
A |
7 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
49 |
A |
8 |
6 |
1. |
12 |
0.5 |
|
|
|
|
|
|
|
50 |
A |
8 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
51 |
A |
7 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
52 |
A |
6 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
53 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
54 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
55 |
A |
4 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
56 |
A |
7 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
57 |
A |
8 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
58 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
59 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
60 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
|
61 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
62 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
63 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
64 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
65 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
66 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
67 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
68 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
69 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
70 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
71 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
72 |
A |
3 |
3 |
1. |
12 |
0.25 |
|
|
|
|
|
|
|
|
|
|
|
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|
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|
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|