Chapter 2
detailed summary tables of results
2.1 List of integrals sorted by grade for each CAS
2.1.1 Rubi
A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 39, 40, 41, 42, 43, 44, 46,
47, 49, 60, 61, 62, 63, 64 }
B grade: { 8, 9, 12, 22, 30, 34, 38, 45, 50 }
C grade: { 2, 23, 24, 25, 31, 32, 33, 35, 36, 37, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59 }
F grade: { }
2.1.3 Maple
A grade: { 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 49, 50, 51, 54, 56, 59, 60, 62 }
B grade: { 6, 7, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 48, 53, 55, 58, 61, 63, 64 }
C grade: { 1, 21, 29, 47, 52 }
F grade: { 57 }
2.1.4 Maxima
A grade: { 1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 16, 17, 18, 59 }
B grade: { 8, 9, 11, 40, 43 }
C grade: { }
F grade: { 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64 }
2.1.5 FriCAS
A grade: { 5, 14, 18, 21, 24, 25, 29, 30, 47, 48, 59 }
B grade: { 1, 2, 3, 4, 6, 7, 8, 9, 10, 15, 16, 17, 19, 20, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64 }
C grade: { 11, 12, 13 }
F grade: { }
2.1.6 Sympy
A grade: { 2, 3, 4, 5, 6, 7, 10, 13, 16, 51, 56, 59 }
B grade: { }
C grade: { }
F grade: { 1, 8, 9, 11, 12, 14, 15, 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, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 64
}
2.1.7 Giac
A grade: { 2, 5, 6, 7, 8, 14, 16, 17, 59, 62, 63 }
B grade: { 3, 4, 20, 21, 23, 24, 25, 34, 35, 36, 37, 38, 41, 43, 44, 46, 47, 50, 55, 58, 60, 61, 64
}
C grade: { 39, 40 }
F grade: { 1, 9, 10, 11, 12, 13, 15, 18, 19, 22, 26, 27, 28, 29, 30, 31, 32, 33, 42, 45, 48, 49, 51, 52, 53, 54, 56,
57 }
2.1.8 Mupad
A grade: { }
B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 26, 28, 29, 34, 38, 39, 40, 42, 43,
44, 46, 47, 49, 51, 52, 54, 56, 57, 59 }
C grade: { }
F grade: { 15, 22, 23, 24, 25, 27, 30, 31, 32, 33, 35, 36, 37, 41, 45, 48, 50, 53, 55, 58, 60, 61, 62, 63, 64
}
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 \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\)
|
|
|
|
|
|
|
|
|
|
Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
B |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 233 | 233 | 148 | 2494 | 179 | 1236 | 0 | 0 | 828 |
normalized size | 1 | 1.00 | 0.64 | 10.70 | 0.77 | 5.30 | 0.00 | 0.00 | 3.55 |
time (sec) |
N/A |
0.299 |
0.813 |
2.507 |
1.973 |
1.149 |
0.000 |
0.000 |
0.925 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 34 | 31 | 23 | 48 | 22 | 40 | 20 |
normalized size | 1 | 1.00 | 1.70 | 1.55 | 1.15 | 2.40 | 1.10 | 2.00 | 1.00 |
time (sec) |
N/A |
0.013 |
0.023 |
0.035 |
1.144 |
0.837 |
0.126 |
0.192 |
0.336 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 71 | 68 | 63 | 127 | 68 | 114 | 45 |
normalized size | 1 | 1.00 | 1.51 | 1.45 | 1.34 | 2.70 | 1.45 | 2.43 | 0.96 |
time (sec) |
N/A |
0.033 |
1.171 |
0.033 |
0.756 |
0.817 |
0.263 |
0.254 |
0.118 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 111 | 116 | 112 | 253 | 126 | 229 | 76 |
normalized size | 1 | 1.00 | 1.42 | 1.49 | 1.44 | 3.24 | 1.62 | 2.94 | 0.97 |
time (sec) |
N/A |
0.047 |
2.769 |
0.036 |
0.928 |
0.489 |
0.552 |
0.371 |
0.454 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 5 |
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 | 49 | 49 | 49 | 64 | 48 | 252 | 279 | 65 | 41 |
normalized size | 1 | 1.00 | 1.00 | 1.31 | 0.98 | 5.14 | 5.69 | 1.33 | 0.84 |
time (sec) |
N/A |
0.073 |
0.057 |
0.326 |
0.885 |
0.461 |
1.461 |
0.187 |
0.124 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 6 |
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 | 97 | 97 | 90 | 173 | 115 | 534 | 2322 | 123 | 119 |
normalized size | 1 | 1.00 | 0.93 | 1.78 | 1.19 | 5.51 | 23.94 | 1.27 | 1.23 |
time (sec) |
N/A |
0.100 |
0.907 |
0.345 |
0.986 |
0.511 |
18.675 |
0.267 |
0.786 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 7 |
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 | 150 | 150 | 138 | 363 | 228 | 1068 | 9629 | 206 | 4866 |
normalized size | 1 | 1.00 | 0.92 | 2.42 | 1.52 | 7.12 | 64.19 | 1.37 | 32.44 |
time (sec) |
N/A |
0.157 |
0.303 |
0.352 |
1.642 |
0.555 |
93.929 |
0.713 |
3.287 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
B |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 51 | 19 | 300 | 91 | 0 | 32 | 18 |
normalized size | 1 | 1.00 | 2.32 | 0.86 | 13.64 | 4.14 | 0.00 | 1.45 | 0.82 |
time (sec) |
N/A |
0.018 |
0.100 |
0.274 |
1.101 |
0.489 |
0.000 |
0.183 |
0.375 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
B |
B |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 5 | 5 | 28 | 6 | 35 | 53 | 0 | 0 | 5 |
normalized size | 1 | 1.00 | 5.60 | 1.20 | 7.00 | 10.60 | 0.00 | 0.00 | 1.00 |
time (sec) |
N/A |
0.014 |
0.013 |
0.229 |
1.003 |
0.518 |
0.000 |
0.000 |
0.318 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 12 | 13 | 10 | 21 | 14 | 0 | 12 |
normalized size | 1 | 1.00 | 1.00 | 1.08 | 0.83 | 1.75 | 1.17 | 0.00 | 1.00 |
time (sec) |
N/A |
0.021 |
0.010 |
0.179 |
0.831 |
0.426 |
0.342 |
0.000 |
0.392 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
C |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 48 | 32 | 284 | 73 | 0 | 0 | 31 |
normalized size | 1 | 1.00 | 1.37 | 0.91 | 8.11 | 2.09 | 0.00 | 0.00 | 0.89 |
time (sec) |
N/A |
0.026 |
0.083 |
0.125 |
1.008 |
0.495 |
0.000 |
0.000 |
0.368 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
C |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 30 | 15 | 17 | 19 | 0 | 0 | 14 |
normalized size | 1 | 1.00 | 2.14 | 1.07 | 1.21 | 1.36 | 0.00 | 0.00 | 1.00 |
time (sec) |
N/A |
0.020 |
0.017 |
0.128 |
1.031 |
0.574 |
0.000 |
0.000 |
0.394 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
C |
A |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 15 | 12 | 14 | 15 | 0 | 13 |
normalized size | 1 | 1.00 | 1.00 | 1.07 | 0.86 | 1.00 | 1.07 | 0.00 | 0.93 |
time (sec) |
N/A |
0.023 |
0.004 |
0.103 |
0.499 |
0.841 |
0.338 |
0.000 |
0.682 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 19 | 29 | 24 | 27 | 0 | 30 | 17 |
normalized size | 1 | 1.00 | 0.68 | 1.04 | 0.86 | 0.96 | 0.00 | 1.07 | 0.61 |
time (sec) |
N/A |
0.096 |
0.024 |
0.296 |
0.745 |
0.430 |
0.000 |
0.180 |
0.586 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 32 | 38 | 27 | 77 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.03 | 1.23 | 0.87 | 2.48 | 0.00 | 0.00 | -0.03 |
time (sec) |
N/A |
0.100 |
0.033 |
0.204 |
0.770 |
0.749 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 10 | 10 | 10 | 11 | 8 | 27 | 12 | 12 | 10 |
normalized size | 1 | 1.00 | 1.00 | 1.10 | 0.80 | 2.70 | 1.20 | 1.20 | 1.00 |
time (sec) |
N/A |
0.048 |
0.011 |
0.186 |
0.547 |
0.522 |
1.196 |
0.203 |
0.476 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 49 | 56 | 52 | 78 | 0 | 42 | 20 |
normalized size | 1 | 1.00 | 1.36 | 1.56 | 1.44 | 2.17 | 0.00 | 1.17 | 0.56 |
time (sec) |
N/A |
0.087 |
0.038 |
0.744 |
0.653 |
0.443 |
0.000 |
0.192 |
0.423 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 19 | 33 | 18 | 35 | 0 | 0 | 34 |
normalized size | 1 | 1.00 | 0.66 | 1.14 | 0.62 | 1.21 | 0.00 | 0.00 | 1.17 |
time (sec) |
N/A |
0.101 |
0.033 |
0.573 |
0.520 |
0.413 |
0.000 |
0.000 |
0.727 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 66 | 66 | 65 | 84 | 0 | 330 | 0 | 0 | 66 |
normalized size | 1 | 1.00 | 0.98 | 1.27 | 0.00 | 5.00 | 0.00 | 0.00 | 1.00 |
time (sec) |
N/A |
0.116 |
0.184 |
0.192 |
0.000 |
0.724 |
0.000 |
0.000 |
3.157 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 48 | 71 | 0 | 248 | 0 | 95 | 53 |
normalized size | 1 | 1.00 | 1.00 | 1.48 | 0.00 | 5.17 | 0.00 | 1.98 | 1.10 |
time (sec) |
N/A |
0.067 |
0.027 |
0.128 |
0.000 |
1.511 |
0.000 |
0.801 |
1.170 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 60 | 591 | 0 | 351 | 0 | 187 | 69 |
normalized size | 1 | 1.00 | 1.00 | 9.85 | 0.00 | 5.85 | 0.00 | 3.12 | 1.15 |
time (sec) |
N/A |
0.097 |
0.027 |
1.074 |
0.000 |
0.502 |
0.000 |
0.736 |
0.477 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 2105 | 174 | 0 | 768 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 23.65 | 1.96 | 0.00 | 8.63 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.125 |
22.398 |
0.184 |
0.000 |
0.494 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 65 | 65 | 167 | 137 | 0 | 515 | 0 | 210 | -1 |
normalized size | 1 | 1.00 | 2.57 | 2.11 | 0.00 | 7.92 | 0.00 | 3.23 | -0.02 |
time (sec) |
N/A |
0.045 |
0.408 |
0.230 |
0.000 |
0.505 |
0.000 |
2.642 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 44 | 752 | 0 | 193 | 0 | 239 | -1 |
normalized size | 1 | 1.00 | 0.86 | 14.75 | 0.00 | 3.78 | 0.00 | 4.69 | -0.02 |
time (sec) |
N/A |
0.089 |
0.097 |
0.988 |
0.000 |
0.577 |
0.000 |
0.495 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 174 | 951 | 0 | 239 | 0 | 476 | -1 |
normalized size | 1 | 1.00 | 2.05 | 11.19 | 0.00 | 2.81 | 0.00 | 5.60 | -0.01 |
time (sec) |
N/A |
0.141 |
1.647 |
0.770 |
0.000 |
0.586 |
0.000 |
0.251 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 88 | 88 | 91 | 150 | 0 | 486 | 0 | 0 | 120 |
normalized size | 1 | 1.00 | 1.03 | 1.70 | 0.00 | 5.52 | 0.00 | 0.00 | 1.36 |
time (sec) |
N/A |
0.137 |
0.503 |
0.132 |
0.000 |
0.607 |
0.000 |
0.000 |
11.132 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 127 | 127 | 253 | 286 | 0 | 1134 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.99 | 2.25 | 0.00 | 8.93 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.230 |
1.194 |
0.129 |
0.000 |
0.681 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 63 | 136 | 0 | 330 | 0 | 0 | 70 |
normalized size | 1 | 1.00 | 0.91 | 1.97 | 0.00 | 4.78 | 0.00 | 0.00 | 1.01 |
time (sec) |
N/A |
0.091 |
0.174 |
0.093 |
0.000 |
0.675 |
0.000 |
0.000 |
3.535 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
A |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 75 | 2628 | 0 | 565 | 0 | 0 | 506 |
normalized size | 1 | 1.00 | 1.00 | 35.04 | 0.00 | 7.53 | 0.00 | 0.00 | 6.75 |
time (sec) |
N/A |
0.128 |
0.073 |
0.759 |
0.000 |
1.779 |
0.000 |
0.000 |
0.540 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 80 | 80 | 222 | 1276 | 0 | 543 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.78 | 15.95 | 0.00 | 6.79 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.125 |
0.741 |
0.691 |
0.000 |
1.433 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 171 | 171 | 259 | 462 | 0 | 1520 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.51 | 2.70 | 0.00 | 8.89 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.186 |
1.690 |
0.455 |
0.000 |
0.527 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 234 | 298 | 0 | 1071 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.86 | 2.37 | 0.00 | 8.50 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.098 |
1.376 |
0.379 |
0.000 |
0.488 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 87 | 87 | 202 | 170 | 0 | 703 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.32 | 1.95 | 0.00 | 8.08 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.055 |
0.611 |
0.420 |
0.000 |
0.938 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-2) |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 111 | 68 | 0 | 239 | 0 | 88 | 41 |
normalized size | 1 | 1.00 | 2.36 | 1.45 | 0.00 | 5.09 | 0.00 | 1.87 | 0.87 |
time (sec) |
N/A |
0.033 |
0.413 |
0.394 |
0.000 |
0.479 |
0.000 |
4.941 |
0.854 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
B |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 231 | 104 | 0 | 526 | 0 | 348 | -1 |
normalized size | 1 | 1.00 | 2.72 | 1.22 | 0.00 | 6.19 | 0.00 | 4.09 | -0.01 |
time (sec) |
N/A |
0.061 |
3.635 |
0.355 |
0.000 |
0.624 |
0.000 |
10.137 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
B |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 135 | 135 | 367 | 176 | 0 | 898 | 0 | 1341 | -1 |
normalized size | 1 | 1.00 | 2.72 | 1.30 | 0.00 | 6.65 | 0.00 | 9.93 | -0.01 |
time (sec) |
N/A |
0.110 |
7.938 |
0.382 |
0.000 |
0.678 |
0.000 |
13.770 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
B |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 190 | 190 | 2553 | 284 | 0 | 1452 | 0 | 3719 | -1 |
normalized size | 1 | 1.00 | 13.44 | 1.49 | 0.00 | 7.64 | 0.00 | 19.57 | -0.01 |
time (sec) |
N/A |
0.194 |
14.676 |
0.381 |
0.000 |
0.629 |
0.000 |
31.532 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 123 | 51 | 0 | 110 | 0 | 257 | 104 |
normalized size | 1 | 1.00 | 2.28 | 0.94 | 0.00 | 2.04 | 0.00 | 4.76 | 1.93 |
time (sec) |
N/A |
0.046 |
0.392 |
0.230 |
0.000 |
1.083 |
0.000 |
0.283 |
0.837 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 62 | 34 | 0 | 68 | 0 | 170 | 88 |
normalized size | 1 | 1.00 | 1.94 | 1.06 | 0.00 | 2.12 | 0.00 | 5.31 | 2.75 |
time (sec) |
N/A |
0.027 |
0.070 |
0.234 |
0.000 |
0.408 |
0.000 |
0.262 |
0.963 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
C |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 42 | 31 | 90 | 56 | 0 | 34 | 85 |
normalized size | 1 | 1.00 | 1.50 | 1.11 | 3.21 | 2.00 | 0.00 | 1.21 | 3.04 |
time (sec) |
N/A |
0.019 |
0.063 |
0.258 |
0.506 |
0.441 |
0.000 |
0.210 |
0.626 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 61 | 61 | 121 | 48 | 0 | 170 | 0 | 179 | -1 |
normalized size | 1 | 1.00 | 1.98 | 0.79 | 0.00 | 2.79 | 0.00 | 2.93 | -0.02 |
time (sec) |
N/A |
0.043 |
0.123 |
0.216 |
0.000 |
0.524 |
0.000 |
0.665 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
F(-1) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 42 | 42 | 60 | 35 | 0 | 123 | 0 | 0 | 34 |
normalized size | 1 | 1.00 | 1.43 | 0.83 | 0.00 | 2.93 | 0.00 | 0.00 | 0.81 |
time (sec) |
N/A |
0.025 |
0.043 |
0.234 |
0.000 |
1.558 |
0.000 |
0.000 |
0.432 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 45 | 21 | 143 | 60 | 0 | 45 | 20 |
normalized size | 1 | 1.00 | 1.73 | 0.81 | 5.50 | 2.31 | 0.00 | 1.73 | 0.77 |
time (sec) |
N/A |
0.017 |
0.034 |
0.247 |
0.499 |
0.524 |
0.000 |
4.068 |
0.502 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 52 | 52 | 52 | 44 | 0 | 284 | 0 | 127 | 44 |
normalized size | 1 | 1.00 | 1.00 | 0.85 | 0.00 | 5.46 | 0.00 | 2.44 | 0.85 |
time (sec) |
N/A |
0.098 |
0.143 |
0.211 |
0.000 |
0.660 |
0.000 |
4.910 |
1.207 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 64 | 64 | 158 | 80 | 0 | 588 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.47 | 1.25 | 0.00 | 9.19 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.090 |
0.272 |
0.192 |
0.000 |
0.637 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 33 | 29 | 0 | 127 | 0 | 70 | 27 |
normalized size | 1 | 1.00 | 1.00 | 0.88 | 0.00 | 3.85 | 0.00 | 2.12 | 0.82 |
time (sec) |
N/A |
0.064 |
0.014 |
0.151 |
0.000 |
0.515 |
0.000 |
0.563 |
0.963 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 60 | 376 | 0 | 419 | 0 | 140 | 93 |
normalized size | 1 | 1.00 | 1.00 | 6.27 | 0.00 | 6.98 | 0.00 | 2.33 | 1.55 |
time (sec) |
N/A |
0.094 |
0.045 |
0.749 |
0.000 |
0.696 |
0.000 |
0.385 |
0.513 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 134 | 328 | 0 | 229 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.48 | 6.07 | 0.00 | 4.24 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.091 |
1.572 |
0.819 |
0.000 |
1.105 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 59 | 68 | 0 | 385 | 0 | 0 | 52 |
normalized size | 1 | 1.00 | 1.00 | 1.15 | 0.00 | 6.53 | 0.00 | 0.00 | 0.88 |
time (sec) |
N/A |
0.111 |
0.213 |
0.166 |
0.000 |
0.453 |
0.000 |
0.000 |
1.917 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-2) |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 137 | 99 | 0 | 388 | 0 | 259 | -1 |
normalized size | 1 | 1.00 | 2.32 | 1.68 | 0.00 | 6.58 | 0.00 | 4.39 | -0.02 |
time (sec) |
N/A |
0.096 |
0.732 |
0.178 |
0.000 |
0.744 |
0.000 |
1.099 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 55 | 55 | 44 | 56 | 0 | 344 | 48 | 0 | 47 |
normalized size | 1 | 1.00 | 0.80 | 1.02 | 0.00 | 6.25 | 0.87 | 0.00 | 0.85 |
time (sec) |
N/A |
0.075 |
0.041 |
0.141 |
0.000 |
0.474 |
10.367 |
0.000 |
1.779 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 75 | 962 | 0 | 863 | 0 | 0 | 1451 |
normalized size | 1 | 1.00 | 0.89 | 11.45 | 0.00 | 10.27 | 0.00 | 0.00 | 17.27 |
time (sec) |
N/A |
0.131 |
0.058 |
0.898 |
0.000 |
1.007 |
0.000 |
0.000 |
0.482 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 92 | 92 | 674 | 421 | 0 | 393 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 7.33 | 4.58 | 0.00 | 4.27 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.154 |
6.897 |
0.955 |
0.000 |
0.905 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 82 | 82 | 69 | 88 | 0 | 698 | 0 | 0 | 88 |
normalized size | 1 | 1.00 | 0.84 | 1.07 | 0.00 | 8.51 | 0.00 | 0.00 | 1.07 |
time (sec) |
N/A |
0.127 |
0.099 |
0.173 |
0.000 |
0.680 |
0.000 |
0.000 |
4.236 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
B |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 94 | 94 | 200 | 166 | 0 | 720 | 0 | 1025 | -1 |
normalized size | 1 | 1.00 | 2.13 | 1.77 | 0.00 | 7.66 | 0.00 | 10.90 | -0.01 |
time (sec) |
N/A |
0.124 |
6.513 |
0.199 |
0.000 |
1.347 |
0.000 |
2.398 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
B |
A |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 47 | 75 | 0 | 627 | 70 | 0 | 82 |
normalized size | 1 | 1.00 | 0.60 | 0.96 | 0.00 | 8.04 | 0.90 | 0.00 | 1.05 |
time (sec) |
N/A |
0.092 |
0.041 |
0.138 |
0.000 |
0.827 |
16.770 |
0.000 |
4.462 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F(-2) |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 118 | 118 | 78 | 0 | 0 | 1531 | 0 | 0 | 2817 |
normalized size | 1 | 1.00 | 0.66 | 0.00 | 0.00 | 12.97 | 0.00 | 0.00 | 23.87 |
time (sec) |
N/A |
0.188 |
0.048 |
0.604 |
0.000 |
0.817 |
0.000 |
0.000 |
1.051 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
B |
F |
B |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 141 | 1450 | 1040 | 0 | 647 | 0 | 1242 | -1 |
normalized size | 1 | 1.00 | 10.28 | 7.38 | 0.00 | 4.59 | 0.00 | 8.81 | -0.01 |
time (sec) |
N/A |
0.241 |
8.017 |
1.299 |
0.000 |
0.701 |
0.000 |
4.941 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 57 | 37 | 33 | 24 | 34 | 34 | 37 |
normalized size | 1 | 1.00 | 1.54 | 1.00 | 0.89 | 0.65 | 0.92 | 0.92 | 1.00 |
time (sec) |
N/A |
0.062 |
0.036 |
0.151 |
0.417 |
0.462 |
0.286 |
0.425 |
0.720 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 90 | 90 | 86 | 139 | 0 | 1063 | 0 | 204 | -1 |
normalized size | 1 | 1.00 | 0.96 | 1.54 | 0.00 | 11.81 | 0.00 | 2.27 | -0.01 |
time (sec) |
N/A |
0.135 |
0.154 |
0.348 |
0.000 |
0.830 |
0.000 |
0.447 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 126 | 126 | 167 | 312 | 0 | 1486 | 0 | 445 | -1 |
normalized size | 1 | 1.00 | 1.33 | 2.48 | 0.00 | 11.79 | 0.00 | 3.53 | -0.01 |
time (sec) |
N/A |
0.202 |
4.396 |
0.286 |
0.000 |
0.851 |
0.000 |
1.799 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 41 | 65 | 0 | 264 | 0 | 58 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.59 | 0.00 | 6.44 | 0.00 | 1.41 | -0.02 |
time (sec) |
N/A |
0.069 |
0.019 |
0.312 |
0.000 |
0.648 |
0.000 |
0.383 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 74 | 74 | 73 | 248 | 0 | 670 | 0 | 111 | -1 |
normalized size | 1 | 1.00 | 0.99 | 3.35 | 0.00 | 9.05 | 0.00 | 1.50 | -0.01 |
time (sec) |
N/A |
0.111 |
0.302 |
0.316 |
0.000 |
1.356 |
0.000 |
0.444 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 117 | 117 | 114 | 602 | 0 | 1365 | 0 | 276 | -1 |
normalized size | 1 | 1.00 | 0.97 | 5.15 | 0.00 | 11.67 | 0.00 | 2.36 | -0.01 |
time (sec) |
N/A |
0.190 |
0.745 |
0.333 |
0.000 |
0.737 |
0.000 |
0.485 |
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 [59] had the largest ratio of
[.7500]
|
|
|
|
|
|
|
| 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 |
15 |
12 |
1.00 |
25 |
0.480 |
|
|
|
|
|
|
|
2 |
A |
3 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
3 |
A |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
4 |
A |
4 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
6 |
A |
5 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
7 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
8 |
A |
4 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
9 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
11 |
A |
5 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
12 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
13 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
14 |
A |
4 |
3 |
1.00 |
17 |
0.176 |
|
|
|
|
|
|
|
15 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
16 |
A |
3 |
3 |
1.00 |
15 |
0.200 |
|
|
|
|
|
|
|
17 |
A |
5 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
18 |
A |
5 |
4 |
1.00 |
17 |
0.235 |
|
|
|
|
|
|
|
19 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
20 |
A |
5 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
21 |
A |
7 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
22 |
A |
7 |
7 |
1.00 |
17 |
0.412 |
|
|
|
|
|
|
|
23 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
24 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
25 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
26 |
A |
7 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
27 |
A |
8 |
8 |
1.00 |
17 |
0.471 |
|
|
|
|
|
|
|
28 |
A |
6 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
29 |
A |
8 |
6 |
1.00 |
15 |
0.400 |
|
|
|
|
|
|
|
30 |
A |
7 |
7 |
1.00 |
17 |
0.412 |
|
|
|
|
|
|
|
31 |
A |
8 |
8 |
1.00 |
16 |
0.500 |
|
|
|
|
|
|
|
32 |
A |
7 |
7 |
1.00 |
16 |
0.438 |
|
|
|
|
|
|
|
33 |
A |
6 |
6 |
1.00 |
16 |
0.375 |
|
|
|
|
|
|
|
34 |
A |
3 |
3 |
1.00 |
16 |
0.188 |
|
|
|
|
|
|
|
35 |
A |
4 |
4 |
1.00 |
16 |
0.250 |
|
|
|
|
|
|
|
36 |
A |
6 |
6 |
1.00 |
16 |
0.375 |
|
|
|
|
|
|
|
37 |
A |
7 |
6 |
1.00 |
16 |
0.375 |
|
|
|
|
|
|
|
38 |
A |
6 |
6 |
1.00 |
12 |
0.500 |
|
|
|
|
|
|
|
39 |
A |
5 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
40 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
41 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
42 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
43 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
44 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
45 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
46 |
A |
4 |
4 |
1.00 |
15 |
0.267 |
|
|
|
|
|
|
|
47 |
A |
7 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
48 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
49 |
A |
5 |
5 |
1.00 |
17 |
0.294 |
|
|
|
|
|
|
|
50 |
A |
4 |
4 |
1.00 |
17 |
0.235 |
|
|
|
|
|
|
|
51 |
A |
5 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
52 |
A |
8 |
6 |
1.00 |
15 |
0.400 |
|
|
|
|
|
|
|
53 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
54 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
55 |
A |
6 |
6 |
1.00 |
17 |
0.353 |
|
|
|
|
|
|
|
56 |
A |
6 |
5 |
1.00 |
15 |
0.333 |
|
|
|
|
|
|
|
57 |
A |
9 |
7 |
1.00 |
15 |
0.467 |
|
|
|
|
|
|
|
58 |
A |
7 |
7 |
1.00 |
17 |
0.412 |
|
|
|
|
|
|
|
59 |
A |
7 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
60 |
A |
8 |
7 |
1.00 |
15 |
0.467 |
|
|
|
|
|
|
|
61 |
A |
9 |
8 |
1.00 |
15 |
0.533 |
|
|
|
|
|
|
|
62 |
A |
4 |
4 |
1.00 |
15 |
0.267 |
|
|
|
|
|
|
|
63 |
A |
6 |
6 |
1.00 |
15 |
0.400 |
|
|
|
|
|
|
|
64 |
A |
7 |
7 |
1.00 |
15 |
0.467 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|