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, 65, 66, 67, 68 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 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, 65, 66 }
B grade: { 3 }
C grade: { 67, 68 }
F grade: { }
2.1.3 Maple
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, 30,
31, 32, 33, 34, 36, 38, 40, 41, 42, 44, 46, 50, 51, 52, 53, 57, 58, 61, 66 }
B grade: { 29, 59, 60, 64, 65 }
C grade: { 35, 47, 54, 55, 56 }
F grade: { 37, 39, 43, 45, 48, 49, 62, 63, 67, 68 }
2.1.4 Maxima
A grade: { 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 64, 65, 66
}
B grade: { 1, 2, 4, 17, 22, 54, 55, 56, 61 }
C grade: { 28, 29 }
F grade: { 43, 45, 62, 63, 67, 68 }
2.1.5 FriCAS
A grade: { 1, 3, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 38, 40, 41, 42,
44, 46, 53, 54, 56, 57, 58, 59, 60, 61, 64, 65, 66 }
B grade: { 2, 6, 9, 13, 16, 20, 21, 22, 30, 34, 36, 50, 51, 52, 55, 62, 63, 67, 68 }
C grade: { }
F grade: { 35, 37, 39, 43, 45, 47, 48, 49 }
2.1.6 Sympy
A grade: { 1, 3, 8, 10, 15, 17, 22, 23, 24, 26, 27, 28, 29, 33, 41, 42, 44, 57, 58, 59, 60, 61, 66
}
B grade: { }
C grade: { }
F grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 16, 18, 19, 20, 21, 25, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 43, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 62, 63, 64, 65, 67, 68 }
2.1.7 Giac
A grade: { 1, 2, 4, 5, 9, 10, 11, 12, 16, 17, 18, 19, 22, 41, 42, 44, 46, 53, 57, 58, 61, 66 }
B grade: { 3, 7, 8, 14, 15, 21, 23, 24, 25, 26, 27, 28, 29, 33, 59, 60, 64, 65 }
C grade: { 54, 55, 56 }
F grade: { 6, 13, 20, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 43, 45, 47, 48, 49, 50, 51, 52, 62, 63, 67, 68
}
2.1.8 Mupad
A grade: { 41, 42, 44, 46, 57, 58 }
B grade: { 1, 3, 8, 10, 15, 17, 22, 23, 24, 27, 28, 29, 33, 61, 66 }
C grade: { }
F grade: { 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 16, 18, 19, 20, 21, 25, 26, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 43,
45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 59, 60, 62, 63, 64, 65, 67, 68 }
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 |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 34 | 34 | 31 | 45 | 80 | 29 | 36 | 48 | 28 |
normalized size | 1 | 1.00 | 0.91 | 1.32 | 2.35 | 0.85 | 1.06 | 1.41 | 0.82 |
time (sec) |
N/A |
0.036 |
0.041 |
0.050 |
0.322 |
0.461 |
0.819 |
0.112 |
0.083 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 67 | 74 | 110 | 189 | 0 | 75 | -1 |
normalized size | 1 | 1.00 | 0.97 | 1.07 | 1.59 | 2.74 | 0.00 | 1.09 | -0.01 |
time (sec) |
N/A |
0.039 |
0.076 |
0.093 |
0.322 |
0.477 |
0.000 |
0.119 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 31 | 14 | 13 | 13 | 19 | 27 | 13 |
normalized size | 1 | 1.00 | 2.07 | 0.93 | 0.87 | 0.87 | 1.27 | 1.80 | 0.87 |
time (sec) |
N/A |
0.015 |
0.010 |
0.041 |
0.320 |
0.453 |
0.206 |
0.113 |
0.048 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 45 | 40 | 87 | 49 | 0 | 41 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.75 | 1.64 | 0.92 | 0.00 | 0.77 | -0.02 |
time (sec) |
N/A |
0.020 |
0.032 |
0.079 |
0.318 |
0.458 |
0.000 |
0.112 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 5 |
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 | 25 | 25 | 23 | 27 | 24 | 39 | 0 | 24 | -1 |
normalized size | 1 | 1.00 | 0.92 | 1.08 | 0.96 | 1.56 | 0.00 | 0.96 | -0.04 |
time (sec) |
N/A |
0.036 |
0.016 |
0.054 |
0.398 |
0.560 |
0.000 |
0.114 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 6 |
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 | 66 | 66 | 70 | 70 | 55 | 183 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.06 | 1.06 | 0.83 | 2.77 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.035 |
0.074 |
0.092 |
0.327 |
0.623 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 7 |
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 | 42 | 42 | 38 | 58 | 40 | 71 | 0 | 108 | -1 |
normalized size | 1 | 1.00 | 0.90 | 1.38 | 0.95 | 1.69 | 0.00 | 2.57 | -0.02 |
time (sec) |
N/A |
0.091 |
0.045 |
0.063 |
0.389 |
0.455 |
0.000 |
0.121 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 40 | 55 | 59 | 60 | 78 | 121 | 42 |
normalized size | 1 | 1.00 | 0.78 | 1.08 | 1.16 | 1.18 | 1.53 | 2.37 | 0.82 |
time (sec) |
N/A |
0.051 |
0.102 |
0.159 |
0.330 |
0.749 |
1.675 |
0.132 |
0.105 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 99 | 99 | 101 | 90 | 95 | 427 | 0 | 97 | -1 |
normalized size | 1 | 1.00 | 1.02 | 0.91 | 0.96 | 4.31 | 0.00 | 0.98 | -0.01 |
time (sec) |
N/A |
0.088 |
0.233 |
0.183 |
0.424 |
0.566 |
0.000 |
0.143 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 10 |
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 | 31 | 31 | 27 | 34 | 38 | 28 | 60 | 54 | 22 |
normalized size | 1 | 1.00 | 0.87 | 1.10 | 1.23 | 0.90 | 1.94 | 1.74 | 0.71 |
time (sec) |
N/A |
0.028 |
0.023 |
0.055 |
0.307 |
0.759 |
0.447 |
0.135 |
0.058 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 11 |
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 | 78 | 78 | 86 | 51 | 56 | 73 | 0 | 58 | -1 |
normalized size | 1 | 1.00 | 1.10 | 0.65 | 0.72 | 0.94 | 0.00 | 0.74 | -0.01 |
time (sec) |
N/A |
0.043 |
0.077 |
0.163 |
0.412 |
0.488 |
0.000 |
0.124 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 12 |
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 | 37 | 37 | 33 | 34 | 31 | 49 | 0 | 35 | -1 |
normalized size | 1 | 1.00 | 0.89 | 0.92 | 0.84 | 1.32 | 0.00 | 0.95 | -0.03 |
time (sec) |
N/A |
0.061 |
0.022 |
0.137 |
0.377 |
0.734 |
0.000 |
0.140 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 13 |
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 | 88 | 88 | 94 | 86 | 61 | 394 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.07 | 0.98 | 0.69 | 4.48 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.071 |
0.246 |
0.172 |
0.385 |
0.680 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 14 |
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 | 57 | 57 | 46 | 69 | 36 | 90 | 0 | 126 | -1 |
normalized size | 1 | 1.00 | 0.81 | 1.21 | 0.63 | 1.58 | 0.00 | 2.21 | -0.02 |
time (sec) |
N/A |
0.126 |
0.099 |
0.147 |
0.382 |
0.598 |
0.000 |
0.146 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 55 | 93 | 100 | 96 | 92 | 185 | 70 |
normalized size | 1 | 1.00 | 0.70 | 1.18 | 1.27 | 1.22 | 1.16 | 2.34 | 0.89 |
time (sec) |
N/A |
0.077 |
0.187 |
0.227 |
0.314 |
0.760 |
2.802 |
0.139 |
0.957 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 160 | 160 | 184 | 157 | 162 | 903 | 0 | 166 | -1 |
normalized size | 1 | 1.00 | 1.15 | 0.98 | 1.01 | 5.64 | 0.00 | 1.04 | -0.01 |
time (sec) |
N/A |
0.143 |
0.325 |
0.264 |
0.407 |
0.511 |
0.000 |
0.139 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 33 | 28 | 62 | 38 | 44 | 56 | 26 |
normalized size | 1 | 1.00 | 1.00 | 0.85 | 1.88 | 1.15 | 1.33 | 1.70 | 0.79 |
time (sec) |
N/A |
0.030 |
0.009 |
0.162 |
0.299 |
0.516 |
0.816 |
0.134 |
0.062 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 18 |
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 | 125 | 125 | 136 | 86 | 91 | 113 | 0 | 95 | -1 |
normalized size | 1 | 1.00 | 1.09 | 0.69 | 0.73 | 0.90 | 0.00 | 0.76 | -0.01 |
time (sec) |
N/A |
0.072 |
0.130 |
0.182 |
0.403 |
0.545 |
0.000 |
0.118 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
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 | 55 | 55 | 49 | 55 | 50 | 83 | 0 | 50 | -1 |
normalized size | 1 | 1.00 | 0.89 | 1.00 | 0.91 | 1.51 | 0.00 | 0.91 | -0.02 |
time (sec) |
N/A |
0.094 |
0.037 |
0.187 |
0.399 |
0.507 |
0.000 |
0.127 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
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 | 136 | 136 | 204 | 149 | 102 | 891 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.50 | 1.10 | 0.75 | 6.55 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.114 |
0.385 |
0.250 |
0.407 |
0.515 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 92 | 120 | 58 | 168 | 0 | 224 | -1 |
normalized size | 1 | 1.00 | 1.01 | 1.32 | 0.64 | 1.85 | 0.00 | 2.46 | -0.01 |
time (sec) |
N/A |
0.219 |
0.089 |
0.214 |
0.403 |
0.469 |
0.000 |
0.151 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
B |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 67 | 52 | 126 | 128 | 94 | 108 | 52 |
normalized size | 1 | 1.00 | 1.00 | 0.78 | 1.88 | 1.91 | 1.40 | 1.61 | 0.78 |
time (sec) |
N/A |
0.044 |
0.028 |
0.169 |
0.304 |
1.009 |
8.157 |
0.138 |
0.988 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 6 | 6 | 5 | 15 | 6 |
normalized size | 1 | 1.00 | 1.00 | 0.88 | 0.75 | 0.75 | 0.62 | 1.88 | 0.75 |
time (sec) |
N/A |
0.011 |
0.003 |
0.040 |
0.297 |
0.501 |
0.285 |
0.113 |
0.879 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 6 | 6 | 8 | 15 | 15 |
normalized size | 1 | 1.00 | 1.00 | 0.88 | 0.75 | 0.75 | 1.00 | 1.88 | 1.88 |
time (sec) |
N/A |
0.012 |
0.005 |
0.037 |
0.300 |
0.482 |
23.176 |
0.115 |
0.903 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
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 | 33 | 33 | 33 | 56 | 37 | 58 | 0 | 173 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.70 | 1.12 | 1.76 | 0.00 | 5.24 | -0.03 |
time (sec) |
N/A |
0.078 |
0.029 |
0.112 |
0.346 |
0.556 |
0.000 |
0.137 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 27 | 24 | 39 | 17 | 43 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.29 | 1.14 | 1.86 | 0.81 | 2.05 | -0.05 |
time (sec) |
N/A |
0.032 |
0.011 |
0.104 |
0.352 |
0.502 |
1.077 |
0.136 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 13 | 14 | 13 | 15 | 15 | 29 | 13 |
normalized size | 1 | 1.00 | 1.00 | 1.08 | 1.00 | 1.15 | 1.15 | 2.23 | 1.00 |
time (sec) |
N/A |
0.015 |
0.005 |
0.045 |
0.301 |
0.872 |
1.024 |
0.136 |
0.875 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
C |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 29 | 44 | 47 | 33 | 29 | 93 | 29 |
normalized size | 1 | 1.00 | 1.00 | 1.52 | 1.62 | 1.14 | 1.00 | 3.21 | 1.00 |
time (sec) |
N/A |
0.030 |
0.030 |
0.077 |
0.378 |
0.416 |
1.865 |
0.142 |
0.885 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
C |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 46 | 46 | 39 | 94 | 48 | 43 | 46 | 216 | 66 |
normalized size | 1 | 1.00 | 0.85 | 2.04 | 1.04 | 0.93 | 1.00 | 4.70 | 1.43 |
time (sec) |
N/A |
0.053 |
0.051 |
0.082 |
0.345 |
0.489 |
3.126 |
0.143 |
0.931 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 67 | 67 | 71 | 70 | 72 | 225 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.06 | 1.04 | 1.07 | 3.36 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.045 |
0.088 |
0.134 |
0.372 |
0.688 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 27 | 24 | 39 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.08 | 0.96 | 1.56 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.032 |
0.017 |
0.102 |
0.367 |
0.531 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 57 | 57 | 49 | 44 | 63 | 53 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.86 | 0.77 | 1.11 | 0.93 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.031 |
0.038 |
0.127 |
0.347 |
0.604 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 15 | 15 | 15 | 14 | 13 | 17 | 22 | 29 | 13 |
normalized size | 1 | 1.00 | 1.00 | 0.93 | 0.87 | 1.13 | 1.47 | 1.93 | 0.87 |
time (sec) |
N/A |
0.017 |
0.005 |
0.041 |
0.296 |
0.503 |
2.909 |
0.142 |
0.901 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
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 | 75 | 75 | 74 | 82 | 63 | 250 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 1.09 | 0.84 | 3.33 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.049 |
0.076 |
0.129 |
0.350 |
0.462 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 77 | 74 | 61 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 1.10 | 0.91 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.022 |
0.078 |
0.123 |
0.402 |
0.409 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
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 | 25 | 25 | 23 | 33 | 30 | 54 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.92 | 1.32 | 1.20 | 2.16 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.038 |
0.028 |
0.094 |
0.410 |
0.428 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 81 | 0 | 68 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.91 | 0.00 | 0.76 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.070 |
0.127 |
0.256 |
0.404 |
0.475 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 43 | 36 | 40 | 37 | 69 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.84 | 0.93 | 0.86 | 1.60 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.063 |
0.036 |
0.294 |
0.395 |
0.496 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 150 | 150 | 138 | 0 | 125 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.92 | 0.00 | 0.83 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.080 |
0.655 |
0.261 |
0.483 |
0.546 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 52 | 67 | 62 | 114 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 1.00 | 0.93 | 1.70 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.100 |
0.060 |
0.414 |
0.427 |
0.641 |
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 |
0.022 |
5.749 |
0.983 |
0.000 |
0.529 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
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 |
0.026 |
8.541 |
0.975 |
0.000 |
0.606 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
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 | 95 | 95 | 94 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.104 |
0.146 |
1.182 |
0.000 |
0.708 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
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 | 39 | 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 | -0.03 |
time (sec) |
N/A |
0.052 |
6.194 |
1.106 |
0.000 |
0.547 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 148 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.187 |
0.383 |
1.244 |
0.000 |
0.515 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
F(-1) |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 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 | -0.02 |
time (sec) |
N/A |
0.061 |
8.797 |
1.072 |
0.000 |
0.556 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 89 | 89 | 100 | 110 | 85 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.12 | 1.24 | 0.96 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.070 |
0.188 |
0.185 |
0.433 |
0.490 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 128 | 128 | 116 | 0 | 101 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.91 | 0.00 | 0.79 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.155 |
0.273 |
0.283 |
0.430 |
0.399 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
A |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 182 | 0 | 173 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.91 | 0.00 | 0.86 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.206 |
1.082 |
0.342 |
0.508 |
0.603 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
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 | 45 | 45 | 46 | 74 | 34 | 140 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.02 | 1.64 | 0.76 | 3.11 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.094 |
0.068 |
0.135 |
0.398 |
0.600 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 54 | 90 | 47 | 182 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.81 | 1.34 | 0.70 | 2.72 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.130 |
0.139 |
0.250 |
0.412 |
0.692 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 113 | 113 | 97 | 152 | 70 | 320 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.86 | 1.35 | 0.62 | 2.83 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.225 |
0.181 |
0.411 |
0.451 |
0.577 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 53 |
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 | 71 | 71 | 60 | 54 | 69 | 98 | 0 | 52 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.76 | 0.97 | 1.38 | 0.00 | 0.73 | -0.01 |
time (sec) |
N/A |
0.045 |
0.782 |
0.114 |
0.397 |
0.479 |
0.000 |
0.201 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
A |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 113 | 113 | 62 | 136 | 818 | 164 | 0 | 137 | -1 |
normalized size | 1 | 1.00 | 0.55 | 1.20 | 7.24 | 1.45 | 0.00 | 1.21 | -0.01 |
time (sec) |
N/A |
0.101 |
0.142 |
0.133 |
0.688 |
0.536 |
0.000 |
0.166 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
B |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 39 | 66 | 650 | 134 | 0 | 99 | -1 |
normalized size | 1 | 1.00 | 0.72 | 1.22 | 12.04 | 2.48 | 0.00 | 1.83 | -0.02 |
time (sec) |
N/A |
0.053 |
0.031 |
0.125 |
0.645 |
0.657 |
0.000 |
0.166 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
B |
A |
F |
C |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 25 | 36 | 478 | 54 | 0 | 39 | -1 |
normalized size | 1 | 1.00 | 0.68 | 0.97 | 12.92 | 1.46 | 0.00 | 1.05 | -0.03 |
time (sec) |
N/A |
0.017 |
0.004 |
0.115 |
0.607 |
1.145 |
0.000 |
0.128 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 57 |
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 | 20 | 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 |
0.036 |
1.585 |
0.140 |
0.000 |
0.520 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 58 |
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 | 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 |
0.040 |
11.052 |
0.165 |
0.000 |
0.663 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 346 | 346 | 104 | 831 | 486 | 104 | 269 | 915 | -1 |
normalized size | 1 | 1.00 | 0.30 | 2.40 | 1.40 | 0.30 | 0.78 | 2.64 | -0.00 |
time (sec) |
N/A |
0.424 |
0.536 |
0.095 |
0.338 |
0.503 |
1.725 |
0.159 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 167 | 167 | 72 | 303 | 291 | 68 | 151 | 300 | -1 |
normalized size | 1 | 1.00 | 0.43 | 1.81 | 1.74 | 0.41 | 0.90 | 1.80 | -0.01 |
time (sec) |
N/A |
0.188 |
0.211 |
0.084 |
0.332 |
0.551 |
0.593 |
0.135 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 50 | 63 | 110 | 44 | 65 | 65 | 43 |
normalized size | 1 | 1.00 | 0.93 | 1.17 | 2.04 | 0.81 | 1.20 | 1.20 | 0.80 |
time (sec) |
N/A |
0.047 |
0.068 |
0.085 |
0.313 |
0.473 |
0.469 |
0.118 |
0.957 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 124 | 124 | 127 | 0 | 0 | 217 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.02 | 0.00 | 0.00 | 1.75 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.290 |
0.526 |
0.100 |
0.000 |
0.924 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 182 | 182 | 199 | 0 | 0 | 317 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.09 | 0.00 | 0.00 | 1.74 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.382 |
1.877 |
0.101 |
0.000 |
0.582 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 537 | 537 | 381 | 1815 | 642 | 181 | 0 | 2163 | -1 |
normalized size | 1 | 1.00 | 0.71 | 3.38 | 1.20 | 0.34 | 0.00 | 4.03 | -0.00 |
time (sec) |
N/A |
0.704 |
0.749 |
0.092 |
0.401 |
1.024 |
0.000 |
0.216 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 261 | 261 | 118 | 659 | 369 | 109 | 0 | 707 | -1 |
normalized size | 1 | 1.00 | 0.45 | 2.52 | 1.41 | 0.42 | 0.00 | 2.71 | -0.00 |
time (sec) |
N/A |
0.319 |
0.428 |
0.086 |
0.323 |
0.559 |
0.000 |
0.161 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 66 |
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 | 85 | 85 | 65 | 133 | 136 | 58 | 94 | 128 | 75 |
normalized size | 1 | 1.00 | 0.76 | 1.56 | 1.60 | 0.68 | 1.11 | 1.51 | 0.88 |
time (sec) |
N/A |
0.080 |
0.090 |
0.087 |
0.325 |
0.465 |
1.224 |
0.124 |
0.984 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 232 | 232 | 231 | 0 | 0 | 503 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 0.00 | 0.00 | 2.17 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.518 |
0.072 |
0.105 |
0.000 |
0.769 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 329 | 329 | 211 | 0 | 0 | 706 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.00 | 0.00 | 2.15 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.692 |
0.548 |
0.110 |
0.000 |
0.520 |
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 [54] 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 |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
2 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
3 |
A |
2 |
2 |
1.00 |
10 |
0.200 |
|
|
|
|
|
|
|
4 |
A |
3 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
6 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
7 |
A |
5 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
8 |
A |
3 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
9 |
A |
6 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
10 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
11 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
12 |
A |
5 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
13 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
14 |
A |
7 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
15 |
A |
4 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
16 |
A |
10 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
17 |
A |
3 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
18 |
A |
8 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
19 |
A |
8 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
20 |
A |
9 |
5 |
1.00 |
14 |
0.357 |
|
|
|
|
|
|
|
21 |
A |
12 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
22 |
A |
3 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
23 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
24 |
A |
2 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
25 |
A |
5 |
5 |
1.00 |
8 |
0.625 |
|
|
|
|
|
|
|
26 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
28 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
29 |
A |
4 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
30 |
A |
5 |
5 |
1.00 |
8 |
0.625 |
|
|
|
|
|
|
|
31 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
32 |
A |
4 |
4 |
1.00 |
12 |
0.333 |
|
|
|
|
|
|
|
33 |
A |
2 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
34 |
A |
5 |
5 |
1.00 |
12 |
0.417 |
|
|
|
|
|
|
|
35 |
A |
3 |
2 |
1.00 |
8 |
0.250 |
|
|
|
|
|
|
|
36 |
A |
3 |
3 |
1.00 |
12 |
0.250 |
|
|
|
|
|
|
|
37 |
A |
5 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
38 |
A |
5 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
39 |
A |
8 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
40 |
A |
8 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
41 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
43 |
A |
3 |
3 |
1.00 |
20 |
0.150 |
|
|
|
|
|
|
|
44 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
45 |
A |
5 |
5 |
1.00 |
22 |
0.227 |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
47 |
A |
3 |
2 |
1.00 |
12 |
0.167 |
|
|
|
|
|
|
|
48 |
A |
5 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
49 |
A |
8 |
3 |
1.00 |
14 |
0.214 |
|
|
|
|
|
|
|
50 |
A |
5 |
5 |
1.00 |
16 |
0.312 |
|
|
|
|
|
|
|
51 |
A |
7 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
52 |
A |
12 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
53 |
A |
4 |
4 |
1.00 |
18 |
0.222 |
|
|
|
|
|
|
|
54 |
A |
12 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
55 |
A |
8 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
56 |
A |
4 |
4 |
1.00 |
8 |
0.500 |
|
|
|
|
|
|
|
57 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
58 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
59 |
A |
16 |
4 |
1.00 |
18 |
0.222 |
|
|
|
|
|
|
|
60 |
A |
10 |
4 |
1.00 |
16 |
0.250 |
|
|
|
|
|
|
|
61 |
A |
4 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
62 |
A |
10 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
63 |
A |
11 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
64 |
A |
23 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
65 |
A |
13 |
5 |
1.00 |
16 |
0.312 |
|
|
|
|
|
|
|
66 |
A |
5 |
4 |
1.00 |
14 |
0.286 |
|
|
|
|
|
|
|
67 |
A |
13 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
68 |
A |
14 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|