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 }
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 37, 38, 39, 40, 41, 42 }
B grade: { }
C grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32,
34, 35, 36 }
F grade: { 15, 26, 27, 33 }
2.1.3 Maple
A grade: { 20, 21, 30, 31, 37, 38, 39, 40, 41, 42 }
B grade: { 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36 }
C grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 32 }
F grade: { 15, 33 }
2.1.4 Maxima
A grade: { }
B grade: { }
C grade: { }
F 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 }
2.1.5 FriCAS
A grade: { 37, 38, 39, 40, 41, 42 }
B grade: { 34, 35, 36 }
C grade: { }
F 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 }
2.1.6 Sympy
A grade: { }
B grade: { }
C grade: { 1, 2, 3, 4, 10, 11 }
F grade: { 5, 6, 7, 8, 9, 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 }
2.1.7 Giac
A grade: { 34, 37, 38, 41 }
B grade: { 42 }
C grade: { }
F 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, 35, 36, 39, 40 }
2.1.8 Mupad
A grade: { }
B grade: { 34 }
C grade: { }
F 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, 35, 36, 37, 38, 39, 40, 41, 42 }
2.2 Detailed conclusion table per each integral for all CAS systems
Detailed conclusion table per each integral is given by table below. The elapsed time is in
seconds. For failed result it is given as F(-1) if the failure was due to timeout. It is given as F(-2)
if the failure was due to an exception being raised, which could indicate a bug in the system. If
the failure was due to integral not being evaluated within the time limit, then it is given just an
F.
In this table,the column normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\)
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Problem 1 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 453 | 878 | 217 | 533 | 0 | 0 | 364 | 0 | -1 |
normalized size | 1 | 1.94 | 0.48 | 1.18 | 0.00 | 0.00 | 0.80 | 0.00 | -0.00 |
time (sec) |
N/A |
0.509 |
0.266 |
0.030 |
0.000 |
0.874 |
9.014 |
0.000 |
0.000 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 367 | 706 | 159 | 403 | 0 | 0 | 262 | 0 | -1 |
normalized size | 1 | 1.92 | 0.43 | 1.10 | 0.00 | 0.00 | 0.71 | 0.00 | -0.00 |
time (sec) |
N/A |
0.396 |
0.168 |
0.009 |
0.000 |
1.069 |
6.609 |
0.000 |
0.000 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 277 | 432 | 121 | 269 | 0 | 0 | 167 | 0 | -1 |
normalized size | 1 | 1.56 | 0.44 | 0.97 | 0.00 | 0.00 | 0.60 | 0.00 | -0.00 |
time (sec) |
N/A |
0.225 |
0.097 |
0.008 |
0.000 |
1.007 |
4.510 |
0.000 |
0.000 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 226 | 226 | 77 | 169 | 0 | 0 | 78 | 0 | -1 |
normalized size | 1 | 1.00 | 0.34 | 0.75 | 0.00 | 0.00 | 0.35 | 0.00 | -0.00 |
time (sec) |
N/A |
0.066 |
0.030 |
0.005 |
0.000 |
1.099 |
1.971 |
0.000 |
0.000 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 369 | 369 | 138 | 192 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.37 | 0.52 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.360 |
0.300 |
0.039 |
0.000 |
60.111 |
0.000 |
0.000 |
0.000 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 641 | 641 | 297 | 679 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.46 | 1.06 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.189 |
1.151 |
0.035 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 875 | 875 | 453 | 1591 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.52 | 1.82 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.784 |
2.133 |
0.036 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 912 | 912 | 222 | 588 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.24 | 0.64 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.726 |
0.281 |
0.041 |
0.000 |
1.142 |
0.000 |
0.000 |
0.000 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 694 | 694 | 166 | 458 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.24 | 0.66 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.532 |
0.172 |
0.010 |
0.000 |
1.401 |
0.000 |
0.000 |
0.000 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 395 | 395 | 126 | 320 | 0 | 0 | 167 | 0 | -1 |
normalized size | 1 | 1.00 | 0.32 | 0.81 | 0.00 | 0.00 | 0.42 | 0.00 | -0.00 |
time (sec) |
N/A |
0.327 |
0.106 |
0.004 |
0.000 |
1.160 |
12.950 |
0.000 |
0.000 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
C |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 262 | 262 | 99 | 217 | 0 | 0 | 78 | 0 | -1 |
normalized size | 1 | 1.00 | 0.38 | 0.83 | 0.00 | 0.00 | 0.30 | 0.00 | -0.00 |
time (sec) |
N/A |
0.096 |
0.044 |
0.005 |
0.000 |
0.914 |
7.186 |
0.000 |
0.000 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 732 | 732 | 432 | 564 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 0.77 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.781 |
0.791 |
0.034 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1494 | 1494 | 427 | 1384 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.29 | 0.93 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
2.007 |
1.572 |
0.043 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2452 | 2452 | 630 | 2326 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.26 | 0.95 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
3.971 |
2.745 |
0.048 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
F |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 169 | 169 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.219 |
0.376 |
0.105 |
0.000 |
1.207 |
0.000 |
0.000 |
0.000 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 48 | 48 | 94 | 81 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.96 | 1.69 | 0.00 | 0.00 | 0.00 | 0.00 | -0.02 |
time (sec) |
N/A |
0.029 |
0.127 |
0.027 |
0.000 |
1.097 |
0.000 |
0.000 |
0.000 |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 755 | 755 | 4473 | 1708 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 5.92 | 2.26 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.357 |
6.477 |
0.019 |
0.000 |
0.898 |
0.000 |
0.000 |
0.000 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 528 | 528 | 674 | 1201 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.28 | 2.27 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.726 |
4.414 |
0.011 |
0.000 |
1.258 |
0.000 |
0.000 |
0.000 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 368 | 368 | 521 | 759 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.42 | 2.06 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.254 |
2.160 |
0.007 |
0.000 |
1.178 |
0.000 |
0.000 |
0.000 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
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grade |
A |
A |
C |
A |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 283 | 283 | 302 | 362 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.07 | 1.28 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.081 |
0.245 |
0.000 |
0.000 |
1.334 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 436 | 436 | 298 | 359 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.68 | 0.82 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.409 |
0.485 |
0.036 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 782 | 782 | 1853 | 1495 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 2.37 | 1.91 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.467 |
3.470 |
0.042 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
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|
|
|
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|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1125 | 1125 | 781 | 4476 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.69 | 3.98 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
3.594 |
5.732 |
0.048 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 859 | 859 | 5432 | 2445 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 6.32 | 2.85 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.380 |
6.653 |
0.037 |
0.000 |
1.558 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 628 | 633 | 766 | 1891 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.01 | 1.22 | 3.01 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.584 |
4.720 |
0.012 |
0.000 |
1.510 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F(-1) |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 481 | 481 | 0 | 1390 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 2.89 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.387 |
0.000 |
0.009 |
0.000 |
1.277 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F(-1) |
B |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 398 | 398 | 0 | 931 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.00 | 2.34 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.195 |
0.000 |
0.006 |
0.000 |
1.310 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 867 | 1045 | 1736 | 3241 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.21 | 2.00 | 3.74 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.424 |
5.057 |
0.036 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F(-1) |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1301 | 2112 | 8031 | 8276 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.62 | 6.17 | 6.36 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
3.386 |
7.801 |
0.059 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-2) |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 273 | 273 | 310 | 365 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.14 | 1.34 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.136 |
0.504 |
0.102 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 271 | 271 | 312 | 369 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.15 | 1.36 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.187 |
0.427 |
0.081 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
C |
F |
F |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 74 | 93 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.70 | 0.88 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.105 |
0.182 |
0.034 |
0.000 |
1.580 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
F |
F |
F |
F |
F(-1) |
F |
F |
verified |
N/A |
Yes |
N/A |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 218 | 218 | 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 |
0.463 |
0.225 |
0.084 |
0.000 |
1.696 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
A |
B |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 106 | 106 | 125 | 296 | 0 | 279 | 0 | 80 | 397 |
normalized size | 1 | 1.00 | 1.18 | 2.79 | 0.00 | 2.63 | 0.00 | 0.75 | 3.75 |
time (sec) |
N/A |
0.129 |
0.201 |
0.030 |
0.000 |
1.634 |
0.000 |
0.652 |
1.845 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 145 | 145 | 441 | 238 | 0 | 603 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 3.04 | 1.64 | 0.00 | 4.16 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.207 |
0.608 |
0.042 |
0.000 |
6.803 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 239 | 239 | 432 | 568 | 0 | 669 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.81 | 2.38 | 0.00 | 2.80 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.184 |
0.685 |
0.108 |
0.000 |
6.822 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 309 | 309 | 214 | 442 | 0 | 469 | 0 | 360 | -1 |
normalized size | 1 | 1.00 | 0.69 | 1.43 | 0.00 | 1.52 | 0.00 | 1.17 | -0.00 |
time (sec) |
N/A |
0.625 |
0.327 |
0.014 |
0.000 |
1.194 |
0.000 |
0.596 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 283 | 283 | 168 | 373 | 0 | 361 | 0 | 265 | -1 |
normalized size | 1 | 1.00 | 0.59 | 1.32 | 0.00 | 1.28 | 0.00 | 0.94 | -0.00 |
time (sec) |
N/A |
0.341 |
0.223 |
0.010 |
0.000 |
1.156 |
0.000 |
0.642 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 286 | 286 | 176 | 251 | 0 | 743 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.62 | 0.88 | 0.00 | 2.60 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.754 |
0.310 |
0.011 |
0.000 |
2.910 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F(-1) |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 171 | 288 | 0 | 731 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.58 | 0.98 | 0.00 | 2.49 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.767 |
0.251 |
0.010 |
0.000 |
2.440 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F(-1) |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 288 | 288 | 177 | 329 | 0 | 749 | 0 | 370 | -1 |
normalized size | 1 | 1.00 | 0.61 | 1.14 | 0.00 | 2.60 | 0.00 | 1.28 | -0.00 |
time (sec) |
N/A |
0.784 |
0.339 |
0.010 |
0.000 |
2.212 |
0.000 |
0.894 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
F(-1) |
B |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 189 | 412 | 0 | 791 | 0 | 720 | -1 |
normalized size | 1 | 1.00 | 0.64 | 1.40 | 0.00 | 2.69 | 0.00 | 2.45 | -0.00 |
time (sec) |
N/A |
0.787 |
0.287 |
0.013 |
0.000 |
2.049 |
0.000 |
1.339 |
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 [14] had the largest ratio of
[.3929]
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| Table 2.1:Rubi specific breakdown of results for each
integral
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# |
grade |
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|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
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1 |
A |
15 |
5 |
1.94 |
28 |
0.179 |
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2 |
A |
12 |
5 |
1.92 |
28 |
0.179 |
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3 |
A |
8 |
5 |
1.56 |
26 |
0.192 |
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4 |
A |
3 |
3 |
1.00 |
19 |
0.158 |
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5 |
A |
3 |
3 |
1.00 |
28 |
0.107 |
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6 |
A |
6 |
6 |
1.00 |
28 |
0.214 |
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7 |
A |
7 |
6 |
1.00 |
28 |
0.214 |
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8 |
A |
12 |
7 |
1.00 |
28 |
0.250 |
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9 |
A |
10 |
6 |
1.00 |
28 |
0.214 |
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10 |
A |
7 |
5 |
1.00 |
26 |
0.192 |
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11 |
A |
4 |
4 |
1.00 |
19 |
0.210 |
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12 |
A |
9 |
7 |
1.00 |
28 |
0.250 |
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13 |
A |
15 |
10 |
1.00 |
28 |
0.357 |
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14 |
A |
22 |
11 |
1.00 |
28 |
0.393 |
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15 |
A |
6 |
3 |
1.00 |
26 |
0.115 |
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16 |
A |
2 |
2 |
1.00 |
27 |
0.074 |
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17 |
A |
6 |
4 |
1.00 |
33 |
0.121 |
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18 |
A |
5 |
4 |
1.00 |
33 |
0.121 |
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19 |
A |
4 |
4 |
1.00 |
31 |
0.129 |
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20 |
A |
3 |
3 |
1.00 |
24 |
0.125 |
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21 |
A |
3 |
3 |
1.00 |
33 |
0.091 |
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22 |
A |
6 |
6 |
1.00 |
33 |
0.182 |
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23 |
A |
7 |
6 |
1.00 |
33 |
0.182 |
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24 |
A |
5 |
5 |
1.00 |
33 |
0.152 |
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25 |
A |
4 |
4 |
1.01 |
33 |
0.121 |
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26 |
A |
4 |
4 |
1.00 |
31 |
0.129 |
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27 |
A |
4 |
4 |
1.00 |
24 |
0.167 |
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28 |
A |
9 |
7 |
1.21 |
33 |
0.212 |
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29 |
A |
15 |
10 |
1.62 |
33 |
0.303 |
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30 |
A |
1 |
1 |
1.00 |
41 |
0.024 |
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31 |
A |
1 |
1 |
1.00 |
41 |
0.024 |
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32 |
A |
4 |
4 |
1.00 |
31 |
0.129 |
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33 |
A |
6 |
3 |
1.00 |
31 |
0.097 |
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34 |
A |
11 |
7 |
1.00 |
28 |
0.250 |
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35 |
A |
4 |
3 |
1.00 |
30 |
0.100 |
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36 |
A |
1 |
1 |
1.00 |
30 |
0.033 |
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37 |
A |
6 |
6 |
1.00 |
38 |
0.158 |
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38 |
A |
6 |
6 |
1.00 |
37 |
0.162 |
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39 |
A |
8 |
7 |
1.00 |
40 |
0.175 |
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40 |
A |
8 |
7 |
1.00 |
40 |
0.175 |
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41 |
A |
8 |
7 |
1.00 |
40 |
0.175 |
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42 |
A |
8 |
7 |
1.00 |
40 |
0.175 |
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