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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 4, 5, 6, 8, 9, 10, 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, 40, 41, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 58, 60, 61 }
B grade: { 3, 7, 11, 37, 38, 39, 42, 43, 47, 48, 57, 59 }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 25, 26, 30, 31, 32, 33, 40, 41, 45, 46, 50, 51, 55, 56,
60, 61 }
B grade: { 1, 2, 3, 6, 7, 11, 12, 13, 22, 23, 24, 27, 28, 29, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 54, 57, 58,
59 }
C grade: { }
F grade: { 34, 35, 36 }
2.1.4 Maxima
A grade: { 4, 5, 9, 19, 20, 21, 25, 26, 30, 31, 40, 46, 50, 51, 55
B grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 54, 57, 58, 59
}
C grade: { }
F grade: { 10, 14, 15, 16, 17, 18, 22, 23, 24, 27, 28, 29, 32, 33, 34, 35, 36, 41, 45, 56, 60, 61
}
2.1.5 FriCAS
A grade: { 4, 5, 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,
40, 41, 45, 46, 50, 51, 55, 56, 60, 61 }
B grade: { 3, 7, 8, 13, 39, 44, 49, 54, 59 }
C grade: { 1, 2, 6, 11, 12, 37, 38, 42, 43, 47, 48, 52, 53, 57, 58 }
F grade: { }
2.1.6 Sympy
A grade: { 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 22, 23, 24, 27, 28, 29, 33, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61
}
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 19, 20, 21, 25, 26, 30, 31, 32, 34, 35, 36, 37, 38, 39, 42, 43, 44, 47, 48, 49,
52, 53, 54, 57, 58, 59 }
2.1.7 Giac
A grade: { 4, 5, 9, 10, 14, 15, 16, 17, 18, 23, 24, 28, 29, 32, 33, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61
}
B grade: { 8, 19, 20, 21, 22, 25, 26, 27, 30, 31 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 34, 35, 36, 37, 38, 39, 42, 43, 44, 47, 48, 49, 52, 53, 54, 57, 58, 59
}
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 |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 101 | 101 | 184 | 240 | 528 | 842 | 0 | 0 |
normalized size | 1 | 1. | 1.82 | 2.38 | 5.23 | 8.34 | 0. | 0. |
time (sec) |
N/A |
0.171 |
0.679 |
0.258 |
1.413 |
1.839 |
0. |
0. |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 136 | 198 | 347 | 664 | 0 | 0 |
normalized size | 1 | 1. | 1.84 | 2.68 | 4.69 | 8.97 | 0. | 0. |
time (sec) |
N/A |
0.14 |
0.418 |
0.242 |
1.411 |
1.767 |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 135 | 150 | 189 | 486 | 0 | 0 |
normalized size | 1 | 1. | 2.55 | 2.83 | 3.57 | 9.17 | 0. | 0. |
time (sec) |
N/A |
0.085 |
3.555 |
0.231 |
1.487 |
1.764 |
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 |
0.016 |
1.769 |
0.184 |
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 |
0.017 |
3.13 |
0.202 |
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 |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 171 | 231 | 1287 | 971 | 0 | 0 |
normalized size | 1 | 1. | 1.76 | 2.38 | 13.27 | 10.01 | 0. | 0. |
time (sec) |
N/A |
0.178 |
0.845 |
0.149 |
2.115 |
1.797 |
0. |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
B |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 153 | 183 | 518 | 749 | 0 | 0 |
normalized size | 1 | 1. | 2.07 | 2.47 | 7. | 10.12 | 0. | 0. |
time (sec) |
N/A |
0.119 |
4.591 |
0.134 |
2.161 |
1.785 |
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 |
B |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 44 | 30 | 363 | 190 | 65 | 1688 |
normalized size | 1 | 1. | 1.42 | 0.97 | 11.71 | 6.13 | 2.1 | 54.45 |
time (sec) |
N/A |
0.023 |
0.199 |
0.043 |
1.688 |
1.719 |
0.526 |
2.157 |
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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 |
0.03 |
3.584 |
0.235 |
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 |
F(-1) |
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 |
3.26 |
0.227 |
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 |
B |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 202 | 202 | 461 | 444 | 2660 | 1473 | 0 | 0 |
normalized size | 1 | 1. | 2.28 | 2.2 | 13.17 | 7.29 | 0. | 0. |
time (sec) |
N/A |
0.301 |
6.71 |
0.283 |
2.201 |
1.871 |
0. |
0. |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 221 | 293 | 1642 | 1077 | 0 | 0 |
normalized size | 1 | 1. | 1.75 | 2.33 | 13.03 | 8.55 | 0. | 0. |
time (sec) |
N/A |
0.186 |
5.143 |
0.267 |
1.698 |
1.871 |
0. |
0. |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
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grade |
A |
A |
A |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 179 | 197 | 798 | 761 | 0 | 0 |
normalized size | 1 | 1. | 1.97 | 2.16 | 8.77 | 8.36 | 0. | 0. |
time (sec) |
N/A |
0.107 |
4.083 |
0.145 |
1.537 |
1.77 |
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 |
F(-1) |
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 |
6.272 |
1.142 |
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 |
F(-1) |
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 |
5.017 |
1.452 |
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 |
A |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 189 | 189 | 246 | 170 | 0 | 343 | 360 | 328 |
normalized size |
1 |
1. |
1.3 |
0.9 |
0. |
1.81 |
1.9 |
1.74 |
time (sec) |
N/A |
0.2 |
0.597 |
0.199 |
0. |
1.612 |
0.772 |
1.218 |
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Problem 17 |
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 | 149 | 108 | 0 | 223 | 226 | 193 |
normalized size |
1 |
1. |
1.09 |
0.79 |
0. |
1.63 |
1.65 |
1.41 |
time (sec) |
N/A |
0.123 |
0.33 |
0.178 |
0. |
1.648 |
0.508 |
1.22 |
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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 | 84 | 84 | 107 | 139 | 0 | 117 | 116 | 90 |
normalized size |
1 |
1. |
1.27 |
1.65 |
0. |
1.39 |
1.38 |
1.07 |
time (sec) |
N/A |
0.054 |
0.23 |
0.143 |
0. |
1.604 |
0.328 |
1.254 |
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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 |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 161 | 161 | 77 | 67 | 150 | 117 | 0 | 495 |
normalized size |
1 |
1. |
0.48 |
0.42 |
0.93 |
0.73 |
0. |
3.07 |
time (sec) |
N/A |
0.278 |
0.274 |
0.142 |
1.387 |
1.686 |
0. |
1.26 |
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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 |
F(-2) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 166 | 166 | 215 | 105 | 163 | 178 | 0 | 797 |
normalized size |
1 |
1. |
1.3 |
0.63 |
0.98 |
1.07 |
0. |
4.8 |
time (sec) |
N/A |
0.227 |
1.199 |
0.158 |
1.45 |
1.645 |
0. |
1.542 |
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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) |
B |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 283 | 143 | 216 | 269 | 0 | 2201 |
normalized size |
1 |
1. |
1.25 |
0.63 |
0.95 |
1.19 |
0. |
9.7 |
time (sec) |
N/A |
0.312 |
1.529 |
0.181 |
1.457 |
1.664 |
0. |
1.269 |
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|
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|
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|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 270 | 270 | 362 | 2261 | 0 | 622 | 653 | 585 |
normalized size |
1 |
1. |
1.34 |
8.37 |
0. |
2.3 |
2.42 |
2.17 |
time (sec) |
N/A |
0.276 |
1.537 |
0.201 |
0. |
1.649 |
1.425 |
1.29 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 202 | 202 | 255 | 1073 | 0 | 383 | 406 | 333 |
normalized size |
1 |
1. |
1.26 |
5.31 |
0. |
1.9 |
2.01 |
1.65 |
time (sec) |
N/A |
0.198 |
0.674 |
0.138 |
0. |
1.63 |
0.941 |
1.283 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 151 | 151 | 165 | 390 | 0 | 194 | 214 | 146 |
normalized size |
1 |
1. |
1.09 |
2.58 |
0. |
1.28 |
1.42 |
0.97 |
time (sec) |
N/A |
0.138 |
0.584 |
0.138 |
0. |
1.58 |
0.58 |
1.27 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
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 | 305 | 305 | 136 | 382 | 262 | 197 | 0 | 1332 |
normalized size |
1 |
1. |
0.45 |
1.25 |
0.86 |
0.65 |
0. |
4.37 |
time (sec) |
N/A |
0.752 |
0.563 |
0.133 |
1.31 |
1.686 |
0. |
1.266 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 434 | 434 | 203 | 537 | 289 | 324 | 0 | 1551 |
normalized size |
1 |
1. |
0.47 |
1.24 |
0.67 |
0.75 |
0. |
3.57 |
time (sec) |
N/A |
0.681 |
0.548 |
0.125 |
1.501 |
1.627 |
0. |
2.247 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 396 | 396 | 664 | 3997 | 0 | 929 | 933 | 841 |
normalized size |
1 |
1. |
1.68 |
10.09 |
0. |
2.35 |
2.36 |
2.12 |
time (sec) |
N/A |
0.381 |
2.411 |
0.133 |
0. |
1.706 |
2.223 |
1.33 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 369 | 1843 | 0 | 560 | 576 | 474 |
normalized size |
1 |
1. |
1.26 |
6.27 |
0. |
1.9 |
1.96 |
1.61 |
time (sec) |
N/A |
0.269 |
0.756 |
0.078 |
0. |
1.611 |
1.49 |
1.264 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 244 | 653 | 0 | 285 | 299 | 204 |
normalized size |
1 |
1. |
1.17 |
3.12 |
0. |
1.36 |
1.43 |
0.98 |
time (sec) |
N/A |
0.218 |
0.611 |
0.074 |
0. |
1.587 |
0.987 |
1.33 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 449 | 449 | 197 | 560 | 371 | 277 | 0 | 2547 |
normalized size |
1 |
1. |
0.44 |
1.25 |
0.83 |
0.62 |
0. |
5.67 |
time (sec) |
N/A |
1.732 |
0.528 |
0.08 |
1.592 |
1.7 |
0. |
1.396 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 712 | 712 | 292 | 755 | 405 | 468 | 0 | 2302 |
normalized size |
1 |
1. |
0.41 |
1.06 |
0.57 |
0.66 |
0. |
3.23 |
time (sec) |
N/A |
1.643 |
0.578 |
0.084 |
1.941 |
1.691 |
0. |
2.852 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
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 |
0.051 |
9.889 |
0.224 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 |
0.029 |
5.703 |
0.178 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 98 | 98 | 190 | 0 | 0 | 209 | 0 | 0 |
normalized size |
1 |
1. |
1.94 |
0. |
0. |
2.13 |
0. |
0. |
time (sec) |
N/A |
0.124 |
1.218 |
0.267 |
0. |
1.671 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
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 | 152 | 0 | 0 | 360 | 0 | 0 |
normalized size |
1 |
1. |
0.89 |
0. |
0. |
2.11 |
0. |
0. |
time (sec) |
N/A |
0.185 |
2.551 |
0.207 |
0. |
1.724 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 238 | 0 | 0 | 513 | 0 | 0 |
normalized size |
1 |
1. |
0.95 |
0. |
0. |
2.04 |
0. |
0. |
time (sec) |
N/A |
0.242 |
3.245 |
0.466 |
0. |
1.78 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
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 | 147 | 147 | 730 | 857 | 1304 | 1524 | 0 | 0 |
normalized size | 1 | 1. | 4.97 | 5.83 | 8.87 | 10.37 | 0. | 0. |
time (sec) |
N/A |
0.255 |
7.446 |
0.396 |
2.542 |
1.92 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
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 | 112 | 112 | 406 | 516 | 702 | 1025 | 0 | 0 |
normalized size | 1 | 1. | 3.62 | 4.61 | 6.27 | 9.15 | 0. | 0. |
time (sec) |
N/A |
0.21 |
2.489 |
0.339 |
1.595 |
1.811 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 83 | 83 | 204 | 240 | 281 | 595 | 0 | 0 |
normalized size | 1 | 1. | 2.46 | 2.89 | 3.39 | 7.17 | 0. | 0. |
time (sec) |
N/A |
0.123 |
4.499 |
0.33 |
1.877 |
1.764 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
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 |
0.029 |
2.119 |
0.348 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
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 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
normalized size |
1 |
0. |
0. |
0. |
0. |
0. |
0. |
0. |
time (sec) |
N/A |
0.028 |
7.817 |
0.375 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 | 295 | 295 | 1611 | 1571 | 5420 | 2641 | 0 | 0 |
normalized size | 1 | 1. | 5.46 | 5.33 | 18.37 | 8.95 | 0. | 0. |
time (sec) |
N/A |
0.531 |
7.378 |
0.584 |
9.784 |
2.212 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 | 227 | 227 | 729 | 899 | 2627 | 1694 | 0 | 0 |
normalized size | 1 | 1. | 3.21 | 3.96 | 11.57 | 7.46 | 0. | 0. |
time (sec) |
N/A |
0.383 |
7.174 |
0.463 |
3.013 |
1.951 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 137 | 137 | 200 | 365 | 1046 | 957 | 0 | 0 |
normalized size | 1 | 1. | 1.46 | 2.66 | 7.64 | 6.99 | 0. | 0. |
time (sec) |
N/A |
0.174 |
2.165 |
0.36 |
1.792 |
1.923 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
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 |
0.053 |
18.78 |
1.752 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
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 |
15.326 |
2.468 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
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 | 603 | 603 | 3045 | 3113 | 15125 | 5689 | 0 | 0 |
normalized size | 1 | 1. | 5.05 | 5.16 | 25.08 | 9.43 | 0. | 0. |
time (sec) |
N/A |
0.981 |
8.475 |
0.777 |
101.748 |
2.827 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
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 | 433 | 433 | 2013 | 1740 | 7318 | 3310 | 0 | 0 |
normalized size | 1 | 1. | 4.65 | 4.02 | 16.9 | 7.64 | 0. | 0. |
time (sec) |
N/A |
0.649 |
7.794 |
0.597 |
16.673 |
2.339 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
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 | 278 | 278 | 276 | 745 | 2730 | 1596 | 0 | 0 |
normalized size | 1 | 1. | 0.99 | 2.68 | 9.82 | 5.74 | 0. | 0. |
time (sec) |
N/A |
0.323 |
3.523 |
0.47 |
3.808 |
1.94 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
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 |
9.156 |
2.582 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
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 |
11.862 |
4.519 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
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 | 242 | 242 | 345 | 1386 | 1332 | 2368 | 0 | 0 |
normalized size | 1 | 1. | 1.43 | 5.73 | 5.5 | 9.79 | 0. | 0. |
time (sec) |
N/A |
0.341 |
1.916 |
0.499 |
3.102 |
2.521 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
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 | 289 | 881 | 971 | 1710 | 0 | 0 |
normalized size | 1 | 1. | 1.6 | 4.87 | 5.36 | 9.45 | 0. | 0. |
time (sec) |
N/A |
0.288 |
1.339 |
0.466 |
2.454 |
2.32 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
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 | 126 | 126 | 182 | 445 | 548 | 1123 | 0 | 0 |
normalized size | 1 | 1. | 1.44 | 3.53 | 4.35 | 8.91 | 0. | 0. |
time (sec) |
N/A |
0.162 |
1.512 |
0.404 |
2.369 |
2.166 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
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 |
0.064 |
1.796 |
0.837 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
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 |
0.059 |
3.623 |
1.929 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 839 | 839 | 1682 | 3446 | 6325 | 7125 | 0 | 0 |
normalized size |
1 |
1. |
2. |
4.11 |
7.54 |
8.49 |
0. |
0. |
time (sec) |
N/A |
1.969 |
10.193 |
0.744 |
15.364 |
2.924 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 650 | 650 | 718 | 2137 | 3487 | 4415 | 0 | 0 |
normalized size |
1 |
1. |
1.1 |
3.29 |
5.36 |
6.79 |
0. |
0. |
time (sec) |
N/A |
1.477 |
8.331 |
0.63 |
5.957 |
2.529 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
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 | 213 | 213 | 730 | 990 | 1598 | 2394 | 0 | 0 |
normalized size |
1 |
1. |
3.43 |
4.65 |
7.5 |
11.24 |
0. |
0. |
time (sec) |
N/A |
0.292 |
7.048 |
0.53 |
3.774 |
2.216 |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
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 |
0.059 |
17.412 |
5.12 |
0. |
0. |
0. |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
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 |
0.056 |
15.699 |
10.492 |
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 |
5 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
17 |
A |
4 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
19 |
A |
7 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
20 |
A |
7 |
4 |
1. |
23 |
0.174 |
|
|
|
|
|
|
|
|
21 |
A |
8 |
5 |
1. |
23 |
0.217 |
|
|
|
|
|
|
|
22 |
A |
10 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
23 |
A |
8 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
24 |
A |
7 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
25 |
A |
21 |
5 |
1. |
23 |
0.217 |
|
|
|
|
|
|
|
26 |
A |
24 |
7 |
1. |
23 |
0.304 |
|
|
|
|
|
|
|
27 |
A |
14 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
28 |
A |
11 |
3 |
1. |
23 |
0.13 |
|
|
|
|
|
|
|
29 |
A |
11 |
3 |
1. |
21 |
0.143 |
|
|
|
|
|
|
|
30 |
A |
53 |
7 |
1. |
23 |
0.304 |
|
|
|
|
|
|
|
31 |
A |
60 |
9 |
1. |
23 |
0.391 |
|
|
|
|
|
|
|
32 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
33 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
35 |
A |
4 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
36 |
A |
5 |
2 |
1. |
23 |
0.087 |
|
|
|
|
|
|
|
37 |
A |
8 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
38 |
A |
7 |
6 |
1. |
18 |
0.333 |
|
|
|
|
|
|
|
39 |
A |
6 |
5 |
1. |
16 |
0.312 |
|
|
|
|
|
|
|
40 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
41 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
42 |
A |
15 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
43 |
A |
13 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
44 |
A |
9 |
7 |
1. |
18 |
0.389 |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
47 |
A |
28 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
48 |
A |
22 |
11 |
1. |
20 |
0.55 |
|
|
|
|
|
|
|
49 |
A |
16 |
9 |
1. |
18 |
0.5 |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
51 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
52 |
A |
6 |
6 |
1. |
20 |
0.3 |
|
|
|
|
|
|
|
53 |
A |
5 |
5 |
1. |
20 |
0.25 |
|
|
|
|
|
|
|
54 |
A |
4 |
4 |
1. |
18 |
0.222 |
|
|
|
|
|
|
|
55 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
56 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
57 |
A |
21 |
9 |
1. |
20 |
0.45 |
|
|
|
|
|
|
|
58 |
A |
18 |
10 |
1. |
20 |
0.5 |
|
|
|
|
|
|
|
59 |
A |
5 |
5 |
1. |
18 |
0.278 |
|
|
|
|
|
|
|
60 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
61 |
A |
0 |
0 |
0. |
0 |
0. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|