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, 3, 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
35, 37, 38, 39, 40, 41, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61 }
B grade: { 6, 11, 12, 34, 42, 43, 47 }
C grade: { 7, 13, 48, 59 }
F grade: { 36 }
2.1.3 Maple
A grade: { 4, 5, 8, 9, 10, 14, 15, 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, 16, 17, 18, 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, 6, 7, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
40, 41, 44, 45, 46, 50, 51, 55, 56, 57, 58, 60, 61 }
B grade: { 1, 2, 3, 8, 12, 13, 37, 38, 42, 43, 47, 48, 49, 52, 53 }
C grade: { }
F grade: { 34, 35, 36, 39, 54, 59 }
2.1.5 FriCAS
A grade: { 4, 5, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 40, 41, 45, 46,
50, 51, 55, 56, 60, 61 }
B grade: { 3, 7, 8, 13, 22, 23, 27, 28, 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, 32, 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, 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, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 40, 41, 45, 46,
50, 51, 55, 56, 60, 61 }
B grade: { 8, 20 }
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
}
2.1.8 Mupad
A grade: { 4, 5, 9, 10, 14, 15, 32, 33, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61 }
B grade: { 8, 16, 17, 18, 22, 23, 24, 27, 28, 29 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 19, 20, 21, 25, 26, 30, 31, 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 |
Mupad |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 91 | 200 | 170 | 216 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.05 | 2.30 | 1.95 | 2.48 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.159 |
0.013 |
0.280 |
1.528 |
2.272 |
0.000 |
0.000 |
0.000 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 63 | 63 | 66 | 166 | 138 | 168 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.05 | 2.63 | 2.19 | 2.67 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.135 |
0.010 |
0.273 |
0.696 |
0.627 |
0.000 |
0.000 |
0.000 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 47 | 122 | 98 | 112 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.04 | 2.71 | 2.18 | 2.49 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.085 |
0.007 |
0.266 |
0.504 |
1.365 |
0.000 |
0.000 |
0.000 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.08 |
time (sec) |
N/A |
0.016 |
13.240 |
0.316 |
0.000 |
0.446 |
0.000 |
0.000 |
0.000 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.08 |
time (sec) |
N/A |
0.017 |
19.121 |
0.293 |
0.000 |
0.459 |
0.000 |
0.000 |
0.000 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
B |
A |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 204 | 198 | 146 | 632 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.34 | 2.28 | 1.68 | 7.26 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.178 |
4.193 |
0.180 |
0.453 |
0.545 |
0.000 |
0.000 |
0.000 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
A |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 163 | 156 | 108 | 453 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.51 | 2.40 | 1.66 | 6.97 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.119 |
5.410 |
0.174 |
0.470 |
0.469 |
0.000 |
0.000 |
0.000 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
B |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 46 | 54 | 115 | 189 | 80 | 98 | 48 |
normalized size | 1 | 1.00 | 1.48 | 1.74 | 3.71 | 6.10 | 2.58 | 3.16 | 1.55 |
time (sec) |
N/A |
0.027 |
0.164 |
0.127 |
0.410 |
0.457 |
2.446 |
0.127 |
1.154 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.07 |
time (sec) |
N/A |
0.028 |
21.532 |
0.266 |
0.000 |
0.455 |
0.000 |
0.000 |
0.000 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.07 |
time (sec) |
N/A |
0.029 |
20.444 |
0.275 |
0.000 |
0.455 |
0.000 |
0.000 |
0.000 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
B |
A |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 179 | 179 | 390 | 375 | 302 | 1985 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.18 | 2.09 | 1.69 | 11.09 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.330 |
4.886 |
0.329 |
0.442 |
0.480 |
0.000 |
0.000 |
0.000 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
B |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 114 | 114 | 295 | 246 | 226 | 1467 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.59 | 2.16 | 1.98 | 12.87 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.207 |
3.444 |
0.311 |
0.416 |
0.523 |
0.000 |
0.000 |
0.000 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 231 | 164 | 149 | 975 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.82 | 2.00 | 1.82 | 11.89 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.122 |
6.151 |
0.258 |
0.416 |
0.520 |
0.000 |
0.000 |
0.000 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.07 |
time (sec) |
N/A |
0.028 |
16.175 |
0.813 |
0.000 |
0.479 |
0.000 |
0.000 |
0.000 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.07 |
time (sec) |
N/A |
0.029 |
12.043 |
0.826 |
0.000 |
0.443 |
0.000 |
0.000 |
0.000 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 169 | 169 | 244 | 959 | 183 | 304 | 864 | 193 | 223 |
normalized size | 1 | 1.00 | 1.44 | 5.67 | 1.08 | 1.80 | 5.11 | 1.14 | 1.32 |
time (sec) |
N/A |
0.185 |
0.438 |
0.432 |
0.460 |
0.512 |
1.743 |
0.131 |
1.451 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 122 | 122 | 169 | 469 | 124 | 192 | 522 | 123 | 186 |
normalized size | 1 | 1.00 | 1.39 | 3.84 | 1.02 | 1.57 | 4.28 | 1.01 | 1.52 |
time (sec) |
N/A |
0.117 |
0.281 |
0.392 |
0.409 |
0.411 |
1.253 |
0.124 |
1.299 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 81 | 175 | 72 | 101 | 250 | 65 | 76 |
normalized size | 1 | 1.00 | 1.09 | 2.36 | 0.97 | 1.36 | 3.38 | 0.88 | 1.03 |
time (sec) |
N/A |
0.051 |
0.289 |
0.375 |
0.409 |
0.387 |
0.890 |
0.130 |
1.257 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 122 | 61 | 48 | 75 | 0 | 50 | -1 |
normalized size | 1 | 1.00 | 0.78 | 0.39 | 0.31 | 0.48 | 0.00 | 0.32 | -0.01 |
time (sec) |
N/A |
0.251 |
0.273 |
1.025 |
0.571 |
0.396 |
0.000 |
0.141 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 159 | 159 | 206 | 91 | 56 | 216 | 0 | 347 | -1 |
normalized size | 1 | 1.00 | 1.30 | 0.57 | 0.35 | 1.36 | 0.00 | 2.18 | -0.01 |
time (sec) |
N/A |
0.213 |
0.810 |
1.050 |
0.803 |
0.399 |
0.000 |
0.177 |
0.000 |
|
|
|
|
|
|
|
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|
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|
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|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 211 | 211 | 265 | 210 | 68 | 342 | 0 | 179 | -1 |
normalized size | 1 | 1.00 | 1.26 | 1.00 | 0.32 | 1.62 | 0.00 | 0.85 | -0.00 |
time (sec) |
N/A |
0.301 |
1.123 |
1.063 |
1.081 |
0.397 |
0.000 |
0.130 |
0.000 |
|
|
|
|
|
|
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|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 230 | 230 | 420 | 1979 | 297 | 571 | 2193 | 383 | 266 |
normalized size | 1 | 1.00 | 1.83 | 8.60 | 1.29 | 2.48 | 9.53 | 1.67 | 1.16 |
time (sec) |
N/A |
0.268 |
1.118 |
0.596 |
0.824 |
0.402 |
2.899 |
0.137 |
1.335 |
|
|
|
|
|
|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 170 | 170 | 207 | 955 | 191 | 359 | 1353 | 227 | 164 |
normalized size | 1 | 1.00 | 1.22 | 5.62 | 1.12 | 2.11 | 7.96 | 1.34 | 0.96 |
time (sec) |
N/A |
0.192 |
1.044 |
0.579 |
0.661 |
0.417 |
2.011 |
0.135 |
1.284 |
|
|
|
|
|
|
|
|
|
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|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 133 | 133 | 114 | 354 | 107 | 190 | 700 | 109 | 88 |
normalized size | 1 | 1.00 | 0.86 | 2.66 | 0.80 | 1.43 | 5.26 | 0.82 | 0.66 |
time (sec) |
N/A |
0.129 |
0.555 |
0.510 |
0.526 |
0.395 |
1.598 |
0.128 |
1.224 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 297 | 297 | 199 | 106 | 81 | 137 | 0 | 78 | -1 |
normalized size | 1 | 1.00 | 0.67 | 0.36 | 0.27 | 0.46 | 0.00 | 0.26 | -0.00 |
time (sec) |
N/A |
0.704 |
0.467 |
1.109 |
1.138 |
0.413 |
0.000 |
0.156 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 420 | 420 | 442 | 164 | 100 | 615 | 0 | 633 | -1 |
normalized size | 1 | 1.00 | 1.05 | 0.39 | 0.24 | 1.46 | 0.00 | 1.51 | -0.00 |
time (sec) |
N/A |
0.733 |
1.482 |
1.174 |
2.156 |
0.410 |
0.000 |
0.212 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 336 | 336 | 615 | 3691 | 406 | 844 | 3918 | 573 | 374 |
normalized size | 1 | 1.00 | 1.83 | 10.99 | 1.21 | 2.51 | 11.66 | 1.71 | 1.11 |
time (sec) |
N/A |
0.370 |
2.476 |
0.710 |
2.856 |
0.404 |
4.282 |
0.148 |
1.437 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 371 | 1767 | 254 | 532 | 2443 | 331 | 234 |
normalized size | 1 | 1.00 | 1.51 | 7.18 | 1.03 | 2.16 | 9.93 | 1.35 | 0.95 |
time (sec) |
N/A |
0.258 |
1.499 |
0.618 |
2.083 |
0.390 |
3.139 |
0.162 |
1.343 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 183 | 183 | 185 | 651 | 138 | 286 | 1287 | 151 | 127 |
normalized size | 1 | 1.00 | 1.01 | 3.56 | 0.75 | 1.56 | 7.03 | 0.83 | 0.69 |
time (sec) |
N/A |
0.205 |
0.769 |
0.632 |
1.164 |
0.387 |
2.397 |
0.141 |
1.256 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 437 | 437 | 312 | 151 | 114 | 195 | 0 | 108 | -1 |
normalized size | 1 | 1.00 | 0.71 | 0.35 | 0.26 | 0.45 | 0.00 | 0.25 | -0.00 |
time (sec) |
N/A |
1.765 |
0.705 |
1.637 |
4.113 |
0.399 |
0.000 |
0.162 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 692 | 692 | 796 | 239 | 140 | 1162 | 0 | 911 | -1 |
normalized size | 1 | 1.00 | 1.15 | 0.35 | 0.20 | 1.68 | 0.00 | 1.32 | -0.00 |
time (sec) |
N/A |
1.762 |
3.117 |
4.387 |
9.928 |
0.407 |
0.000 |
0.239 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.050 |
36.108 |
0.301 |
0.000 |
0.411 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.05 |
time (sec) |
N/A |
0.027 |
14.584 |
0.376 |
0.000 |
0.392 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 186 | 0 | 0 | 148 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.11 | 0.00 | 0.00 | 1.68 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.117 |
1.198 |
0.580 |
0.000 |
0.399 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 152 | 194 | 0 | 0 | 248 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.28 | 0.00 | 0.00 | 1.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.172 |
8.575 |
0.467 |
0.000 |
0.429 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F(-1) |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 223 | 223 | 0 | 0 | 0 | 345 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 0.00 | 0.00 | 1.55 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.233 |
180.002 |
0.549 |
0.000 |
0.414 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 133 | 133 | 249 | 748 | 419 | 488 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.87 | 5.62 | 3.15 | 3.67 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.260 |
0.389 |
0.446 |
0.478 |
0.434 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 101 | 101 | 149 | 447 | 240 | 303 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.48 | 4.43 | 2.38 | 3.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.217 |
0.237 |
0.349 |
0.487 |
0.432 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 87 | 201 | 0 | 156 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.16 | 2.68 | 0.00 | 2.08 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.126 |
0.056 |
0.310 |
0.000 |
0.412 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.05 |
time (sec) |
N/A |
0.029 |
5.317 |
0.580 |
0.000 |
0.394 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.05 |
time (sec) |
N/A |
0.029 |
21.373 |
0.633 |
0.000 |
0.425 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 271 | 271 | 1138 | 1393 | 781 | 3239 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.20 | 5.14 | 2.88 | 11.95 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.546 |
13.643 |
0.677 |
0.501 |
0.500 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 478 | 793 | 494 | 1854 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.29 | 3.79 | 2.36 | 8.87 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.404 |
10.334 |
0.591 |
0.477 |
0.453 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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|
|
|
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|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 192 | 318 | 244 | 851 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.51 | 2.50 | 1.92 | 6.70 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.184 |
1.982 |
0.519 |
0.496 |
0.440 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
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|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.055 |
45.487 |
1.062 |
0.000 |
0.409 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
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|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.053 |
32.086 |
1.292 |
0.000 |
0.426 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
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|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 556 | 556 | 2043 | 2777 | 1531 | 11137 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.67 | 4.99 | 2.75 | 20.03 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.049 |
14.520 |
0.877 |
0.534 |
0.664 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 401 | 401 | 2029 | 1542 | 997 | 6356 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 5.06 | 3.85 | 2.49 | 15.85 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.698 |
13.387 |
0.753 |
0.505 |
0.620 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 266 | 651 | 528 | 2907 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.02 | 2.49 | 2.02 | 11.14 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.350 |
3.768 |
0.645 |
0.468 |
0.536 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.055 |
65.838 |
1.396 |
0.000 |
0.430 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.053 |
60.837 |
1.777 |
0.000 |
0.457 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 210 | 210 | 239 | 1145 | 521 | 730 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.14 | 5.45 | 2.48 | 3.48 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.345 |
3.354 |
0.839 |
0.519 |
0.451 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 156 | 156 | 191 | 722 | 332 | 492 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.22 | 4.63 | 2.13 | 3.15 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.284 |
3.694 |
0.854 |
0.493 |
0.441 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 108 | 108 | 144 | 357 | 0 | 300 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.33 | 3.31 | 0.00 | 2.78 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.163 |
3.181 |
0.751 |
0.000 |
0.455 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.063 |
15.095 |
0.686 |
0.000 |
0.394 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.060 |
23.939 |
0.690 |
0.000 |
0.403 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 638 | 638 | 1022 | 2423 | 1056 | 6171 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.60 | 3.80 | 1.66 | 9.67 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.274 |
12.710 |
1.066 |
1.035 |
0.571 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 475 | 475 | 509 | 1449 | 751 | 3702 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.07 | 3.05 | 1.58 | 7.79 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.714 |
8.706 |
1.007 |
0.838 |
0.502 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 196 | 196 | 477 | 524 | 0 | 1797 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.43 | 2.67 | 0.00 | 9.17 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.296 |
6.234 |
0.951 |
0.000 |
0.460 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.062 |
155.010 |
1.833 |
0.000 |
0.405 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.059 |
125.762 |
1.760 |
0.000 |
0.415 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
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
[.8333]
|
|
|
|
|
|
|
| Table 2.1:Rubi 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 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
2 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
3 |
A |
4 |
4 |
1.00 |
8 |
0.500 |
|
|
|
|
|
|
|
4 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
5 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
6 |
A |
7 |
7 |
1.00 |
12 |
0.583 |
|
|
|
|
|
|
|
7 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
8 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
9 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
10 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
11 |
A |
13 |
10 |
1.00 |
12 |
0.833 |
|
|
|
|
|
|
|
12 |
A |
9 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
13 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
15 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
16 |
A |
5 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
17 |
A |
4 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
18 |
A |
3 |
3 |
1.00 |
18 |
0.167 |
|
|
|
|
|
|
|
19 |
A |
7 |
4 |
1.00 |
20 |
0.200 |
|
|
|
|
|
|
|
20 |
A |
7 |
4 |
1.00 |
20 |
0.200 |
|
|
|
|
|
|
|
21 |
A |
8 |
5 |
1.00 |
20 |
0.250 |
|
|
|
|
|
|
|
22 |
A |
10 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
23 |
A |
8 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
24 |
A |
7 |
3 |
1.00 |
18 |
0.167 |
|
|
|
|
|
|
|
25 |
A |
21 |
5 |
1.00 |
20 |
0.250 |
|
|
|
|
|
|
|
26 |
A |
24 |
7 |
1.00 |
20 |
0.350 |
|
|
|
|
|
|
|
27 |
A |
14 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
28 |
A |
11 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
29 |
A |
11 |
3 |
1.00 |
18 |
0.167 |
|
|
|
|
|
|
|
30 |
A |
53 |
7 |
1.00 |
20 |
0.350 |
|
|
|
|
|
|
|
31 |
A |
60 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
32 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
33 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
34 |
A |
2 |
2 |
1.00 |
20 |
0.100 |
|
|
|
|
|
|
|
35 |
A |
4 |
2 |
1.00 |
20 |
0.100 |
|
|
|
|
|
|
|
36 |
A |
5 |
2 |
1.00 |
20 |
0.100 |
|
|
|
|
|
|
|
37 |
A |
8 |
7 |
1.00 |
18 |
0.389 |
|
|
|
|
|
|
|
38 |
A |
7 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
39 |
A |
6 |
5 |
1.00 |
16 |
0.312 |
|
|
|
|
|
|
|
40 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
41 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
42 |
A |
15 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
43 |
A |
13 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
44 |
A |
9 |
7 |
1.00 |
18 |
0.389 |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
47 |
A |
28 |
11 |
1.00 |
20 |
0.550 |
|
|
|
|
|
|
|
48 |
A |
22 |
11 |
1.00 |
20 |
0.550 |
|
|
|
|
|
|
|
49 |
A |
16 |
9 |
1.00 |
18 |
0.500 |
|
|
|
|
|
|
|
50 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
51 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
52 |
A |
6 |
6 |
1.00 |
20 |
0.300 |
|
|
|
|
|
|
|
53 |
A |
5 |
5 |
1.00 |
20 |
0.250 |
|
|
|
|
|
|
|
54 |
A |
4 |
4 |
1.00 |
18 |
0.222 |
|
|
|
|
|
|
|
55 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
56 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
57 |
A |
28 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
58 |
A |
24 |
11 |
1.00 |
20 |
0.550 |
|
|
|
|
|
|
|
59 |
A |
5 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
60 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
61 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|