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 }
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, 15, 16, 18, 19, 20, 21, 23, 27, 28, 29, 30, 33, 34, 35, 43, 44,
45, 46, 47, 48, 49, 50 }
B grade: { 14, 22, 31, 32, 36, 38, 39, 40, 41, 42 }
C grade: { 17, 24, 25, 26 }
F grade: { 37 }
2.1.3 Maple
A grade: { 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 21, 22, 23, 27, 28, 29, 30, 32, 33, 34, 38, 39, 40, 42, 50
}
B grade: { 11, 12, 18, 19, 20, 24, 25, 26 }
C grade: { }
F grade: { 1, 31, 35, 36, 37, 41, 43, 44, 45, 46, 47, 48, 49 }
2.1.4 Maxima
A grade: { 2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 16, 22, 38, 39, 40, 41 }
B grade: { 8, 9 }
C grade: { }
F grade: { 1, 6, 13, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 42, 43, 44, 45,
46, 47, 48, 49, 50 }
2.1.5 FriCAS
A grade: { 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 15, 18, 19, 20, 21, 25, 26, 38, 39, 40, 41, 47, 48, 49 }
B grade: { 14, 16, 22, 24 }
C grade: { }
F grade: { 1, 6, 13, 17, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 42, 43, 44, 45, 46, 50 }
2.1.6 Sympy
A grade: { 10, 11, 12 }
B grade: { }
C grade: { 2, 3, 4, 5, 7, 8, 9, 14, 15, 16 }
F grade: { 1, 6, 13, 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 }
2.1.7 Giac
A grade: { 7, 8, 9, 10, 11, 12, 15, 16, 21, 24, 38, 39, 40, 41, 47, 50 }
B grade: { 2, 3, 4, 5, 14, 18, 19, 20, 22, 25, 26 }
C grade: { }
F grade: { 1, 6, 13, 17, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 42, 43, 44, 45, 46, 48, 49
}
2.1.8 Mupad
A grade: { }
B grade: { 5, 7, 11, 12, 14, 22, 38, 39, 40, 41 }
C grade: { }
F grade: { 1, 2, 3, 4, 6, 8, 9, 10, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 42, 43, 44, 45, 46, 47, 48, 49, 50 }
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 |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 62 | 62 | 56 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.087 |
0.038 |
0.180 |
0.000 |
2.017 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
C |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 40 | 43 | 66 | 32 | 119 | 152 | -1 |
normalized size | 1 | 1.00 | 0.69 | 0.74 | 1.14 | 0.55 | 2.05 | 2.62 | -0.02 |
time (sec) |
N/A |
0.020 |
0.028 |
0.052 |
0.332 |
0.488 |
73.945 |
0.134 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
C |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 35 | 38 | 52 | 27 | 90 | 116 | -1 |
normalized size | 1 | 1.00 | 0.74 | 0.81 | 1.11 | 0.57 | 1.91 | 2.47 | -0.02 |
time (sec) |
N/A |
0.018 |
0.022 |
0.049 |
0.333 |
0.772 |
25.692 |
0.138 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
C |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 36 | 36 | 28 | 31 | 38 | 20 | 58 | 80 | -1 |
normalized size | 1 | 1.00 | 0.78 | 0.86 | 1.06 | 0.56 | 1.61 | 2.22 | -0.03 |
time (sec) |
N/A |
0.011 |
0.018 |
0.048 |
0.328 |
0.891 |
8.944 |
0.150 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
C |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 18 | 18 | 25 | 21 | 14 | 29 | 41 | 21 |
normalized size | 1 | 1.00 | 1.00 | 1.39 | 1.17 | 0.78 | 1.61 | 2.28 | 1.17 |
time (sec) |
N/A |
0.004 |
0.006 |
0.048 |
0.332 |
0.932 |
4.423 |
0.127 |
1.149 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 54 | 63 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 1.12 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.079 |
0.029 |
0.115 |
0.000 |
0.605 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
C |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 32 | 46 | 51 | 19 | 76 | 30 | 28 |
normalized size | 1 | 1.00 | 0.84 | 1.21 | 1.34 | 0.50 | 2.00 | 0.79 | 0.74 |
time (sec) |
N/A |
0.016 |
0.022 |
0.051 |
0.419 |
1.586 |
20.047 |
0.128 |
0.684 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 55 | 57 | 80 | 29 | 144 | 44 | -1 |
normalized size | 1 | 1.00 | 1.02 | 1.06 | 1.48 | 0.54 | 2.67 | 0.81 | -0.02 |
time (sec) |
N/A |
0.019 |
0.034 |
0.049 |
0.437 |
1.691 |
53.237 |
0.142 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 68 | 68 | 45 | 67 | 106 | 35 | 180 | 58 | -1 |
normalized size | 1 | 1.00 | 0.66 | 0.99 | 1.56 | 0.51 | 2.65 | 0.85 | -0.01 |
time (sec) |
N/A |
0.024 |
0.055 |
0.050 |
0.428 |
0.537 |
132.622 |
0.148 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 56 | 56 | 42 | 66 | 54 | 39 | 51 | 47 | -1 |
normalized size | 1 | 1.00 | 0.75 | 1.18 | 0.96 | 0.70 | 0.91 | 0.84 | -0.02 |
time (sec) |
N/A |
0.042 |
0.035 |
0.081 |
0.433 |
1.089 |
0.488 |
0.136 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 47 | 44 | 93 | 46 | 38 | 41 | 39 | 38 |
normalized size | 1 | 1.00 | 0.94 | 1.98 | 0.98 | 0.81 | 0.87 | 0.83 | 0.81 |
time (sec) |
N/A |
0.020 |
0.025 |
0.067 |
0.426 |
0.877 |
0.221 |
0.122 |
0.628 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 51 | 24 | 27 | 22 | 28 | 24 |
normalized size | 1 | 1.00 | 1.00 | 1.96 | 0.92 | 1.04 | 0.85 | 1.08 | 0.92 |
time (sec) |
N/A |
0.009 |
0.012 |
0.060 |
0.340 |
0.909 |
0.146 |
0.127 |
0.109 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 59 | 59 | 59 | 76 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.29 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.060 |
0.024 |
0.118 |
0.000 |
0.916 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
C |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 93 | 41 | 52 | 107 | 29 | 61 | 29 |
normalized size | 1 | 1.00 | 3.00 | 1.32 | 1.68 | 3.45 | 0.94 | 1.97 | 0.94 |
time (sec) |
N/A |
0.030 |
0.142 |
0.044 |
0.340 |
2.117 |
2.173 |
0.141 |
0.605 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 36 | 54 | 32 | 39 | 53 | 61 | -1 |
normalized size | 1 | 1.00 | 0.95 | 1.42 | 0.84 | 1.03 | 1.39 | 1.61 | -0.03 |
time (sec) |
N/A |
0.021 |
0.022 |
0.056 |
0.422 |
0.646 |
1.250 |
0.176 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
B |
C |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 60 | 60 | 69 | 98 | 64 | 142 | 100 | 80 | -1 |
normalized size | 1 | 1.00 | 1.15 | 1.63 | 1.07 | 2.37 | 1.67 | 1.33 | -0.02 |
time (sec) |
N/A |
0.042 |
0.044 |
0.056 |
0.420 |
0.943 |
2.515 |
0.125 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 60 | 98 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.87 | 1.42 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.093 |
0.093 |
0.161 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 197 | 197 | 173 | 509 | 0 | 152 | 0 | 409 | -1 |
normalized size | 1 | 1.00 | 0.88 | 2.58 | 0.00 | 0.77 | 0.00 | 2.08 | -0.01 |
time (sec) |
N/A |
0.232 |
0.186 |
0.088 |
0.000 |
0.751 |
0.000 |
0.180 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 150 | 359 | 0 | 130 | 0 | 299 | -1 |
normalized size | 1 | 1.00 | 0.97 | 2.32 | 0.00 | 0.84 | 0.00 | 1.93 | -0.01 |
time (sec) |
N/A |
0.139 |
0.287 |
0.057 |
0.000 |
2.065 |
0.000 |
0.187 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 116 | 116 | 131 | 273 | 0 | 117 | 0 | 204 | -1 |
normalized size | 1 | 1.00 | 1.13 | 2.35 | 0.00 | 1.01 | 0.00 | 1.76 | -0.01 |
time (sec) |
N/A |
0.090 |
0.178 |
0.057 |
0.000 |
2.182 |
0.000 |
0.170 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 78 | 78 | 110 | 126 | 0 | 104 | 0 | 133 | -1 |
normalized size | 1 | 1.00 | 1.41 | 1.62 | 0.00 | 1.33 | 0.00 | 1.71 | -0.01 |
time (sec) |
N/A |
0.053 |
0.123 |
0.053 |
0.000 |
1.443 |
0.000 |
0.156 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
B |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 37 | 37 | 115 | 51 | 55 | 73 | 0 | 82 | 35 |
normalized size | 1 | 1.00 | 3.11 | 1.38 | 1.49 | 1.97 | 0.00 | 2.22 | 0.95 |
time (sec) |
N/A |
0.023 |
0.144 |
0.049 |
0.333 |
1.160 |
0.000 |
0.171 |
0.862 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 200 | 200 | 284 | 374 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.42 | 1.87 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.309 |
0.323 |
0.836 |
0.000 |
0.974 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 70 | 70 | 112 | 154 | 0 | 281 | 0 | 94 | -1 |
normalized size | 1 | 1.00 | 1.60 | 2.20 | 0.00 | 4.01 | 0.00 | 1.34 | -0.01 |
time (sec) |
N/A |
0.097 |
0.314 |
0.061 |
0.000 |
1.013 |
0.000 |
0.193 |
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 |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 125 | 125 | 198 | 452 | 0 | 427 | 0 | 216 | -1 |
normalized size | 1 | 1.00 | 1.58 | 3.62 | 0.00 | 3.42 | 0.00 | 1.73 | -0.01 |
time (sec) |
N/A |
0.192 |
1.052 |
0.064 |
0.000 |
2.712 |
0.000 |
0.221 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
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 | 181 | 181 | 241 | 760 | 0 | 548 | 0 | 451 | -1 |
normalized size | 1 | 1.00 | 1.33 | 4.20 | 0.00 | 3.03 | 0.00 | 2.49 | -0.01 |
time (sec) |
N/A |
0.288 |
0.431 |
0.069 |
0.000 |
1.928 |
0.000 |
0.223 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 381 | 381 | 667 | 734 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.75 | 1.93 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.302 |
9.418 |
4.002 |
0.000 |
2.013 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 288 | 288 | 473 | 540 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.64 | 1.88 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.232 |
6.582 |
1.686 |
0.000 |
1.030 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 142 | 264 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.92 | 1.71 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.131 |
0.147 |
0.330 |
0.000 |
1.685 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 94 | 94 | 111 | 179 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.18 | 1.90 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.063 |
0.113 |
0.282 |
0.000 |
1.348 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 310 | 310 | 813 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.62 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.485 |
2.378 |
1.450 |
0.000 |
3.071 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-1) |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 244 | 244 | 686 | 341 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.81 | 1.40 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.395 |
2.293 |
0.844 |
0.000 |
2.409 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 494 | 494 | 442 | 770 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 1.56 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.397 |
0.525 |
1.789 |
0.000 |
3.721 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 278 | 278 | 248 | 429 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.89 | 1.54 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.259 |
0.438 |
1.532 |
0.000 |
1.189 |
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 |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 160 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.104 |
0.099 |
0.620 |
0.000 |
0.768 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 430 | 430 | 1058 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.46 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.533 |
4.055 |
1.563 |
0.000 |
1.760 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F(-1) |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 362 | 362 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.595 |
180.002 |
1.441 |
0.000 |
2.226 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 148 | 81 | 71 | 96 | 0 | 100 | 52 |
normalized size | 1 | 1.00 | 2.55 | 1.40 | 1.22 | 1.66 | 0.00 | 1.72 | 0.90 |
time (sec) |
N/A |
0.073 |
0.237 |
0.103 |
0.347 |
0.600 |
0.000 |
0.377 |
1.031 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 148 | 81 | 71 | 96 | 0 | 100 | 52 |
normalized size | 1 | 1.00 | 2.55 | 1.40 | 1.22 | 1.66 | 0.00 | 1.72 | 0.90 |
time (sec) |
N/A |
0.076 |
0.155 |
0.101 |
0.360 |
0.798 |
0.000 |
0.401 |
0.774 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 58 | 58 | 137 | 81 | 71 | 96 | 0 | 100 | 52 |
normalized size | 1 | 1.00 | 2.36 | 1.40 | 1.22 | 1.66 | 0.00 | 1.72 | 0.90 |
time (sec) |
N/A |
0.078 |
0.383 |
0.101 |
0.375 |
1.213 |
0.000 |
0.382 |
0.777 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
A |
A |
F(-1) |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 49 | 49 | 130 | 0 | 66 | 92 | 0 | 75 | 44 |
normalized size | 1 | 1.00 | 2.65 | 0.00 | 1.35 | 1.88 | 0.00 | 1.53 | 0.90 |
time (sec) |
N/A |
0.072 |
0.340 |
0.156 |
0.359 |
1.010 |
0.000 |
0.279 |
1.084 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
F(-1) |
F(-2) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 85 | 85 | 280 | 116 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.29 | 1.36 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.081 |
0.855 |
0.168 |
0.000 |
0.000 |
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 |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 99 | 99 | 95 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.117 |
0.312 |
0.054 |
0.000 |
1.339 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 107 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.18 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.107 |
0.197 |
0.051 |
0.000 |
0.955 |
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 |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 91 | 91 | 54 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.092 |
0.060 |
0.049 |
0.000 |
0.973 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 79 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.76 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.058 |
0.054 |
0.050 |
0.000 |
0.991 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 39 | 39 | 34 | 0 | 0 | 23 | 0 | 51 | -1 |
normalized size | 1 | 1.00 | 0.87 | 0.00 | 0.00 | 0.59 | 0.00 | 1.31 | -0.03 |
time (sec) |
N/A |
0.030 |
0.043 |
0.051 |
0.000 |
1.420 |
0.000 |
0.205 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 30 | 0 | 0 | 30 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.73 | 0.00 | 0.00 | 0.73 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.042 |
0.045 |
0.053 |
0.000 |
1.053 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 84 | 84 | 54 | 0 | 0 | 40 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 0.00 | 0.00 | 0.48 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.065 |
0.139 |
0.051 |
0.000 |
1.840 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
F |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 69 | 69 | 59 | 92 | 0 | 0 | 0 | 115 | -1 |
normalized size | 1 | 1.00 | 0.86 | 1.33 | 0.00 | 0.00 | 0.00 | 1.67 | -0.01 |
time (sec) |
N/A |
0.091 |
0.050 |
0.351 |
0.000 |
0.527 |
0.000 |
0.591 |
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 [34] had the largest ratio of
[1.200]
|
|
|
|
|
|
|
| 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 |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
2 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
3 |
A |
4 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
4 |
A |
4 |
3 |
1.00 |
8 |
0.375 |
|
|
|
|
|
|
|
5 |
A |
3 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
6 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
7 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
8 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
9 |
A |
7 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
10 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
11 |
A |
4 |
4 |
1.00 |
8 |
0.500 |
|
|
|
|
|
|
|
12 |
A |
3 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
13 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
14 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
15 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
16 |
A |
6 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
17 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
18 |
A |
9 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
19 |
A |
8 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
20 |
A |
7 |
6 |
1.00 |
10 |
0.600 |
|
|
|
|
|
|
|
21 |
A |
6 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
22 |
A |
5 |
5 |
1.00 |
6 |
0.833 |
|
|
|
|
|
|
|
23 |
A |
14 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
24 |
A |
5 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
25 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
26 |
A |
8 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
27 |
A |
20 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
28 |
A |
17 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
29 |
A |
11 |
8 |
1.00 |
10 |
0.800 |
|
|
|
|
|
|
|
30 |
A |
8 |
6 |
1.00 |
8 |
0.750 |
|
|
|
|
|
|
|
31 |
A |
17 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
32 |
A |
12 |
8 |
1.00 |
12 |
0.667 |
|
|
|
|
|
|
|
33 |
A |
25 |
14 |
1.00 |
12 |
1.167 |
|
|
|
|
|
|
|
34 |
A |
16 |
12 |
1.00 |
10 |
1.200 |
|
|
|
|
|
|
|
35 |
A |
10 |
7 |
1.00 |
8 |
0.875 |
|
|
|
|
|
|
|
36 |
A |
20 |
10 |
1.00 |
12 |
0.833 |
|
|
|
|
|
|
|
37 |
A |
14 |
9 |
1.00 |
12 |
0.750 |
|
|
|
|
|
|
|
38 |
A |
7 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
39 |
A |
7 |
6 |
1.00 |
16 |
0.375 |
|
|
|
|
|
|
|
40 |
A |
7 |
6 |
1.00 |
16 |
0.375 |
|
|
|
|
|
|
|
41 |
A |
6 |
6 |
1.00 |
14 |
0.429 |
|
|
|
|
|
|
|
42 |
A |
7 |
7 |
1.00 |
10 |
0.700 |
|
|
|
|
|
|
|
43 |
A |
6 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
44 |
A |
6 |
4 |
1.00 |
8 |
0.500 |
|
|
|
|
|
|
|
45 |
A |
5 |
3 |
1.00 |
6 |
0.500 |
|
|
|
|
|
|
|
46 |
A |
6 |
5 |
1.00 |
10 |
0.500 |
|
|
|
|
|
|
|
47 |
A |
3 |
3 |
1.00 |
10 |
0.300 |
|
|
|
|
|
|
|
48 |
A |
5 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
49 |
A |
6 |
4 |
1.00 |
10 |
0.400 |
|
|
|
|
|
|
|
50 |
A |
8 |
8 |
1.00 |
19 |
0.421 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|