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
}
B grade: { }
C grade: { }
F grade: { }
2.1.2 Mathematica
A grade: { 27, 28, 29, 30, 31, 36, 37, 38, 39, 40, 47, 48, 49, 50 }
B grade: { }
C 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, 32, 33, 34,
35, 41, 42, 43, 44, 45, 46, 51 }
F grade: { }
2.1.3 Maple
A grade: { 29, 32, 33, 36, 37, 38, 41, 42, 43 }
B grade: { 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 45, 46, 47, 48, 51
}
C grade: { 7, 16 }
F grade: { 25, 26, 30, 31, 34, 35, 39, 40, 44, 49, 50 }
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, 43, 44, 45, 46, 47, 48, 49, 50, 51
}
2.1.5 FriCAS
A grade: { 27, 28, 29, 30, 31, 36, 37, 38, 39, 40 }
B grade: { 45, 46, 47, 48, 49, 50 }
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, 32, 33, 34,
35, 41, 42, 43, 44, 51 }
2.1.6 Sympy
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, 43, 44, 45, 46, 47, 48, 49, 50, 51
}
2.1.7 Giac
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, 43, 44, 45, 46, 47, 48, 49, 50, 51
}
2.1.8 Mupad
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, 43, 44, 45, 46, 47, 48, 49, 50, 51
}
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 |
B |
F |
F(-1) |
F |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 975 | 975 | 690 | 17767396 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.71 | 18222.97 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
24.463 |
6.246 |
2.127 |
0.000 |
0.000 |
0.000 |
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 |
B |
F |
F(-1) |
F |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 889 | 889 | 582 | 17247437 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.65 | 19400.94 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.995 |
5.412 |
0.971 |
0.000 |
0.000 |
0.000 |
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 |
B |
F |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 748 | 748 | 451 | 17766851 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 23752.47 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.630 |
2.527 |
0.997 |
0.000 |
0.000 |
0.000 |
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 |
B |
F |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 676 | 676 | 405 | 17246975 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 25513.28 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
26.571 |
0.642 |
0.946 |
0.000 |
0.000 |
0.000 |
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 |
B |
F |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 601 | 601 | 250 | 17766655 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.42 | 29561.82 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.288 |
0.307 |
0.778 |
0.000 |
0.000 |
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 |
B |
F(-2) |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 574 | 574 | 228 | 17246812 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.40 | 30046.71 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.269 |
0.198 |
0.831 |
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(-2) |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 571 | 571 | 223 | 53065 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.39 | 92.93 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.583 |
0.361 |
9.102 |
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 |
B |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 612 | 612 | 261 | 1165398 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.43 | 1904.25 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.744 |
1.219 |
27.166 |
0.000 |
0.000 |
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 |
B |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 690 | 690 | 289 | 2844525 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.42 | 4122.50 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
23.733 |
1.920 |
152.533 |
0.000 |
0.000 |
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 |
B |
F(-1) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 548 | 548 | 456 | 9581348 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.83 | 17484.21 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
1.085 |
6.135 |
0.844 |
0.000 |
0.000 |
0.000 |
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 |
B |
F(-1) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 495 | 495 | 283 | 7491876 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.57 | 15135.10 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.820 |
2.757 |
0.823 |
0.000 |
0.000 |
0.000 |
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 |
B |
F |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 383 | 383 | 252 | 9581103 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.66 | 25015.93 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.731 |
0.942 |
0.813 |
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 |
B |
F |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 352 | 352 | 228 | 7491751 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.65 | 21283.38 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.373 |
0.205 |
0.790 |
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 |
B |
F(-2) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 294 | 294 | 173 | 9338543 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.59 | 31763.75 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.286 |
0.115 |
0.834 |
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 |
C |
B |
F(-2) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 298 | 298 | 173 | 7300213 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.58 | 24497.36 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.311 |
0.141 |
0.786 |
0.000 |
0.000 |
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(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 350 | 350 | 223 | 28519 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.64 | 81.48 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.701 |
0.404 |
4.291 |
0.000 |
0.000 |
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(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 395 | 395 | 264 | 636822 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.67 | 1612.21 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.765 |
1.575 |
21.380 |
0.000 |
0.000 |
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(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 500 | 500 | 315 | 2045168 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.63 | 4090.34 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.800 |
5.976 |
52.267 |
0.000 |
0.000 |
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(-1) |
F(-1) |
F |
F(-2) |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1190 | 1190 | 884 | 13067521 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.74 | 10981.11 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
6.507 |
25.135 |
0.924 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 864 | 864 | 611 | 13066795 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.71 | 15123.61 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
4.790 |
21.301 |
0.890 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 686 | 686 | 286 | 13066487 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.42 | 19047.36 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
4.514 |
17.333 |
0.858 |
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(-2) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 638 | 638 | 328 | 11847956 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.51 | 18570.46 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
3.950 |
4.992 |
0.827 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 635 | 635 | 318 | 13066297 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.50 | 20576.85 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
3.791 |
4.751 |
0.805 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-1) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 750 | 750 | 450 | 15830985 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.60 | 21107.98 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
4.578 |
5.192 |
168.926 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F(-1) |
F(-1) |
F(-1) |
F |
F(-2) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 829 | 829 | 583 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.70 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
4.939 |
6.171 |
180.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F(-1) |
F(-1) |
F(-1) |
F |
F(-2) |
F(-1) |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1007 | 1007 | 786 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.78 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
4.928 |
6.208 |
180.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 270 | 270 | 467 | 684 | 0 | 1405 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.73 | 2.53 | 0.00 | 5.20 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.613 |
6.133 |
0.674 |
0.000 |
8.732 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 209 | 209 | 208 | 467 | 0 | 1199 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 2.23 | 0.00 | 5.74 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.349 |
1.566 |
0.435 |
0.000 |
6.366 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 179 | 179 | 180 | 289 | 0 | 1057 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.01 | 1.61 | 0.00 | 5.91 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.224 |
0.300 |
0.396 |
0.000 |
3.292 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 30 |
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 | 203 | 203 | 192 | 0 | 0 | 2097 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.95 | 0.00 | 0.00 | 10.33 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.292 |
1.178 |
1.494 |
0.000 |
1.953 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 31 |
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 | 435 | 435 | 187 | 0 | 0 | 1186 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.43 | 0.00 | 0.00 | 2.73 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.544 |
1.199 |
1.371 |
0.000 |
5.566 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F |
F |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 1254 | 1254 | 639 | 1945 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.51 | 1.55 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.993 |
29.608 |
0.417 |
0.000 |
1.672 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 33 |
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 | 829 | 829 | 428 | 1497 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.52 | 1.81 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.552 |
1.981 |
0.480 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 861 | 861 | 1258 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.46 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.585 |
27.042 |
1.374 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 943 | 943 | 1590 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.69 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.717 |
31.415 |
1.316 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-1) |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 182 | 182 | 173 | 240 | 0 | 1226 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.95 | 1.32 | 0.00 | 6.74 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.353 |
2.488 |
0.476 |
0.000 |
2.923 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 141 | 141 | 136 | 155 | 0 | 993 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 1.10 | 0.00 | 7.04 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.210 |
0.251 |
0.437 |
0.000 |
2.201 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 79 | 79 | 79 | 102 | 0 | 299 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.00 | 1.29 | 0.00 | 3.78 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.114 |
0.114 |
0.394 |
0.000 |
0.820 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F |
A |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 142 | 142 | 136 | 0 | 0 | 1015 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.96 | 0.00 | 0.00 | 7.15 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.228 |
0.752 |
1.733 |
0.000 |
1.924 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-1) |
A |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 249 | 249 | 188 | 0 | 0 | 1350 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.76 | 0.00 | 0.00 | 5.42 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.318 |
4.247 |
1.721 |
0.000 |
3.025 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
A |
F(-1) |
F(-1) |
F |
F |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 662 | 662 | 533 | 646 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.81 | 0.98 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.439 |
23.158 |
0.456 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
|
|
|
|
|
|
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Problem 42 |
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 | 311 | 402 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.71 | 0.92 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.341 |
11.188 |
0.424 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
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Problem 43 |
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 | 235 | 231 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.54 | 0.53 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.306 |
0.739 |
0.573 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
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Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
F |
F(-1) |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 707 | 707 | 683 | 0 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.97 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.728 |
28.157 |
1.475 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
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Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 235 | 235 | 182725 | 826 | 0 | 3773 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 777.55 | 3.51 | 0.00 | 16.06 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.549 |
34.860 |
0.552 |
0.000 |
6.310 |
0.000 |
0.000 |
0.000 |
|
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Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 159 | 159 | 57597 | 601 | 0 | 1095 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 362.25 | 3.78 | 0.00 | 6.89 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.394 |
36.240 |
0.446 |
0.000 |
1.277 |
0.000 |
0.000 |
0.000 |
|
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 154 | 154 | 155 | 508 | 0 | 1077 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.01 | 3.30 | 0.00 | 6.99 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.285 |
2.895 |
0.451 |
0.000 |
1.151 |
0.000 |
0.000 |
0.000 |
|
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
F |
B |
F |
F(-1) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 155 | 155 | 156 | 417 | 0 | 1099 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.01 | 2.69 | 0.00 | 7.09 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A |
0.218 |
2.978 |
0.380 |
0.000 |
1.063 |
0.000 |
0.000 |
0.000 |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-2) |
B |
F |
F(-2) |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 280 | 280 | 278 | 0 | 0 | 3951 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.99 | 0.00 | 0.00 | 14.11 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.387 |
3.449 |
1.549 |
0.000 |
7.467 |
0.000 |
0.000 |
0.000 |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
F |
F(-1) |
B |
F |
F(-1) |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 477 | 477 | 555 | 0 | 0 | 5189 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 1.16 | 0.00 | 0.00 | 10.88 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.552 |
6.069 |
1.580 |
0.000 |
9.152 |
0.000 |
0.000 |
0.000 |
|
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Mupad |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F |
F(-1) |
F |
F(-1) |
F |
verified |
N/A |
Yes |
NO |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 981 | 981 | 831 | 3598 | 0 | 0 | 0 | 0 | -1 |
normalized size | 1 | 1.00 | 0.85 | 3.67 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A |
0.876 |
34.453 |
0.560 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
|
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|
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 [1] had the largest ratio of
[.4242]
|
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|
| Table 2.1:Rubi specific breakdown of results for each
integral
|
|
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|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size | |
|
\(\frac {\text {number of rules}}{\text {integrand leaf size}}\) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
A |
21 |
14 |
1.00 |
33 |
0.424 |
|
|
|
|
|
|
|
2 |
A |
19 |
12 |
1.00 |
33 |
0.364 |
|
|
|
|
|
|
|
3 |
A |
16 |
12 |
1.00 |
33 |
0.364 |
|
|
|
|
|
|
|
4 |
A |
10 |
9 |
1.00 |
33 |
0.273 |
|
|
|
|
|
|
|
5 |
A |
10 |
9 |
1.00 |
31 |
0.290 |
|
|
|
|
|
|
|
6 |
A |
9 |
7 |
1.00 |
24 |
0.292 |
|
|
|
|
|
|
|
7 |
A |
18 |
13 |
1.00 |
31 |
0.419 |
|
|
|
|
|
|
|
8 |
A |
17 |
12 |
1.00 |
33 |
0.364 |
|
|
|
|
|
|
|
9 |
A |
21 |
14 |
1.00 |
33 |
0.424 |
|
|
|
|
|
|
|
10 |
A |
15 |
10 |
1.00 |
33 |
0.303 |
|
|
|
|
|
|
|
11 |
A |
14 |
9 |
1.00 |
33 |
0.273 |
|
|
|
|
|
|
|
12 |
A |
11 |
8 |
1.00 |
33 |
0.242 |
|
|
|
|
|
|
|
13 |
A |
9 |
7 |
1.00 |
33 |
0.212 |
|
|
|
|
|
|
|
14 |
A |
6 |
4 |
1.00 |
31 |
0.129 |
|
|
|
|
|
|
|
15 |
A |
6 |
3 |
1.00 |
24 |
0.125 |
|
|
|
|
|
|
|
16 |
A |
10 |
7 |
1.00 |
31 |
0.226 |
|
|
|
|
|
|
|
17 |
A |
11 |
8 |
1.00 |
33 |
0.242 |
|
|
|
|
|
|
|
18 |
A |
14 |
9 |
1.00 |
33 |
0.273 |
|
|
|
|
|
|
|
19 |
A |
20 |
12 |
1.00 |
33 |
0.364 |
|
|
|
|
|
|
|
20 |
A |
14 |
11 |
1.00 |
33 |
0.333 |
|
|
|
|
|
|
|
21 |
A |
10 |
7 |
1.00 |
33 |
0.212 |
|
|
|
|
|
|
|
22 |
A |
7 |
5 |
1.00 |
33 |
0.152 |
|
|
|
|
|
|
|
23 |
A |
7 |
5 |
1.00 |
31 |
0.161 |
|
|
|
|
|
|
|
24 |
A |
13 |
10 |
1.00 |
31 |
0.323 |
|
|
|
|
|
|
|
25 |
A |
13 |
10 |
1.00 |
33 |
0.303 |
|
|
|
|
|
|
|
26 |
A |
18 |
12 |
1.00 |
33 |
0.364 |
|
|
|
|
|
|
|
27 |
A |
9 |
8 |
1.00 |
35 |
0.229 |
|
|
|
|
|
|
|
28 |
A |
8 |
7 |
1.00 |
35 |
0.200 |
|
|
|
|
|
|
|
29 |
A |
8 |
7 |
1.00 |
33 |
0.212 |
|
|
|
|
|
|
|
30 |
A |
10 |
7 |
1.00 |
33 |
0.212 |
|
|
|
|
|
|
|
31 |
A |
22 |
9 |
1.00 |
35 |
0.257 |
|
|
|
|
|
|
|
32 |
A |
14 |
9 |
1.00 |
35 |
0.257 |
|
|
|
|
|
|
|
33 |
A |
8 |
6 |
1.00 |
26 |
0.231 |
|
|
|
|
|
|
|
34 |
A |
9 |
8 |
1.00 |
35 |
0.229 |
|
|
|
|
|
|
|
35 |
A |
10 |
8 |
1.00 |
35 |
0.229 |
|
|
|
|
|
|
|
36 |
A |
8 |
7 |
1.00 |
35 |
0.200 |
|
|
|
|
|
|
|
37 |
A |
7 |
6 |
1.00 |
35 |
0.171 |
|
|
|
|
|
|
|
38 |
A |
4 |
4 |
1.00 |
33 |
0.121 |
|
|
|
|
|
|
|
39 |
A |
8 |
5 |
1.00 |
33 |
0.152 |
|
|
|
|
|
|
|
40 |
A |
11 |
6 |
1.00 |
35 |
0.171 |
|
|
|
|
|
|
|
41 |
A |
5 |
5 |
1.00 |
35 |
0.143 |
|
|
|
|
|
|
|
42 |
A |
4 |
4 |
1.00 |
35 |
0.114 |
|
|
|
|
|
|
|
43 |
A |
4 |
3 |
1.00 |
26 |
0.115 |
|
|
|
|
|
|
|
44 |
A |
7 |
7 |
1.00 |
35 |
0.200 |
|
|
|
|
|
|
|
45 |
A |
8 |
7 |
1.00 |
35 |
0.200 |
|
|
|
|
|
|
|
46 |
A |
6 |
6 |
1.00 |
35 |
0.171 |
|
|
|
|
|
|
|
47 |
A |
6 |
6 |
1.00 |
35 |
0.171 |
|
|
|
|
|
|
|
48 |
A |
6 |
6 |
1.00 |
33 |
0.182 |
|
|
|
|
|
|
|
49 |
A |
12 |
7 |
1.00 |
33 |
0.212 |
|
|
|
|
|
|
|
50 |
A |
16 |
8 |
1.00 |
35 |
0.229 |
|
|
|
|
|
|
|
51 |
A |
9 |
8 |
1.00 |
35 |
0.229 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|