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
}
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, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51
}
B grade: { }
C grade: { }
F grade: { }
2.1.3 Maple
A grade: { 4, 50, 51 }
B grade: { 1, 2, 3, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 8, 9, 10, 11, 16, 17, 18, 50, 51 }
B grade: { 15 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.5 FriCAS
A grade: { 4, 50, 51 }
B grade: { 1, 2, 3, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.6 Sympy
A grade: { 1, 2, 3, 4, 8, 9, 10, 11, 16, 17, 18, 50, 51 }
B grade: { }
C grade: { 5, 6, 12, 19, 22, 23, 24, 25, 26, 32, 33, 40 }
F grade: { 7, 13, 14, 15, 20, 21, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49
}
2.1.7 Giac
A grade: { 50, 51 }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 18, 50, 51 }
C grade: { }
F grade: { 5, 6, 7, 12, 13, 14, 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 }
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 |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 189 | 189 | 151 | 1229 | 338 | 911 | 6156 | 1708 | 469 |
normalized size | 1 | 1.00 | 0.80 | 6.50 | 1.79 | 4.82 | 32.57 | 9.04 | 2.48 |
time (sec) |
N/A |
0.227 |
0.323 |
0.009 |
1.878 |
1.670 |
11.007 |
0.570 |
1.438 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 144 | 144 | 113 | 711 | 242 | 532 | 3373 | 1009 | 305 |
normalized size | 1 | 1.00 | 0.78 | 4.94 | 1.68 | 3.69 | 23.42 | 7.01 | 2.12 |
time (sec) |
N/A |
0.121 |
0.167 |
0.006 |
1.814 |
1.229 |
7.627 |
0.561 |
1.197 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 97 | 97 | 73 | 321 | 146 | 235 | 1515 | 478 | 185 |
normalized size | 1 | 1.00 | 0.75 | 3.31 | 1.51 | 2.42 | 15.62 | 4.93 | 1.91 |
time (sec) |
N/A |
0.061 |
0.078 |
0.005 |
1.627 |
0.978 |
2.600 |
0.444 |
1.028 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 43 | 111 | 70 | 94 | 459 | 167 | 97 |
normalized size | 1 | 1.00 | 0.72 | 1.85 | 1.17 | 1.57 | 7.65 | 2.78 | 1.62 |
time (sec) |
N/A |
0.031 |
0.043 |
0.004 |
1.448 |
1.040 |
1.935 |
0.499 |
0.956 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 118 | 118 | 93 | 0 | 0 | 0 | 428 | 0 | -1 |
normalized size | 1 | 1.00 | 0.79 | 0.00 | 0.00 | 0.00 | 3.63 | 0.00 | -0.01 |
time (sec) |
N/A |
0.098 |
0.105 |
0.063 |
0.000 |
1.133 |
11.545 |
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 |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 108 | 0 | 0 | 0 | 2076 | 0 | -1 |
normalized size | 1 | 1.00 | 0.63 | 0.00 | 0.00 | 0.00 | 12.14 | 0.00 | -0.01 |
time (sec) |
N/A |
0.241 |
0.156 |
0.065 |
0.000 |
1.179 |
94.482 |
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 |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 133 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.296 |
0.165 |
0.068 |
0.000 |
0.844 |
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 |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 247 | 2443 | 550 | 1711 | 12199 | 3283 | 694 |
normalized size | 1 | 1.00 | 0.85 | 8.37 | 1.88 | 5.86 | 41.78 | 11.24 | 2.38 |
time (sec) |
N/A |
0.287 |
0.587 |
0.010 |
2.268 |
1.176 |
13.413 |
0.814 |
1.741 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 216 | 216 | 178 | 1471 | 396 | 1043 | 7019 | 2010 | 499 |
normalized size | 1 | 1.00 | 0.82 | 6.81 | 1.83 | 4.83 | 32.50 | 9.31 | 2.31 |
time (sec) |
N/A |
0.247 |
0.354 |
0.009 |
2.060 |
0.959 |
10.021 |
0.677 |
1.388 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 144 | 144 | 113 | 711 | 242 | 495 | 3373 | 1009 | 305 |
normalized size | 1 | 1.00 | 0.78 | 4.94 | 1.68 | 3.44 | 23.42 | 7.01 | 2.12 |
time (sec) |
N/A |
0.134 |
0.173 |
0.008 |
1.801 |
0.728 |
5.664 |
0.638 |
1.182 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
B |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 67 | 263 | 116 | 217 | 1137 | 380 | 179 |
normalized size | 1 | 1.00 | 0.74 | 2.89 | 1.27 | 2.38 | 12.49 | 4.18 | 1.97 |
time (sec) |
N/A |
0.069 |
0.054 |
0.007 |
1.510 |
1.107 |
3.167 |
0.454 |
1.048 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 178 | 178 | 146 | 0 | 0 | 0 | 666 | 0 | -1 |
normalized size | 1 | 1.00 | 0.82 | 0.00 | 0.00 | 0.00 | 3.74 | 0.00 | -0.01 |
time (sec) |
N/A |
0.189 |
0.264 |
0.066 |
0.000 |
1.130 |
20.593 |
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 |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 247 | 247 | 156 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.442 |
0.230 |
0.069 |
0.000 |
1.040 |
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 |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 165 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.57 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.408 |
0.209 |
0.077 |
0.000 |
0.966 |
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 |
A |
A |
A |
B |
B |
B |
F(-1) |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 379 | 379 | 327 | 3953 | 762 | 2657 | 0 | 5234 | 933 |
normalized size | 1 | 1.00 | 0.86 | 10.43 | 2.01 | 7.01 | 0.00 | 13.81 | 2.46 |
time (sec) |
N/A |
0.401 |
0.825 |
0.010 |
2.493 |
0.907 |
0.000 |
1.049 |
2.099 |
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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 |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 284 | 284 | 239 | 2443 | 550 | 1690 | 12199 | 3283 | 694 |
normalized size | 1 | 1.00 | 0.84 | 8.60 | 1.94 | 5.95 | 42.95 | 11.56 | 2.44 |
time (sec) |
N/A |
0.284 |
0.415 |
0.010 |
2.317 |
1.219 |
13.849 |
0.946 |
1.745 |
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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 |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 189 | 189 | 151 | 1229 | 338 | 837 | 6156 | 1708 | 469 |
normalized size | 1 | 1.00 | 0.80 | 6.50 | 1.79 | 4.43 | 32.57 | 9.04 | 2.48 |
time (sec) |
N/A |
0.176 |
0.279 |
0.009 |
1.964 |
1.028 |
8.211 |
0.655 |
1.432 |
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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 |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 121 | 121 | 90 | 475 | 162 | 381 | 2220 | 673 | 280 |
normalized size | 1 | 1.00 | 0.74 | 3.93 | 1.34 | 3.15 | 18.35 | 5.56 | 2.31 |
time (sec) |
N/A |
0.079 |
0.079 |
0.006 |
1.637 |
0.871 |
8.511 |
0.507 |
1.133 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 258 | 258 | 217 | 0 | 0 | 0 | 911 | 0 | -1 |
normalized size | 1 | 1.00 | 0.84 | 0.00 | 0.00 | 0.00 | 3.53 | 0.00 | -0.00 |
time (sec) |
N/A |
0.282 |
0.492 |
0.069 |
0.000 |
0.983 |
29.054 |
0.000 |
0.000 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 347 | 347 | 209 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.661 |
0.434 |
0.063 |
0.000 |
1.002 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 480 | 480 | 218 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.45 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.073 |
0.391 |
0.074 |
0.000 |
0.931 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 363 | 363 | 315 | 0 | 0 | 0 | 1132 | 0 | -1 |
normalized size | 1 | 1.00 | 0.87 | 0.00 | 0.00 | 0.00 | 3.12 | 0.00 | -0.00 |
time (sec) |
N/A |
0.371 |
0.518 |
0.065 |
0.000 |
0.697 |
61.959 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 260 | 260 | 219 | 0 | 0 | 0 | 911 | 0 | -1 |
normalized size | 1 | 1.00 | 0.84 | 0.00 | 0.00 | 0.00 | 3.50 | 0.00 | -0.00 |
time (sec) |
N/A |
0.295 |
0.348 |
0.066 |
0.000 |
0.709 |
31.105 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 180 | 180 | 147 | 0 | 0 | 0 | 666 | 0 | -1 |
normalized size | 1 | 1.00 | 0.82 | 0.00 | 0.00 | 0.00 | 3.70 | 0.00 | -0.01 |
time (sec) |
N/A |
0.188 |
0.218 |
0.069 |
0.000 |
1.209 |
21.309 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 120 | 120 | 93 | 0 | 0 | 0 | 428 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 0.00 | 0.00 | 0.00 | 3.57 | 0.00 | -0.01 |
time (sec) |
N/A |
0.096 |
0.102 |
0.066 |
0.000 |
0.953 |
13.661 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 77 | 77 | 56 | 0 | 0 | 0 | 204 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 0.00 | 0.00 | 0.00 | 2.65 | 0.00 | -0.01 |
time (sec) |
N/A |
0.038 |
0.065 |
0.065 |
0.000 |
0.607 |
6.863 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 125 | 125 | 100 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.80 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.137 |
0.136 |
0.068 |
0.000 |
0.685 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 206 | 206 | 149 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.385 |
0.204 |
0.079 |
0.000 |
0.908 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 342 | 342 | 195 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.57 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.775 |
0.275 |
0.084 |
0.000 |
0.931 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 340 | 340 | 212 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.716 |
0.476 |
0.066 |
0.000 |
1.068 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 246 | 246 | 158 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.405 |
0.310 |
0.084 |
0.000 |
1.090 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 108 | 0 | 0 | 0 | 2076 | 0 | -1 |
normalized size | 1 | 1.00 | 0.63 | 0.00 | 0.00 | 0.00 | 12.14 | 0.00 | -0.01 |
time (sec) |
N/A |
0.227 |
0.175 |
0.063 |
0.000 |
0.943 |
83.999 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 81 | 0 | 0 | 0 | 954 | 0 | -1 |
normalized size | 1 | 1.00 | 0.79 | 0.00 | 0.00 | 0.00 | 9.26 | 0.00 | -0.01 |
time (sec) |
N/A |
0.047 |
0.082 |
0.066 |
0.000 |
0.820 |
42.975 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 205 | 205 | 147 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.72 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.383 |
0.171 |
0.072 |
0.000 |
1.065 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 304 | 304 | 207 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.802 |
0.268 |
0.069 |
0.000 |
1.399 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 491 | 491 | 265 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.54 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.425 |
0.362 |
0.081 |
0.000 |
1.348 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 433 | 433 | 222 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.51 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.138 |
0.351 |
0.070 |
0.000 |
0.955 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 292 | 292 | 169 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.58 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.407 |
0.201 |
0.081 |
0.000 |
0.935 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 208 | 208 | 133 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.298 |
0.142 |
0.074 |
0.000 |
0.946 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 103 | 103 | 81 | 0 | 0 | 0 | 3172 | 0 | -1 |
normalized size | 1 | 1.00 | 0.79 | 0.00 | 0.00 | 0.00 | 30.80 | 0.00 | -0.01 |
time (sec) |
N/A |
0.047 |
0.063 |
0.066 |
0.000 |
0.939 |
141.881 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 333 | 333 | 197 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.719 |
0.230 |
0.073 |
0.000 |
0.881 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 452 | 452 | 266 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.338 |
0.355 |
0.079 |
0.000 |
0.963 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
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Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 665 | 665 | 329 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.179 |
0.492 |
0.081 |
0.000 |
0.807 |
0.000 |
0.000 |
0.000 |
|
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1059 | 1047 | 248 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.99 | 0.23 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.512 |
0.469 |
0.089 |
0.000 |
1.020 |
0.000 |
0.000 |
0.000 |
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 495 | 464 | 198 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.94 | 0.40 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.749 |
0.268 |
0.070 |
0.000 |
0.854 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 253 | 238 | 147 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.94 | 0.58 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.227 |
0.131 |
0.072 |
0.000 |
0.947 |
0.000 |
0.000 |
0.000 |
|
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 162 | 162 | 118 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.159 |
0.209 |
0.071 |
0.000 |
0.622 |
0.000 |
0.000 |
0.000 |
|
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 295 | 295 | 128 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.43 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.413 |
0.245 |
0.071 |
0.000 |
0.975 |
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 |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 483 | 483 | 128 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.27 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.039 |
0.509 |
0.080 |
0.000 |
1.001 |
0.000 |
0.000 |
0.000 |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 91 | 112 | 100 | 231 | 97 | 113 | 101 |
normalized size | 1 | 1.00 | 1.08 | 1.33 | 1.19 | 2.75 | 1.15 | 1.35 | 1.20 |
time (sec) |
N/A |
0.077 |
0.140 |
0.010 |
1.377 |
0.800 |
59.288 |
0.432 |
1.510 |
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 91 | 112 | 100 | 217 | 97 | 113 | 101 |
normalized size | 1 | 1.00 | 1.08 | 1.33 | 1.19 | 2.58 | 1.15 | 1.35 | 1.20 |
time (sec) |
N/A |
0.077 |
0.160 |
0.011 |
1.366 |
0.661 |
54.080 |
0.569 |
1.483 |
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|
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 [48] had the largest ratio of
[.1935]
|
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|
| Table 2.1:Rubi specific breakdown of results for each
integral
|
|
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|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
2 |
1 |
1.00 |
29 |
0.034 |
|
|
|
|
|
|
|
2 |
A |
2 |
1 |
1.00 |
29 |
0.034 |
|
|
|
|
|
|
|
3 |
A |
2 |
1 |
1.00 |
27 |
0.037 |
|
|
|
|
|
|
|
4 |
A |
2 |
1 |
1.00 |
20 |
0.050 |
|
|
|
|
|
|
|
5 |
A |
3 |
2 |
1.00 |
29 |
0.069 |
|
|
|
|
|
|
|
6 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
|
|
|
|
|
|
|
7 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
|
|
|
|
|
|
|
8 |
A |
2 |
1 |
1.00 |
31 |
0.032 |
|
|
|
|
|
|
|
9 |
A |
2 |
1 |
1.00 |
31 |
0.032 |
|
|
|
|
|
|
|
10 |
A |
2 |
1 |
1.00 |
29 |
0.034 |
|
|
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|
|
|
|
11 |
A |
2 |
1 |
1.00 |
22 |
0.045 |
|
|
|
|
|
|
|
12 |
A |
3 |
2 |
1.00 |
31 |
0.065 |
|
|
|
|
|
|
|
13 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
14 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
15 |
A |
2 |
1 |
1.00 |
31 |
0.032 |
|
|
|
|
|
|
|
16 |
A |
2 |
1 |
1.00 |
31 |
0.032 |
|
|
|
|
|
|
|
17 |
A |
2 |
1 |
1.00 |
29 |
0.034 |
|
|
|
|
|
|
|
18 |
A |
2 |
1 |
1.00 |
22 |
0.045 |
|
|
|
|
|
|
|
19 |
A |
3 |
2 |
1.00 |
31 |
0.065 |
|
|
|
|
|
|
|
20 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
21 |
A |
5 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
22 |
A |
3 |
2 |
1.00 |
31 |
0.065 |
|
|
|
|
|
|
|
23 |
A |
3 |
2 |
1.00 |
31 |
0.065 |
|
|
|
|
|
|
|
24 |
A |
3 |
2 |
1.00 |
31 |
0.065 |
|
|
|
|
|
|
|
25 |
A |
3 |
2 |
1.00 |
29 |
0.069 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1.00 |
22 |
0.091 |
|
|
|
|
|
|
|
27 |
A |
4 |
2 |
1.00 |
31 |
0.065 |
|
|
|
|
|
|
|
28 |
A |
5 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
29 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
30 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
31 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
32 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
|
|
|
|
|
|
|
33 |
A |
2 |
2 |
1.00 |
22 |
0.091 |
|
|
|
|
|
|
|
34 |
A |
5 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
35 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
36 |
A |
7 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
37 |
A |
5 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
38 |
A |
4 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
39 |
A |
3 |
3 |
1.00 |
29 |
0.103 |
|
|
|
|
|
|
|
40 |
A |
2 |
2 |
1.00 |
22 |
0.091 |
|
|
|
|
|
|
|
41 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
42 |
A |
7 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
43 |
A |
8 |
3 |
1.00 |
31 |
0.097 |
|
|
|
|
|
|
|
44 |
A |
6 |
4 |
0.99 |
31 |
0.129 |
|
|
|
|
|
|
|
45 |
A |
5 |
4 |
0.94 |
31 |
0.129 |
|
|
|
|
|
|
|
46 |
A |
4 |
4 |
0.94 |
29 |
0.138 |
|
|
|
|
|
|
|
47 |
A |
6 |
5 |
1.00 |
31 |
0.161 |
|
|
|
|
|
|
|
48 |
A |
7 |
6 |
1.00 |
31 |
0.194 |
|
|
|
|
|
|
|
49 |
A |
8 |
6 |
1.00 |
31 |
0.194 |
|
|
|
|
|
|
|
50 |
A |
5 |
5 |
1.00 |
29 |
0.172 |
|
|
|
|
|
|
|
51 |
A |
5 |
5 |
1.00 |
29 |
0.172 |
|
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|