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 }
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, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 62
}
B grade: { 37, 41, 45, 49, 56, 61 }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 1, 2, 3, 4, 7, 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, 54, 55 }
B grade: { 6, 50, 51, 52 }
C grade: { 5, 53 }
F grade: { 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 56, 57, 58, 59, 60, 61, 62 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 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: { 5 }
C grade: { }
F grade: { 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
}
2.1.5 FriCAS
A grade: { 1, 2, 3, 4, 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: { 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
}
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 }
2.1.7 Giac
A grade: { 6, 7, 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62
}
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(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 83 | 86 | 104 | 194 | 0 | 0 |
normalized size |
1 |
1. |
0.7 |
0.72 |
0.87 |
1.63 |
0. |
0. |
time (sec) |
N/A |
0.07 |
0.269 |
0.754 |
1.948 |
1.577 |
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 |
A |
A |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 71 | 73 | 81 | 161 | 0 | 0 |
normalized size |
1 |
1. |
0.8 |
0.82 |
0.91 |
1.81 |
0. |
0. |
time (sec) |
N/A |
0.049 |
0.117 |
0.711 |
2.102 |
1.566 |
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 |
A |
A |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 55 | 58 | 51 | 117 | 0 | 0 |
normalized size |
1 |
1. |
0.93 |
0.98 |
0.86 |
1.98 |
0. |
0. |
time (sec) |
N/A |
0.028 |
0.064 |
0.717 |
2.704 |
1.532 |
0. |
0. |
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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 |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 29 | 43 | 27 | 84 | 0 | 0 |
normalized size | 1 | 1. | 1.12 | 1.65 | 1.04 | 3.23 | 0. | 0. |
time (sec) |
N/A |
0.013 |
0.03 |
0.491 |
2.02 |
1.562 |
0. |
0. |
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Problem 5 |
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 | 46 | 46 | 40 | 54 | 122 | 347 | 0 | 0 |
normalized size | 1 | 1. | 0.87 | 1.17 | 2.65 | 7.54 | 0. | 0. |
time (sec) |
N/A |
0.021 |
0.016 |
0.112 |
2.992 |
1.635 |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
F(-1) |
B |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 63 | 138 | 0 | 409 | 0 | 109 |
normalized size |
1 |
1. |
0.82 |
1.79 |
0. |
5.31 |
0. |
1.42 |
time (sec) |
N/A |
0.041 |
0.081 |
1.136 |
0. |
1.643 |
0. |
1.612 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
F(-1) |
B |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 107 | 107 | 65 | 174 | 0 | 501 | 0 | 139 |
normalized size |
1 |
1. |
0.61 |
1.63 |
0. |
4.68 |
0. |
1.3 |
time (sec) |
N/A |
0.063 |
0.183 |
1.234 |
0. |
1.694 |
0. |
1.747 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 69 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.03 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.036 |
0.1 |
0.133 |
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 | 69 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.05 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.034 |
0.065 |
0.21 |
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 |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 69 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.06 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.03 |
0.064 |
0.167 |
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 | 65 | 65 | 67 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.03 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.054 |
0.187 |
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 |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 67 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.03 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.031 |
0.056 |
0.125 |
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 | 68 | 68 | 69 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.032 |
0.064 |
0.129 |
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 | 73 | 73 | 74 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.01 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.035 |
0.074 |
0.527 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 75 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.033 |
0.078 |
0.483 |
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 | 59 | 59 | 77 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.31 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.018 |
0.093 |
0.368 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 74 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.23 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.016 |
0.072 |
0.39 |
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 | 20 | 18 | 32 | 73 | 41 | 41 |
normalized size | 1 | 1. | 0.65 | 0.58 | 1.03 | 2.35 | 1.32 | 1.32 |
time (sec) |
N/A |
0.012 |
0.027 |
0.029 |
1.872 |
1.635 |
1.09 |
1.145 |
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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 | 43 | 48 | 92 | 163 | 187 | 80 |
normalized size | 1 | 1. | 0.77 | 0.86 | 1.64 | 2.91 | 3.34 | 1.43 |
time (sec) |
N/A |
0.035 |
0.089 |
0.034 |
1.825 |
1.626 |
2.882 |
1.204 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 81 | 81 | 56 | 79 | 150 | 258 | 359 | 101 |
normalized size | 1 | 1. | 0.69 | 0.98 | 1.85 | 3.19 | 4.43 | 1.25 |
time (sec) |
N/A |
0.062 |
0.189 |
0.034 |
1.76 |
1.663 |
6.101 |
1.229 |
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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 |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 66 | 110 | 204 | 362 | 592 | 119 |
normalized size | 1 | 1. | 0.62 | 1.04 | 1.92 | 3.42 | 5.58 | 1.12 |
time (sec) |
N/A |
0.093 |
0.255 |
0.041 |
2.099 |
1.744 |
15.903 |
1.243 |
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Problem 22 |
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 | 33 | 33 | 20 | 18 | 32 | 73 | 41 | 41 |
normalized size | 1 | 1. | 0.61 | 0.55 | 0.97 | 2.21 | 1.24 | 1.24 |
time (sec) |
N/A |
0.013 |
0.026 |
0.029 |
1.884 |
1.585 |
0.814 |
1.15 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 43 | 48 | 93 | 163 | 192 | 82 |
normalized size | 1 | 1. | 0.74 | 0.83 | 1.6 | 2.81 | 3.31 | 1.41 |
time (sec) |
N/A |
0.036 |
0.08 |
0.038 |
1.857 |
1.626 |
2.133 |
1.181 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 65 | 83 | 151 | 258 | 364 | 104 |
normalized size | 1 | 1. | 0.78 | 1. | 1.82 | 3.11 | 4.39 | 1.25 |
time (sec) |
N/A |
0.063 |
0.137 |
0.038 |
1.99 |
1.643 |
5.795 |
1.124 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
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 | 108 | 108 | 66 | 116 | 205 | 362 | 597 | 122 |
normalized size | 1 | 1. | 0.61 | 1.07 | 1.9 | 3.35 | 5.53 | 1.13 |
time (sec) |
N/A |
0.096 |
0.227 |
0.04 |
2.194 |
1.62 |
15.149 |
1.221 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 20 | 18 | 32 | 72 | 42 | 41 |
normalized size | 1 | 1. | 0.61 | 0.55 | 0.97 | 2.18 | 1.27 | 1.24 |
time (sec) |
N/A |
0.013 |
0.024 |
0.033 |
1.873 |
1.611 |
0.776 |
1.226 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 43 | 48 | 93 | 163 | 192 | 82 |
normalized size | 1 | 1. | 0.74 | 0.83 | 1.6 | 2.81 | 3.31 | 1.41 |
time (sec) |
N/A |
0.033 |
0.023 |
0.034 |
1.956 |
1.655 |
2.217 |
1.2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
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 | 83 | 83 | 65 | 83 | 151 | 257 | 366 | 104 |
normalized size | 1 | 1. | 0.78 | 1. | 1.82 | 3.1 | 4.41 | 1.25 |
time (sec) |
N/A |
0.064 |
0.106 |
0.036 |
1.938 |
1.655 |
5.659 |
1.138 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
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 | 108 | 108 | 66 | 116 | 205 | 362 | 597 | 122 |
normalized size | 1 | 1. | 0.61 | 1.07 | 1.9 | 3.35 | 5.53 | 1.13 |
time (sec) |
N/A |
0.095 |
0.026 |
0.04 |
2.089 |
1.659 |
14.676 |
1.157 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 20 | 18 | 32 | 72 | 44 | 41 |
normalized size | 1 | 1. | 0.65 | 0.58 | 1.03 | 2.32 | 1.42 | 1.32 |
time (sec) |
N/A |
0.013 |
0.026 |
0.029 |
2.127 |
1.619 |
0.779 |
1.137 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 43 | 48 | 92 | 163 | 190 | 80 |
normalized size | 1 | 1. | 0.77 | 0.86 | 1.64 | 2.91 | 3.39 | 1.43 |
time (sec) |
N/A |
0.029 |
0.071 |
0.037 |
1.977 |
1.642 |
2.437 |
1.17 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
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 | 81 | 81 | 65 | 79 | 150 | 257 | 362 | 101 |
normalized size | 1 | 1. | 0.8 | 0.98 | 1.85 | 3.17 | 4.47 | 1.25 |
time (sec) |
N/A |
0.061 |
0.121 |
0.037 |
2.018 |
1.659 |
6.492 |
1.311 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 106 | 106 | 66 | 110 | 204 | 362 | 595 | 119 |
normalized size | 1 | 1. | 0.62 | 1.04 | 1.92 | 3.42 | 5.61 | 1.12 |
time (sec) |
N/A |
0.096 |
0.026 |
0.039 |
2.015 |
1.668 |
15.884 |
1.313 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 65 | 65 | 65 | 36 | 65 | 136 | 41 | 46 |
normalized size | 1 | 1. | 1. | 0.55 | 1. | 2.09 | 0.63 | 0.71 |
time (sec) |
N/A |
0.019 |
0.026 |
0.035 |
1.207 |
1.59 |
0.734 |
1.222 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 90 | 90 | 143 | 72 | 123 | 258 | 228 | 84 |
normalized size | 1 | 1. | 1.59 | 0.8 | 1.37 | 2.87 | 2.53 | 0.93 |
time (sec) |
N/A |
0.041 |
0.078 |
0.039 |
1.416 |
1.67 |
2.272 |
1.165 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 115 | 115 | 217 | 108 | 182 | 379 | 474 | 105 |
normalized size | 1 | 1. | 1.89 | 0.94 | 1.58 | 3.3 | 4.12 | 0.91 |
time (sec) |
N/A |
0.075 |
0.194 |
0.042 |
1.163 |
1.628 |
4.954 |
1.179 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
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 | 140 | 140 | 296 | 144 | 235 | 521 | 813 | 123 |
normalized size | 1 | 1. | 2.11 | 1.03 | 1.68 | 3.72 | 5.81 | 0.88 |
time (sec) |
N/A |
0.113 |
0.246 |
0.04 |
1.417 |
1.665 |
13.631 |
1.19 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 63 | 63 | 63 | 40 | 68 | 140 | 44 | 51 |
normalized size | 1 | 1. | 1. | 0.63 | 1.08 | 2.22 | 0.7 | 0.81 |
time (sec) |
N/A |
0.018 |
0.025 |
0.033 |
1.638 |
1.591 |
0.727 |
1.387 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 88 | 88 | 139 | 80 | 127 | 259 | 240 | 92 |
normalized size | 1 | 1. | 1.58 | 0.91 | 1.44 | 2.94 | 2.73 | 1.05 |
time (sec) |
N/A |
0.042 |
0.079 |
0.04 |
1.591 |
1.585 |
1.901 |
1.292 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 113 | 113 | 211 | 120 | 185 | 383 | 490 | 113 |
normalized size | 1 | 1. | 1.87 | 1.06 | 1.64 | 3.39 | 4.34 | 1. |
time (sec) |
N/A |
0.072 |
0.145 |
0.041 |
1.586 |
1.691 |
5.185 |
1.218 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 | 138 | 138 | 288 | 160 | 239 | 522 | 831 | 131 |
normalized size | 1 | 1. | 2.09 | 1.16 | 1.73 | 3.78 | 6.02 | 0.95 |
time (sec) |
N/A |
0.113 |
0.231 |
0.043 |
1.847 |
1.682 |
14.099 |
1.155 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 63 | 63 | 63 | 40 | 68 | 139 | 44 | 51 |
normalized size | 1 | 1. | 1. | 0.63 | 1.08 | 2.21 | 0.7 | 0.81 |
time (sec) |
N/A |
0.017 |
0.021 |
0.035 |
2.116 |
1.646 |
1.206 |
1.159 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 88 | 88 | 139 | 80 | 127 | 259 | 240 | 92 |
normalized size | 1 | 1. | 1.58 | 0.91 | 1.44 | 2.94 | 2.73 | 1.05 |
time (sec) |
N/A |
0.04 |
0.021 |
0.039 |
1.549 |
1.652 |
2.305 |
1.162 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 113 | 113 | 211 | 120 | 185 | 382 | 490 | 113 |
normalized size | 1 | 1. | 1.87 | 1.06 | 1.64 | 3.38 | 4.34 | 1. |
time (sec) |
N/A |
0.072 |
0.131 |
0.038 |
1.522 |
1.637 |
5.178 |
1.346 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 | 138 | 138 | 288 | 160 | 239 | 522 | 831 | 131 |
normalized size | 1 | 1. | 2.09 | 1.16 | 1.73 | 3.78 | 6.02 | 0.95 |
time (sec) |
N/A |
0.108 |
0.024 |
0.04 |
1.579 |
1.681 |
13.817 |
1.237 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 65 | 65 | 65 | 36 | 65 | 138 | 42 | 46 |
normalized size | 1 | 1. | 1. | 0.55 | 1. | 2.12 | 0.65 | 0.71 |
time (sec) |
N/A |
0.018 |
0.024 |
0.033 |
1.373 |
1.615 |
0.728 |
1.202 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 90 | 90 | 143 | 72 | 123 | 258 | 231 | 84 |
normalized size | 1 | 1. | 1.59 | 0.8 | 1.37 | 2.87 | 2.57 | 0.93 |
time (sec) |
N/A |
0.041 |
0.022 |
0.04 |
1.027 |
1.596 |
1.889 |
1.167 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
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 | 217 | 108 | 182 | 381 | 478 | 105 |
normalized size | 1 | 1. | 1.89 | 0.94 | 1.58 | 3.31 | 4.16 | 0.91 |
time (sec) |
N/A |
0.072 |
0.107 |
0.04 |
1.509 |
1.694 |
5.003 |
1.155 |
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
Problem 49 |
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 | 140 | 140 | 296 | 144 | 235 | 521 | 816 | 123 |
normalized size | 1 | 1. | 2.11 | 1.03 | 1.68 | 3.72 | 5.83 | 0.88 |
time (sec) |
N/A |
0.12 |
0.025 |
0.042 |
1.481 |
1.672 |
12.666 |
1.154 |
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
F |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 197 | 197 | 177 | 662 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.9 |
3.36 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.282 |
0.767 |
2.738 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 157 | 157 | 134 | 450 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.85 |
2.87 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.171 |
0.521 |
2.338 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
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 | 57 | 57 | 57 | 170 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 2.98 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.037 |
0.069 |
2.023 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
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 | 57 | 57 | 57 | 75 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1. | 1.32 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.039 |
0.049 |
0.405 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 106 | 106 | 83 | 217 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 0.78 | 2.05 | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.068 |
0.195 |
2.919 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
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 | 221 | 221 | 158 | 489 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
0.71 |
2.21 |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.233 |
0.893 |
5.29 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
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 | 108 | 108 | 246 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
2.28 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.077 |
1.858 |
0.129 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
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 | 105 | 105 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.12 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.067 |
0.182 |
0.279 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
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 | 105 | 105 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.12 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.063 |
0.182 |
0.289 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 105 | 105 | 118 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.12 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.066 |
0.206 |
0.213 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 | 105 | 105 | 116 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 1.1 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.064 |
0.176 |
0.205 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
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 | 110 | 110 | 268 | 0 | 0 | 0 | 0 | 0 |
normalized size | 1 | 1. | 2.44 | 0. | 0. | 0. | 0. | 0. |
time (sec) |
N/A |
0.069 |
2.039 |
0.127 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
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 | 103 | 103 | 121 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
1. |
1.17 |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.067 |
0.236 |
0.58 |
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 [50] had the largest ratio of [ 0.5
]
|
|
|
|
|
|
|
| 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 |
4 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
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 |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
35 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
36 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
37 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
38 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
39 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
40 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
|
41 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
42 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
43 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
44 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
45 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
46 |
A |
2 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
47 |
A |
4 |
4 |
1. |
12 |
0.333 |
|
|
|
|
|
|
|
48 |
A |
5 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
49 |
A |
6 |
5 |
1. |
12 |
0.417 |
|
|
|
|
|
|
|
50 |
A |
7 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
51 |
A |
6 |
6 |
1. |
14 |
0.429 |
|
|
|
|
|
|
|
52 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
53 |
A |
2 |
2 |
1. |
14 |
0.143 |
|
|
|
|
|
|
|
54 |
A |
4 |
4 |
1. |
14 |
0.286 |
|
|
|
|
|
|
|
55 |
A |
7 |
7 |
1. |
14 |
0.5 |
|
|
|
|
|
|
|
56 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
57 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
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. |
12 |
0.25 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|