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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 1, 2, 3, 4, 5, 7, 10, 11, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 41, 42, 43, 44, 45, 46 }
B grade: { 6, 8, 12, 16, 17, 39, 40 }
C grade: { }
F grade: { 9 }
2.1.3 Maple
A grade: { 4, 5, 9, 10, 13, 14, 15, 18, 19, 20, 21, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 45, 46
}
B grade: { 1, 2, 3, 6, 7, 8, 11, 12, 16, 17, 24, 25, 26, 29, 30, 31, 36, 41 }
C grade: { }
F grade: { 34, 35, 39, 40 }
2.1.4 Maxima
A grade: { 9, 10, 14, 19, 20, 21, 22, 23, 32, 33, 42, 43, 44, 45, 46 }
B grade: { 1, 2, 6, 7, 11, 12, 13, 16, 17, 18, 24, 25, 29, 30 }
C grade: { }
F grade: { 3, 4, 5, 8, 15, 26, 27, 28, 31, 34, 35, 36, 37, 38, 39, 40, 41 }
2.1.5 FriCAS
A grade: { 4, 5, 9, 10, 13, 14, 15, 18, 19, 20, 21, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 45, 46
}
B grade: { 3, 8, 12, 17, 26, 31, 36, 41 }
C grade: { 1, 2, 6, 7, 11, 16, 24, 25, 29, 30, 34, 35, 39, 40 }
F grade: { }
2.1.6 Sympy
A grade: { 4, 5, 9, 10, 14, 15, 19, 20, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 45, 46 }
B grade: { }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 16, 17, 18, 21, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41, 44
}
2.1.7 Giac
A grade: { 4, 5, 9, 14, 15, 19, 20, 21, 22, 23, 27, 28, 32, 37, 38, 42, 44, 45, 46 }
B grade: { 13, 18 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 8, 10, 11, 12, 16, 17, 24, 25, 26, 29, 30, 31, 33, 34, 35, 36, 39, 40, 41, 43
}
2.1.8 Mupad
A grade: { 4, 5, 9, 10, 14, 15, 19, 20, 21, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 44, 45, 46 }
B grade: { 13, 18 }
C grade: { }
F grade: { 1, 2, 3, 6, 7, 8, 11, 12, 16, 17, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41 }
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 |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 218 | 747 | 926 | 1081 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 3.29 | 4.08 | 4.76 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.205 |
0.115 |
1.014 |
1.388 |
1.254 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 151 | 431 | 510 | 671 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 2.75 | 3.25 | 4.27 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.140 |
0.146 |
0.962 |
1.122 |
0.639 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-1) |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 93 | 93 | 87 | 208 | 0 | 343 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.94 | 2.24 | 0.00 | 3.69 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.067 |
0.058 |
0.266 |
0.000 |
0.646 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.05 |
time (sec) |
N/A |
0.027 |
6.485 |
1.232 |
0.000 |
0.618 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.026 |
6.470 |
1.227 |
0.000 |
0.774 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 371 | 371 | 811 | 1506 | 3403 | 1883 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.19 | 4.06 | 9.17 | 5.08 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.443 |
9.087 |
1.731 |
1.998 |
1.055 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 262 | 262 | 505 | 668 | 1707 | 1096 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.93 | 2.55 | 6.52 | 4.18 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.313 |
5.881 |
1.447 |
0.838 |
0.762 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
F(-1) |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 134 | 134 | 330 | 279 | 0 | 525 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.46 | 2.08 | 0.00 | 3.92 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.115 |
5.605 |
0.528 |
0.000 |
0.853 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
F(-1) |
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 |
0.051 |
0.000 |
4.233 |
0.000 |
1.562 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
F(-1) |
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 |
0.048 |
29.124 |
5.145 |
0.000 |
0.680 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 152 | 152 | 216 | 406 | 1284 | 514 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.42 | 2.67 | 8.45 | 3.38 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.329 |
1.801 |
0.897 |
0.786 |
0.925 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 119 | 119 | 528 | 225 | 376 | 290 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.44 | 1.89 | 3.16 | 2.44 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.245 |
6.490 |
0.818 |
1.139 |
1.450 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 67 | 67 | 104 | 76 | 273 | 99 | 0 | 290 | 79 |
normalized size | 1 | 1.00 | 1.55 | 1.13 | 4.07 | 1.48 | 0.00 | 4.33 | 1.18 |
time (sec) |
N/A |
0.097 |
0.750 |
0.526 |
1.042 |
0.792 |
0.000 |
1.027 |
0.757 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 14 |
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 |
0.058 |
7.134 |
1.000 |
0.000 |
0.641 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.052 |
5.258 |
1.115 |
0.000 |
1.263 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
C |
F |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 288 | 288 | 1447 | 799 | 4261 | 929 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 5.02 | 2.77 | 14.80 | 3.23 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.719 |
7.554 |
1.186 |
3.094 |
1.577 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
B |
B |
F |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 229 | 229 | 925 | 442 | 1033 | 491 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 4.04 | 1.93 | 4.51 | 2.14 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.498 |
6.855 |
0.966 |
2.430 |
0.838 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
B |
A |
F |
B |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 140 | 140 | 172 | 138 | 1058 | 183 | 0 | 900 | 247 |
normalized size | 1 | 1.00 | 1.23 | 0.99 | 7.56 | 1.31 | 0.00 | 6.43 | 1.76 |
time (sec) |
N/A |
0.196 |
1.649 |
0.637 |
0.615 |
1.349 |
0.000 |
3.245 |
5.983 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.055 |
15.240 |
5.734 |
0.000 |
0.537 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
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 |
0.052 |
17.608 |
6.844 |
0.000 |
0.824 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
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 | 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 |
0.047 |
0.855 |
0.874 |
0.000 |
0.689 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
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 |
0.025 |
6.668 |
0.755 |
0.000 |
0.782 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.053 |
0.991 |
0.713 |
0.000 |
1.063 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 227 | 227 | 365 | 747 | 926 | 1081 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.61 | 3.29 | 4.08 | 4.76 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.213 |
0.493 |
1.216 |
1.260 |
1.070 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 157 | 157 | 203 | 431 | 510 | 671 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.29 | 2.75 | 3.25 | 4.27 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.141 |
0.268 |
0.936 |
1.228 |
0.799 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-1) |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 93 | 93 | 104 | 208 | 0 | 343 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.12 | 2.24 | 0.00 | 3.69 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.071 |
0.013 |
0.267 |
0.000 |
0.913 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.028 |
1.003 |
1.247 |
0.000 |
0.751 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
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 |
0.028 |
1.385 |
1.309 |
0.000 |
0.795 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 364 | 364 | 646 | 1478 | 3267 | 1819 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.77 | 4.06 | 8.98 | 5.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.455 |
2.533 |
1.736 |
2.263 |
0.975 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
B |
C |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 356 | 651 | 1644 | 1056 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.39 | 2.53 | 6.40 | 4.11 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.308 |
1.460 |
1.357 |
1.056 |
0.794 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-1) |
B |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 131 | 131 | 151 | 271 | 0 | 505 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 2.07 | 0.00 | 3.85 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.120 |
0.527 |
0.518 |
0.000 |
0.652 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.051 |
42.669 |
4.184 |
0.000 |
0.647 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
F(-1) |
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 |
0.049 |
32.068 |
5.034 |
0.000 |
0.735 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 526 | 526 | 449 | 0 | 0 | 2353 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 0.00 | 0.00 | 4.47 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.043 |
1.133 |
2.567 |
0.000 |
0.912 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
C |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 394 | 394 | 338 | 0 | 0 | 1663 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.86 | 0.00 | 0.00 | 4.22 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.867 |
0.807 |
2.158 |
0.000 |
1.060 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 257 | 257 | 214 | 516 | 0 | 1073 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.83 | 2.01 | 0.00 | 4.18 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.490 |
0.479 |
0.953 |
0.000 |
0.890 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.059 |
1.587 |
1.088 |
0.000 |
0.733 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
F(-1) |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.055 |
11.692 |
1.170 |
0.000 |
1.042 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F(-2) |
C |
F |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1523 | 1523 | 8176 | 0 | 0 | 7050 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 5.37 | 0.00 | 0.00 | 4.63 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.809 |
26.195 |
3.475 |
0.000 |
1.765 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
F |
F(-2) |
C |
F |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1117 | 1117 | 11147 | 0 | 0 | 4310 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 9.98 | 0.00 | 0.00 | 3.86 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.111 |
22.405 |
2.862 |
0.000 |
1.408 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F(-2) |
B |
F |
F |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 582 | 582 | 1037 | 1289 | 0 | 2112 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.78 | 2.21 | 0.00 | 3.63 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.047 |
10.003 |
1.072 |
0.000 |
0.820 |
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 |
0.058 |
30.147 |
4.730 |
0.000 |
0.569 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
F(-1) |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.055 |
45.462 |
6.086 |
0.000 |
0.705 |
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 |
F(-1) |
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 |
0.051 |
1.797 |
0.907 |
0.000 |
0.766 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
N/A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 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 |
0.026 |
0.536 |
0.728 |
0.000 |
0.728 |
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 |
A |
A |
A |
verified |
N/A |
N/A |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.04 |
time (sec) |
N/A |
0.055 |
0.739 |
0.553 |
0.000 |
0.739 |
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 [40] had the largest ratio of
[.5500]
|
|
|
|
|
|
|
| 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 |
11 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
2 |
A |
9 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
3 |
A |
7 |
4 |
1.00 |
16 |
0.250 |
|
|
|
|
|
|
|
4 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
5 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
6 |
A |
17 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
7 |
A |
14 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
8 |
A |
9 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
9 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
10 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
11 |
A |
9 |
8 |
1.00 |
20 |
0.400 |
|
|
|
|
|
|
|
12 |
A |
8 |
7 |
1.00 |
20 |
0.350 |
|
|
|
|
|
|
|
13 |
A |
5 |
4 |
1.00 |
18 |
0.222 |
|
|
|
|
|
|
|
14 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
15 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
16 |
A |
19 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
17 |
A |
17 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
18 |
A |
9 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
19 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
20 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
21 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
22 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
23 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
24 |
A |
11 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
25 |
A |
9 |
5 |
1.00 |
18 |
0.278 |
|
|
|
|
|
|
|
26 |
A |
7 |
4 |
1.00 |
16 |
0.250 |
|
|
|
|
|
|
|
27 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
28 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
29 |
A |
17 |
9 |
1.00 |
20 |
0.450 |
|
|
|
|
|
|
|
30 |
A |
14 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
31 |
A |
9 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
32 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
33 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
34 |
A |
14 |
8 |
1.00 |
20 |
0.400 |
|
|
|
|
|
|
|
35 |
A |
12 |
7 |
1.00 |
20 |
0.350 |
|
|
|
|
|
|
|
36 |
A |
10 |
6 |
1.00 |
18 |
0.333 |
|
|
|
|
|
|
|
37 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
38 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
39 |
A |
36 |
10 |
1.00 |
20 |
0.500 |
|
|
|
|
|
|
|
40 |
A |
30 |
11 |
1.00 |
20 |
0.550 |
|
|
|
|
|
|
|
41 |
A |
21 |
9 |
1.00 |
18 |
0.500 |
|
|
|
|
|
|
|
42 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
43 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
44 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
45 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
46 |
A |
0 |
0 |
0.00 |
0 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|