2 detailed summary tables of results
2.1 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 |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 18 | 14 | 22 | 65 | 19 | 46 | 14 |
normalized size | 1 | 1. | 1.5 | 1.17 | 1.83 | 5.42 | 1.58 | 3.83 | 1.17 |
time (sec) |
N/A |
0.017 |
0.005 |
0. |
1.49 |
0.221 |
0.052 |
0.21 |
0.502 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 13 | 12 | 20 | 26 | 14 | 20 | 12 |
normalized size | 1 | 1. | 1. | 0.92 | 1.54 | 2. | 1.08 | 1.54 | 0.92 |
time (sec) |
N/A |
0.017 |
0.006 |
0.007 |
1.484 |
0.21 |
0.117 |
0.202 |
2.248 |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 14 | 11 | 15 | 19 | 15 | 15 | 15 |
normalized size | 1 | 1. | 0.74 | 0.58 | 0.79 | 1. | 0.79 | 0.79 | 0.79 |
time (sec) |
N/A |
0.008 |
0.004 |
0.003 |
1.331 |
0.211 |
0.272 |
0.198 |
1.025 |
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Problem 4 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2 | 2 | 2 | 3 | 3 | 3 | 2 | 3 | 2 |
normalized size | 1 | 1. | 1. | 1.5 | 1.5 | 1.5 | 1. | 1.5 | 1. |
time (sec) |
N/A |
0.005 |
0.002 |
0.001 |
1.372 |
0.208 |
0.03 |
0.2 |
0.023 |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 9 | 9 | 9 | 7 | 8 | 8 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 0.78 | 0.89 | 0.89 | 0.56 | 0.89 | 0.56 |
time (sec) |
N/A |
0.01 |
0.002 |
0. |
1.37 |
0.205 |
0.06 |
0.2 |
1.036 |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 8 | 7 | 8 | 0 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 1. | 0.88 | 1. | 0. |
time (sec) |
N/A |
0.017 |
0.003 |
0.009 |
1.326 |
0.224 |
0.051 |
0.198 |
0. |
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Problem 7 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 13 | 10 | 12 | 93 | 22 | 12 | 8 |
normalized size | 1 | 1. | 1. | 0.77 | 0.92 | 7.15 | 1.69 | 0.92 | 0.62 |
time (sec) |
N/A |
0.005 |
0.003 |
0.003 |
1.34 |
0.234 |
0.216 |
0.204 |
0.763 |
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Problem 8 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 14 | 14 | 15 | 18 | 15 | 15 | 15 |
normalized size | 1 | 1. | 0.74 | 0.74 | 0.79 | 0.95 | 0.79 | 0.79 | 0.79 |
time (sec) |
N/A |
0.013 |
0.016 |
0.003 |
1.353 |
0.226 |
0.358 |
0.2 |
1.193 |
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Problem 9 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 19 | 8 | 8 | 8 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 2.38 | 1. | 1. | 1. |
time (sec) |
N/A |
0.021 |
0.004 |
0.018 |
1.357 |
0.22 |
0.051 |
0.201 |
1.111 |
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Problem 10 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
F |
F |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 4 | 4 | 4 | 4 | 20 | 0 | 0 | 4 | 3 |
normalized size | 1 | 1. | 1. | 1. | 5. | 0. | 0. | 1. | 0.75 |
time (sec) |
N/A |
0.019 |
0.009 |
0.004 |
1.52 |
0. |
0. |
0.202 |
1.36 |
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Problem 11 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
F |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2 | 2 | 2 | 3 | 18 | 0 | 2 | 3 | 2 |
normalized size | 1 | 1. | 1. | 1.5 | 9. | 0. | 1. | 1.5 | 1. |
time (sec) |
N/A |
0.018 |
0.008 |
0.001 |
1.487 |
0. |
0.829 |
0.199 |
0.697 |
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Problem 12 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 8 | 9 | 9 | 5 | 9 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.12 | 1.12 | 0.62 | 1.12 | 0.62 |
time (sec) |
N/A |
0.008 |
0.005 |
0.001 |
1.355 |
0.236 |
0.059 |
0.198 |
0.476 |
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Problem 13 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 7 | 7 | 7 | 7 | 8 | 8 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.14 | 1.14 | 0.71 | 1.14 | 0.71 |
time (sec) |
N/A |
0.032 |
0.004 |
0.003 |
1.349 |
0.202 |
0.069 |
0.209 |
1.617 |
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Problem 14 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 26 | 22 | 26 | 26 | 20 | 26 | 0 |
normalized size | 1 | 1. | 0.93 | 0.79 | 0.93 | 0.93 | 0.71 | 0.93 | 0. |
time (sec) |
N/A |
0.037 |
0.009 |
0.003 |
1.377 |
0.203 |
0.094 |
0.214 |
0. |
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Problem 15 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 22 | 17 | 22 | 22 | 15 | 22 | 15 |
normalized size | 1 | 1. | 1. | 0.77 | 1. | 1. | 0.68 | 1. | 0.68 |
time (sec) |
N/A |
0.011 |
0.004 |
0.004 |
1.392 |
0.214 |
0.083 |
0.211 |
0.637 |
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Problem 16 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 8 | 5 | 8 | 0 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 1. | 0.62 | 1. | 0. |
time (sec) |
N/A |
0.012 |
0.002 |
0. |
1.344 |
0.226 |
0.034 |
0.215 |
0. |
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Problem 17 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 9 | 9 | 9 | 7 | 8 | 8 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 0.78 | 0.89 | 0.89 | 0.56 | 0.89 | 0.56 |
time (sec) |
N/A |
0.012 |
0.001 |
0. |
1.352 |
0.212 |
0.059 |
0.224 |
1.061 |
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Problem 18 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 13 | 10 | 12 | 93 | 22 | 12 | 8 |
normalized size | 1 | 1. | 1. | 0.77 | 0.92 | 7.15 | 1.69 | 0.92 | 0.62 |
time (sec) |
N/A |
0.005 |
0.001 |
0. |
1.333 |
0.244 |
0.23 |
0.222 |
0.759 |
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Problem 19 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 6 | 6 | 6 | 6 | 7 | 7 | 5 | 7 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.17 | 1.17 | 0.83 | 1.17 | 0.83 |
time (sec) |
N/A |
0.026 |
0.002 |
0.003 |
1.328 |
0.198 |
0.058 |
0.212 |
2.742 |
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Problem 20 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 9 | 9 | 9 | 6 | 7 | 7 | 7 | 7 | 7 |
normalized size | 1 | 1. | 1. | 0.67 | 0.78 | 0.78 | 0.78 | 0.78 | 0.78 |
time (sec) |
N/A |
0.003 |
0.001 |
0.003 |
1.335 |
0.203 |
0.028 |
0.201 |
0.454 |
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Problem 21 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 10 | 10 | 10 | 9 | 11 | 11 | 7 | 11 | 7 |
normalized size | 1 | 1. | 1. | 0.9 | 1.1 | 1.1 | 0.7 | 1.1 | 0.7 |
time (sec) |
N/A |
0.007 |
0.003 |
0.007 |
1.366 |
0.215 |
0.122 |
0.198 |
0.515 |
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Problem 22 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 8 | 9 | 9 | 7 | 9 | 7 |
normalized size | 1 | 1. | 1. | 1. | 1.12 | 1.12 | 0.88 | 1.12 | 0.88 |
time (sec) |
N/A |
0.007 |
0.001 |
0.003 |
1.347 |
0.234 |
0.072 |
0.2 |
0.839 |
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Problem 23 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 10 | 10 | 19 | 8 | 9 | 9 | 8 | 9 | 8 |
normalized size | 1 | 1. | 1.9 | 0.8 | 0.9 | 0.9 | 0.8 | 0.9 | 0.8 |
time (sec) |
N/A |
0.039 |
0.017 |
0.007 |
1.372 |
0.231 |
2.943 |
0.198 |
6.09 |
|
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Problem 24 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 23 | 16 | 13 | 20 | 20 | 34 | 20 | 19 |
normalized size | 1 | 1. | 0.7 | 0.57 | 0.87 | 0.87 | 1.48 | 0.87 | 0.83 |
time (sec) |
N/A |
0.012 |
0.005 |
0.004 |
1.34 |
0.205 |
1.449 |
0.198 |
0.968 |
|
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Problem 25 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 25 | 10 | 23 | 23 | 17 | 26 | 10 |
normalized size | 1 | 1. | 1.92 | 0.77 | 1.77 | 1.77 | 1.31 | 2. | 0.77 |
time (sec) |
N/A |
0.008 |
0.005 |
0.001 |
1.528 |
0.205 |
0.162 |
0.199 |
0.62 |
|
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Problem 26 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 18 | 18 | 18 | 14 | 18 | 18 | 15 | 18 | 17 |
normalized size | 1 | 1. | 1. | 0.78 | 1. | 1. | 0.83 | 1. | 0.94 |
time (sec) |
N/A |
0.036 |
0.013 |
0.005 |
1.55 |
0.207 |
0.113 |
0.207 |
3.154 |
|
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Problem 27 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F(-2) |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 31 | 31 | 31 | 20 | 0 | 1 | 22 | 26 | 27 |
normalized size | 1 | 1. | 1. | 0.65 | 0. | 0.03 | 0.71 | 0.84 | 0.87 |
time (sec) |
N/A |
0.059 |
0.012 |
0.01 |
0. |
0.215 |
0.171 |
0.203 |
4.474 |
|
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Problem 28 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 9 | 11 | 15 | 7 | 11 | 7 |
normalized size | 1 | 1. | 1. | 1.12 | 1.38 | 1.88 | 0.88 | 1.38 | 0.88 |
time (sec) |
N/A |
0.049 |
0.002 |
0.003 |
1.355 |
0.222 |
0.08 |
0.2 |
2.739 |
|
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Problem 29 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 12 | 13 | 16 | 16 | 8 | 16 | 8 |
normalized size | 1 | 1. | 1. | 1.08 | 1.33 | 1.33 | 0.67 | 1.33 | 0.67 |
time (sec) |
N/A |
0.011 |
0.005 |
0.02 |
1.345 |
0.222 |
0.074 |
0.199 |
0.487 |
|
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Problem 30 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 20 | 17 | 22 | 22 | 15 | 22 | 15 |
normalized size | 1 | 1. | 1. | 0.85 | 1.1 | 1.1 | 0.75 | 1.1 | 0.75 |
time (sec) |
N/A |
0.027 |
0.005 |
0. |
1.335 |
0.229 |
0.776 |
0.198 |
1.413 |
|
|
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Problem 31 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 49 | 49 | 260 | 41 | 54 | 63 | 46 | 54 | 42 |
normalized size | 1 | 1. | 5.31 | 0.84 | 1.1 | 1.29 | 0.94 | 1.1 | 0.86 |
time (sec) |
N/A |
0.076 |
0.198 |
0.001 |
1.535 |
0.21 |
0.258 |
0.208 |
3.955 |
|
|
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Problem 32 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
C |
F |
F |
F(-2) |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 115 | 115 | 142 | 41 | 0 | 0 | 0 | 0 | 87 |
normalized size | 1 | 1. | 1.23 | 0.36 | 0. | 0. | 0. | 0. | 0.76 |
time (sec) |
N/A |
0.099 |
0.457 |
0.149 |
0. |
0. |
0. |
0. |
5.274 |
|
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|
Problem 33 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 22 | 17 | 22 | 22 | 20 | 22 | 48 |
normalized size | 1 | 1. | 1. | 0.77 | 1. | 1. | 0.91 | 1. | 2.18 |
time (sec) |
N/A |
0.018 |
0.01 |
0.012 |
1.339 |
0.24 |
0.405 |
0.2 |
2.596 |
|
|
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Problem 34 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 23 | 16 | 13 | 20 | 20 | 34 | 20 | 19 |
normalized size | 1 | 1. | 0.7 | 0.57 | 0.87 | 0.87 | 1.48 | 0.87 | 0.83 |
time (sec) |
N/A |
0.011 |
0.005 |
0. |
1.34 |
0.209 |
1.445 |
0.199 |
0.981 |
|
|
|
|
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|
Problem 35 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 32 | 92 | 32 | 32 | 0 | 32 | 0 |
normalized size | 1 | 1. | 1. | 2.88 | 1. | 1. | 0. | 1. | 0. |
time (sec) |
N/A |
0.027 |
0.013 |
0.001 |
1.352 |
0.207 |
0. |
0.206 |
0. |
|
|
|
|
|
|
|
|
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|
Problem 36 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
F |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 44 | 44 | 66 | 75 | 108 | 82 | 0 | 82 | 48 |
normalized size | 1 | 1. | 1.5 | 1.7 | 2.45 | 1.86 | 0. | 1.86 | 1.09 |
time (sec) |
N/A |
0.033 |
0.063 |
0.014 |
1.519 |
0.21 |
0. |
0.215 |
1.928 |
|
|
|
|
|
|
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|
Problem 37 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 26 | 30 | 59 | 182 | 105 | 39 | 26 |
normalized size | 1 | 1. | 0.74 | 0.86 | 1.69 | 5.2 | 3. | 1.11 | 0.74 |
time (sec) |
N/A |
0.028 |
0.045 |
0.009 |
1.5 |
0.392 |
3.934 |
0.207 |
2.5 |
|
|
|
|
|
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|
|
Problem 38 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 75 | 75 | 41 | 70 | 153 | 266 | 190 | 111 | 66 |
normalized size | 1 | 1. | 0.55 | 0.93 | 2.04 | 3.55 | 2.53 | 1.48 | 0.88 |
time (sec) |
N/A |
0.037 |
0.04 |
0.006 |
1.348 |
0.214 |
39.165 |
0.29 |
2.669 |
|
|
|
|
|
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|
Problem 39 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 26 | 30 | 59 | 182 | 105 | 39 | 26 |
normalized size | 1 | 1. | 0.74 | 0.86 | 1.69 | 5.2 | 3. | 1.11 | 0.74 |
time (sec) |
N/A |
0.028 |
0.011 |
0. |
1.493 |
0.214 |
4.139 |
0.207 |
2.478 |
|
|
|
|
|
|
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|
Problem 40 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
A |
F |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 51 | 51 | 134 | 262 | 0 | 174 | 0 | 0 | 46 |
normalized size | 1 | 1. | 2.63 | 5.14 | 0. | 3.41 | 0. | 0. | 0.9 |
time (sec) |
N/A |
0.077 |
0.032 |
0.035 |
0. |
0.25 |
0. |
0. |
12.011 |
|
|
|
|
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|
Problem 41 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 11 | 14 | 14 | 10 | 14 | 10 |
normalized size | 1 | 1. | 1. | 0.79 | 1. | 1. | 0.71 | 1. | 0.71 |
time (sec) |
N/A |
0.01 |
0.003 |
0. |
1.339 |
0.221 |
0.036 |
0.236 |
0.488 |
|
|
|
|
|
|
|
|
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|
Problem 42 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
C |
F(-2) |
A |
F(-1) |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 49 | 57 | 99 | 149 | 0 | 378 | 0 | 0 | 54 |
normalized size | 1 | 1.16 | 2.02 | 3.04 | 0. | 7.71 | 0. | 0. | 1.1 |
time (sec) |
N/A |
0.159 |
0.132 |
0.156 |
0. |
0.33 |
0. |
0. |
15.247 |
|
|
|
|
|
|
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|
|
Problem 43 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 9 | 9 | 6 | 5 | 12 | 12 | 3 | 41 | 7 |
normalized size | 1 | 1. | 0.67 | 0.56 | 1.33 | 1.33 | 0.33 | 4.56 | 0.78 |
time (sec) |
N/A |
0.015 |
0.006 |
0. |
1.425 |
0.211 |
0.212 |
0.206 |
0.483 |
|
|
|
|
|
|
|
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|
|
Problem 44 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 7 | 7 | 8 | 8 | 5 | 8 | 7 |
normalized size | 1 | 1. | 0.64 | 0.64 | 0.73 | 0.73 | 0.45 | 0.73 | 0.64 |
time (sec) |
N/A |
0.011 |
0.002 |
0. |
1.4 |
0.228 |
0.058 |
0.226 |
1.018 |
|
|
|
|
|
|
|
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|
|
Problem 45 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
F |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 9 | 9 | 9 | 9 | 11 | 11 | 5 | 0 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.22 | 1.22 | 0.56 | 0. | 0.56 |
time (sec) |
N/A |
0.038 |
0.005 |
0.005 |
1.349 |
0.202 |
0.076 |
0. |
2.003 |
|
|
|
|
|
|
|
|
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|
|
Problem 46 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 7 | 7 | 7 | 7 | 8 | 8 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.14 | 1.14 | 0.71 | 1.14 | 0.71 |
time (sec) |
N/A |
0.047 |
0.004 |
0.003 |
1.358 |
0.199 |
0.068 |
0.209 |
2.778 |
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Problem 47 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
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|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 11 | 8 | 12 | 9 | 8 | 12 | 8 |
normalized size | 1 | 1. | 1. | 0.73 | 1.09 | 0.82 | 0.73 | 1.09 | 0.73 |
time (sec) |
N/A |
0.005 |
0.002 |
0.002 |
1.36 |
0.219 |
0.301 |
0.215 |
0.495 |
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Problem 48 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2 | 2 | 2 | 8 | 3 | 3 | 2 | 3 | 2 |
normalized size | 1 | 1. | 1. | 4. | 1.5 | 1.5 | 1. | 1.5 | 1. |
time (sec) |
N/A |
0.014 |
0.002 |
0.003 |
1.421 |
0.212 |
1.271 |
0.226 |
1.32 |
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Problem 49 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
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|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 41 | 41 | 40 | 35 | 46 | 55 | 41 | 47 | 37 |
normalized size | 1 | 1. | 0.98 | 0.85 | 1.12 | 1.34 | 1. | 1.15 | 0.9 |
time (sec) |
N/A |
0.044 |
0.013 |
0. |
1.486 |
0.208 |
0.173 |
0.224 |
3.185 |
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Problem 50 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
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|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 47 | 73 | 75 | 66 | 88 | 101 | 83 | 90 | 68 |
normalized size | 1 | 1.55 | 1.6 | 1.4 | 1.87 | 2.15 | 1.77 | 1.91 | 1.45 |
time (sec) |
N/A |
0.189 |
0.021 |
0. |
1.527 |
0.203 |
0.358 |
0.225 |
16.996 |
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Problem 51 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 44 | 53 | 36 | 70 | 55 | 12 |
normalized size | 1 | 1. | 1. | 2.1 | 2.52 | 1.71 | 3.33 | 2.62 | 0.57 |
time (sec) |
N/A |
0.02 |
0.006 |
0.009 |
1.349 |
0.198 |
0.4 |
0.209 |
1.311 |
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Problem 52 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 14 | 18 | 18 | 12 | 18 | 0 |
normalized size | 1 | 1. | 1. | 0.82 | 1.06 | 1.06 | 0.71 | 1.06 | 0. |
time (sec) |
N/A |
0.008 |
0.001 |
0.002 |
12.108 |
0.202 |
0.07 |
0.233 |
0. |
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Problem 53 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 40 | 40 | 29 | 34 | 45 | 32 | 32 | 51 | 27 |
normalized size | 1 | 1. | 0.72 | 0.85 | 1.12 | 0.8 | 0.8 | 1.27 | 0.68 |
time (sec) |
N/A |
0.049 |
0.02 |
0. |
1.498 |
0.221 |
0.461 |
0.212 |
3.12 |
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Problem 54 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 7 | 7 | 7 | 8 | 9 | 9 | 5 | 9 | 15 |
normalized size | 1 | 1. | 1. | 1.14 | 1.29 | 1.29 | 0.71 | 1.29 | 2.14 |
time (sec) |
N/A |
0.005 |
0.001 |
0.001 |
1.327 |
0.208 |
0.062 |
0.227 |
0.561 |
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Problem 55 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 9 | 11 | 11 | 5 | 11 | 0 |
normalized size | 1 | 1. | 1. | 1.12 | 1.38 | 1.38 | 0.62 | 1.38 | 0. |
time (sec) |
N/A |
0.063 |
0.005 |
0.051 |
1.451 |
0.217 |
0.077 |
0.227 |
0. |
|
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Problem 56 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 3 | 3 | 3 | 4 | 4 | 4 | 15 | 4 | 3 |
normalized size | 1 | 1. | 1. | 1.33 | 1.33 | 1.33 | 5. | 1.33 | 1. |
time (sec) |
N/A |
0.028 |
0.006 |
0.005 |
1.509 |
0.221 |
0.142 |
0.232 |
3.264 |
|
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Problem 57 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
B |
A |
F |
F |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2 | 2 | 2 | 9 | 4 | 0 | 0 | 4 | 2 |
normalized size | 1 | 1. | 1. | 4.5 | 2. | 0. | 0. | 2. | 1. |
time (sec) |
N/A |
0.004 |
0.003 |
0. |
1.415 |
0. |
0. |
0.214 |
0.026 |
|
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Problem 58 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 15 | 19 | 19 | 15 | 19 | 0 |
normalized size | 1 | 1. | 1. | 1.07 | 1.36 | 1.36 | 1.07 | 1.36 | 0. |
time (sec) |
N/A |
0.032 |
0.008 |
0.016 |
1.358 |
0.209 |
0.191 |
0.222 |
0. |
|
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Problem 59 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 13 | 16 | 15 | 19 | 8 | 15 | 0 |
normalized size | 1 | 1. | 0.76 | 0.94 | 0.88 | 1.12 | 0.47 | 0.88 | 0. |
time (sec) |
N/A |
0.05 |
0.01 |
0.003 |
1.352 |
0.22 |
0.07 |
0.229 |
0. |
|
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Problem 60 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 38 | 38 | 29 | 29 | 32 | 32 | 26 | 32 | 0 |
normalized size | 1 | 1. | 0.76 | 0.76 | 0.84 | 0.84 | 0.68 | 0.84 | 0. |
time (sec) |
N/A |
0.049 |
0.016 |
0.002 |
1.357 |
0.213 |
0.084 |
0.214 |
0. |
|
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Problem 61 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 7 | 7 | 7 | 8 | 9 | 9 | 7 | 9 | 7 |
normalized size | 1 | 1. | 1. | 1.14 | 1.29 | 1.29 | 1. | 1.29 | 1. |
time (sec) |
N/A |
0.015 |
0.004 |
0. |
1.341 |
0.219 |
0.181 |
0.206 |
0.767 |
|
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Problem 62 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 22 | 22 | 22 | 17 | 22 | 22 | 20 | 22 | 48 |
normalized size | 1 | 1. | 1. | 0.77 | 1. | 1. | 0.91 | 1. | 2.18 |
time (sec) |
N/A |
0.017 |
0.01 |
0. |
1.345 |
0.23 |
0.401 |
0.229 |
2.612 |
|
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Problem 63 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 7 | 7 | 7 | 8 | 9 | 9 | 7 | 9 | 7 |
normalized size | 1 | 1. | 1. | 1.14 | 1.29 | 1.29 | 1. | 1.29 | 1. |
time (sec) |
N/A |
0.014 |
0.003 |
0. |
1.313 |
0.233 |
0.181 |
0.226 |
0.784 |
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Problem 64 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 28 | 28 | 28 | 23 | 23 | 30 | 22 | 30 | 0 |
normalized size | 1 | 1. | 1. | 0.82 | 0.82 | 1.07 | 0.79 | 1.07 | 0. |
time (sec) |
N/A |
0.016 |
0.002 |
0. |
1.346 |
0.225 |
0.093 |
0.21 |
0. |
|
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Problem 65 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 11 | 11 | 11 | 10 | 12 | 20 | 8 | 12 | 8 |
normalized size | 1 | 1. | 1. | 0.91 | 1.09 | 1.82 | 0.73 | 1.09 | 0.73 |
time (sec) |
N/A |
0.019 |
0.005 |
0.023 |
1.344 |
0.238 |
0.816 |
0.207 |
1.17 |
|
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Problem 66 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 3 | 3 | 3 | 4 | 4 | 4 | 15 | 4 | 3 |
normalized size | 1 | 1. | 1. | 1.33 | 1.33 | 1.33 | 5. | 1.33 | 1. |
time (sec) |
N/A |
0.029 |
0.005 |
0.002 |
1.497 |
0.223 |
0.141 |
0.206 |
3.182 |
|
|
|
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Problem 67 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
F |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 3 | 3 | 3 | 4 | 0 | 4 | 3 | 4 | 3 |
normalized size | 1 | 1. | 1. | 1.33 | 0. | 1.33 | 1. | 1.33 | 1. |
time (sec) |
N/A |
0.075 |
0.007 |
0.009 |
0. |
0.23 |
0.714 |
0.212 |
4.847 |
|
|
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Problem 68 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 16 | 21 | 72 | 20 | 12 | 28 | 12 |
normalized size | 1 | 1. | 1. | 1.31 | 4.5 | 1.25 | 0.75 | 1.75 | 0.75 |
time (sec) |
N/A |
0.058 |
0.011 |
0.036 |
1.502 |
0.226 |
0.157 |
0.232 |
4.644 |
|
|
|
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Problem 69 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
C |
B |
F(-2) |
A |
F |
F(-1) |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 53 | 53 | 127 | 262 | 0 | 173 | 0 | 0 | 48 |
normalized size | 1 | 1. | 2.4 | 4.94 | 0. | 3.26 | 0. | 0. | 0.91 |
time (sec) |
N/A |
0.129 |
0.086 |
0.016 |
0. |
0.243 |
0. |
0. |
14.425 |
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Problem 70 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
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|
|
|
grade |
A |
A |
B |
B |
A |
A |
F(-1) |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 35 | 127 | 93 | 90 | 0 | 112 | 17 |
normalized size | 1 | 1. | 2.19 | 7.94 | 5.81 | 5.62 | 0. | 7. | 1.06 |
time (sec) |
N/A |
0.136 |
0.09 |
0.056 |
1.509 |
0.271 |
0. |
0.257 |
14.036 |
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Problem 71 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
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|
|
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|
|
|
grade |
A |
A |
B |
B |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 35 | 121 | 92 | 35 | 422 | 111 | 0 |
normalized size | 1 | 1. | 2.19 | 7.56 | 5.75 | 2.19 | 26.38 | 6.94 | 0. |
time (sec) |
N/A |
0.199 |
0.028 |
0.012 |
1.506 |
0.214 |
2.324 |
0.222 |
0. |
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Problem 72 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
B |
B |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 35 | 121 | 88 | 35 | 422 | 107 | 14 |
normalized size | 1 | 1. | 2.19 | 7.56 | 5.5 | 2.19 | 26.38 | 6.69 | 0.88 |
time (sec) |
N/A |
0.041 |
0.02 |
0.01 |
1.505 |
0.216 |
2.275 |
0.209 |
9.155 |
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Problem 73 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 26 | 30 | 59 | 182 | 105 | 39 | 26 |
normalized size | 1 | 1. | 0.74 | 0.86 | 1.69 | 5.2 | 3. | 1.11 | 0.74 |
time (sec) |
N/A |
0.029 |
0.044 |
0. |
1.479 |
0.212 |
4.076 |
0.207 |
2.561 |
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Problem 74 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 18 | 13 | 16 | 35 | 19 | 16 | 19 |
normalized size | 1 | 1. | 1.29 | 0.93 | 1.14 | 2.5 | 1.36 | 1.14 | 1.36 |
time (sec) |
N/A |
0.014 |
0.01 |
0.011 |
1.503 |
0.229 |
0.047 |
0.207 |
2.583 |
|
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Problem 75 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 13 | 13 | 13 | 12 | 15 | 15 | 8 | 15 | 8 |
normalized size | 1 | 1. | 1. | 0.92 | 1.15 | 1.15 | 0.62 | 1.15 | 0.62 |
time (sec) |
N/A |
0.015 |
0.006 |
0.002 |
1.496 |
0.199 |
0.075 |
0.203 |
1.564 |
|
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Problem 76 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 7 | 7 | 7 | 7 | 8 | 8 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.14 | 1.14 | 0.71 | 1.14 | 0.71 |
time (sec) |
N/A |
0.046 |
0.004 |
0. |
1.332 |
0.209 |
0.068 |
0.205 |
2.783 |
|
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Problem 77 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F(-2) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 19 | 24 | 31 | 31 | 20 | 41 | 0 |
normalized size | 1 | 1. | 0.79 | 1. | 1.29 | 1.29 | 0.83 | 1.71 | 0. |
time (sec) |
N/A |
0.493 |
0.021 |
0.007 |
1.631 |
0.236 |
0.116 |
0.204 |
0. |
|
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Problem 78 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 9 | 9 | 9 | 9 | 11 | 11 | 7 | 11 | 7 |
normalized size | 1 | 1. | 1. | 1. | 1.22 | 1.22 | 0.78 | 1.22 | 0.78 |
time (sec) |
N/A |
0.006 |
0.002 |
0.001 |
1.339 |
0.231 |
0.053 |
0.203 |
0.522 |
|
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Problem 79 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 20 | 32 | 26 | 19 | 26 | 0 |
normalized size | 1 | 1. | 1. | 0.8 | 1.28 | 1.04 | 0.76 | 1.04 | 0. |
time (sec) |
N/A |
0.048 |
0.002 |
0.006 |
1.33 |
0.212 |
0.081 |
0.205 |
0. |
|
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Problem 80 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 8 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 1. | 0.62 | 1. | 0.62 |
time (sec) |
N/A |
0.009 |
0.005 |
0.003 |
1.54 |
0.204 |
0.088 |
0.205 |
1.064 |
|
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Problem 81 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 16 | 16 | 16 | 13 | 16 | 16 | 10 | 16 | 10 |
normalized size | 1 | 1. | 1. | 0.81 | 1. | 1. | 0.62 | 1. | 0.62 |
time (sec) |
N/A |
0.018 |
0.005 |
0.003 |
1.529 |
0.197 |
0.09 |
0.203 |
2.225 |
|
|
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Problem 82 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 17 | 22 | 27 | 36 | 22 | 10 |
normalized size | 1 | 1. | 1. | 1.21 | 1.57 | 1.93 | 2.57 | 1.57 | 0.71 |
time (sec) |
N/A |
0.019 |
0.004 |
0.011 |
1.48 |
0.217 |
0.725 |
0.209 |
0.564 |
|
|
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Problem 83 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 26 | 30 | 59 | 182 | 105 | 39 | 26 |
normalized size | 1 | 1. | 0.74 | 0.86 | 1.69 | 5.2 | 3. | 1.11 | 0.74 |
time (sec) |
N/A |
0.03 |
0.012 |
0. |
1.49 |
0.224 |
4.099 |
0.209 |
2.543 |
|
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Problem 84 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 16 | 20 | 86 | 34 | 28 | 12 |
normalized size | 1 | 1. | 1. | 0.94 | 1.18 | 5.06 | 2. | 1.65 | 0.71 |
time (sec) |
N/A |
0.017 |
0.024 |
0.007 |
1.485 |
0.243 |
0.97 |
0.209 |
1.668 |
|
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Problem 85 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 14 | 14 | 15 | 18 | 15 | 15 | 15 |
normalized size | 1 | 1. | 0.74 | 0.74 | 0.79 | 0.95 | 0.79 | 0.79 | 0.79 |
time (sec) |
N/A |
0.014 |
0.016 |
0. |
1.341 |
0.257 |
0.365 |
0.203 |
1.207 |
|
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Problem 86 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 2 | 2 | 2 | 3 | 3 | 3 | 2 | 4 | 2 |
normalized size | 1 | 1. | 1. | 1.5 | 1.5 | 1.5 | 1. | 2. | 1. |
time (sec) |
N/A |
0.002 |
0. |
0. |
1.332 |
0.227 |
0.025 |
0.201 |
0.022 |
|
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|
Problem 87 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 45 | 45 | 73 | 31 | 346 | 96 | 0 | 45 | 0 |
normalized size | 1 | 1. | 1.62 | 0.69 | 7.69 | 2.13 | 0. | 1. | 0. |
time (sec) |
N/A |
0.198 |
0.254 |
0.194 |
1.695 |
0.272 |
0. |
0.22 |
0. |
|
|
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Problem 88 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 14 | 11 | 14 | 14 | 10 | 22 | 10 |
normalized size | 1 | 1. | 1. | 0.79 | 1. | 1. | 0.71 | 1.57 | 0.71 |
time (sec) |
N/A |
0.011 |
0.003 |
0. |
1.359 |
0.247 |
0.044 |
0.209 |
0.493 |
|
|
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|
Problem 89 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 14 | 11 | 15 | 14 | 14 | 15 | 14 |
normalized size | 1 | 1. | 0.82 | 0.65 | 0.88 | 0.82 | 0.82 | 0.88 | 0.82 |
time (sec) |
N/A |
0.007 |
0.004 |
0.006 |
1.324 |
0.232 |
0.27 |
0.203 |
0.977 |
|
|
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|
Problem 90 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 23 | 23 | 23 | 20 | 26 | 107 | 0 | 36 | 31 |
normalized size | 1 | 1. | 1. | 0.87 | 1.13 | 4.65 | 0. | 1.57 | 1.35 |
time (sec) |
N/A |
0.023 |
0.013 |
0.008 |
1.496 |
0.233 |
0. |
0.213 |
1.722 |
|
|
|
|
|
|
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|
|
Problem 91 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 14 | 5 | 8 | 5 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 1.75 | 0.62 | 1. | 0.62 |
time (sec) |
N/A |
0.019 |
0.002 |
0.003 |
1.339 |
0.273 |
0.039 |
0.201 |
1.093 |
|
|
|
|
|
|
|
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|
|
Problem 92 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 6 | 6 | 6 | 6 | 7 | 7 | 5 | 7 | 5 |
normalized size | 1 | 1. | 1. | 1. | 1.17 | 1.17 | 0.83 | 1.17 | 0.83 |
time (sec) |
N/A |
0.026 |
0.002 |
0. |
1.363 |
0.22 |
0.06 |
0.204 |
2.757 |
|
|
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Problem 93 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 12 | 11 | 14 | 14 | 8 | 14 | 8 |
normalized size | 1 | 1. | 1. | 0.92 | 1.17 | 1.17 | 0.67 | 1.17 | 0.67 |
time (sec) |
N/A |
0.036 |
0.005 |
0.001 |
1.369 |
0.234 |
0.076 |
0.204 |
3.397 |
|
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Problem 94 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 12 | 12 | 8 | 9 | 14 | 14 | 7 | 11 | 8 |
normalized size | 1 | 1. | 0.67 | 0.75 | 1.17 | 1.17 | 0.58 | 0.92 | 0.67 |
time (sec) |
N/A |
0.015 |
0.008 |
0.003 |
1.33 |
0.223 |
0.656 |
0.212 |
0.511 |
|
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Problem 95 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 8 | 7 | 8 | 0 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 1. | 0.88 | 1. | 0. |
time (sec) |
N/A |
0.019 |
0.003 |
0. |
1.373 |
0.256 |
0.052 |
0.202 |
0. |
|
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Problem 96 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 17 | 17 | 17 | 14 | 18 | 18 | 12 | 18 | 0 |
normalized size | 1 | 1. | 1. | 0.82 | 1.06 | 1.06 | 0.71 | 1.06 | 0. |
time (sec) |
N/A |
0.008 |
0.001 |
0.001 |
1.339 |
0.232 |
0.069 |
0.201 |
0. |
|
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Problem 97 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 8 | 8 | 8 | 7 | 8 | 8 | 5 | 8 | 0 |
normalized size | 1 | 1. | 1. | 0.88 | 1. | 1. | 0.62 | 1. | 0. |
time (sec) |
N/A |
0.012 |
0.002 |
0. |
1.337 |
0.221 |
0.033 |
0.201 |
0. |
|
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Problem 98 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
F |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 45 | 21 | 30 | 43 | 0 | 27 | 15 |
normalized size | 1 | 1. | 1.88 | 0.88 | 1.25 | 1.79 | 0. | 1.12 | 0.62 |
time (sec) |
N/A |
0.024 |
0.027 |
0.008 |
1.503 |
0.248 |
0. |
0.212 |
1.901 |
|
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Problem 99 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 20 | 14 | 18 | 26 | 15 | 18 | 17 |
normalized size | 1 | 1. | 1. | 0.7 | 0.9 | 1.3 | 0.75 | 0.9 | 0.85 |
time (sec) |
N/A |
0.038 |
0.014 |
0.003 |
1.485 |
0.219 |
0.115 |
0.21 |
3.228 |
|
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Problem 100 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 35 | 35 | 26 | 30 | 59 | 182 | 105 | 39 | 26 |
normalized size | 1 | 1. | 0.74 | 0.86 | 1.69 | 5.2 | 3. | 1.11 | 0.74 |
time (sec) |
N/A |
0.029 |
0.041 |
0. |
1.495 |
0.246 |
4.03 |
0.209 |
2.551 |
|
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Problem 101 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 20 | 20 | 20 | 15 | 19 | 19 | 24 | 19 | 14 |
normalized size | 1 | 1. | 1. | 0.75 | 0.95 | 0.95 | 1.2 | 0.95 | 0.7 |
time (sec) |
N/A |
0.04 |
0.009 |
0.006 |
1.484 |
0.222 |
0.12 |
0.201 |
4.071 |
|
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Problem 102 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 33 | 33 | 33 | 27 | 35 | 54 | 31 | 35 | 31 |
normalized size | 1 | 1. | 1. | 0.82 | 1.06 | 1.64 | 0.94 | 1.06 | 0.94 |
time (sec) |
N/A |
0.024 |
0.019 |
0. |
1.486 |
0.222 |
0.14 |
0.206 |
1.897 |
|
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Problem 103 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 21 | 21 | 21 | 20 | 26 | 26 | 15 | 26 | 0 |
normalized size | 1 | 1. | 1. | 0.95 | 1.24 | 1.24 | 0.71 | 1.24 | 0. |
time (sec) |
N/A |
0.03 |
0. |
0.002 |
1.344 |
0.227 |
0.034 |
0.203 |
0. |
|
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Problem 104 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 26 | 26 | 26 | 25 | 32 | 32 | 19 | 31 | 0 |
normalized size | 1 | 1. | 1. | 0.96 | 1.23 | 1.23 | 0.73 | 1.19 | 0. |
time (sec) |
N/A |
0.036 |
0. |
0.001 |
1.343 |
0.202 |
0.037 |
0.201 |
0. |
|
|
|
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Problem 105 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 27 | 26 | 34 | 34 | 22 | 34 | 0 |
normalized size | 1 | 1. | 1. | 0.96 | 1.26 | 1.26 | 0.81 | 1.26 | 0. |
time (sec) |
N/A |
0.04 |
0. |
0.001 |
1.337 |
0.226 |
0.037 |
0.211 |
0. |
|
|
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Problem 106 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 32 | 32 | 32 | 31 | 41 | 41 | 26 | 39 | 0 |
normalized size | 1 | 1. | 1. | 0.97 | 1.28 | 1.28 | 0.81 | 1.22 | 0. |
time (sec) |
N/A |
0.047 |
0. |
0.001 |
1.348 |
0.201 |
0.04 |
0.2 |
0. |
|
|
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Problem 107 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 25 | 24 | 31 | 31 | 20 | 31 | 0 |
normalized size | 1 | 1. | 1.04 | 1. | 1.29 | 1.29 | 0.83 | 1.29 | 0. |
time (sec) |
N/A |
0.052 |
0. |
0.002 |
1.368 |
0.203 |
0.037 |
0.206 |
0. |
|
|
|
|
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Problem 108 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 29 | 29 | 30 | 29 | 38 | 38 | 24 | 36 | 0 |
normalized size | 1 | 1. | 1.03 | 1. | 1.31 | 1.31 | 0.83 | 1.24 | 0. |
time (sec) |
N/A |
0.059 |
0. |
0.002 |
1.364 |
0.203 |
0.04 |
0.208 |
0. |
|
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Problem 109 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 19 | 19 | 18 | 23 | 23 | 15 | 23 | 0 |
normalized size | 1 | 1. | 1. | 0.95 | 1.21 | 1.21 | 0.79 | 1.21 | 0. |
time (sec) |
N/A |
0.019 |
0. |
0.002 |
1.439 |
0.199 |
0.033 |
0.203 |
0. |
|
|
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Problem 110 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 24 | 24 | 24 | 23 | 30 | 30 | 19 | 28 | 0 |
normalized size | 1 | 1. | 1. | 0.96 | 1.25 | 1.25 | 0.79 | 1.17 | 0. |
time (sec) |
N/A |
0.023 |
0. |
0.002 |
1.345 |
0.207 |
0.036 |
0.203 |
0. |
|
|
|
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|
Problem 111 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 25 | 25 | 24 | 31 | 31 | 22 | 31 | 0 |
normalized size | 1 | 1. | 1. | 0.96 | 1.24 | 1.24 | 0.88 | 1.24 | 0. |
time (sec) |
N/A |
0.025 |
0. |
0.001 |
1.361 |
0.206 |
0.036 |
0.207 |
0. |
|
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|
Problem 112 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 30 | 30 | 30 | 29 | 38 | 38 | 26 | 36 | 0 |
normalized size | 1 | 1. | 1. | 0.97 | 1.27 | 1.27 | 0.87 | 1.2 | 0. |
time (sec) |
N/A |
0.03 |
0. |
0.002 |
1.351 |
0.202 |
0.04 |
0.207 |
0. |
|
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Problem 113 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
|
|
|
|
|
|
|
|
|
|
grade |
A |
A |
A |
A |
A |
A |
A |
A |
F |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 27 | 27 | 28 | 27 | 35 | 35 | 24 | 34 | 0 |
normalized size | 1 | 1. | 1.04 | 1. | 1.3 | 1.3 | 0.89 | 1.26 | 0. |
time (sec) |
N/A |
0.018 |
0. |
0.001 |
1.422 |
0.198 |
0.039 |
0.208 |
0. |
|
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|
2.2 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 [50] had the largest ratio of [ 0.8571
]
|
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|
|
| Table 1: Rubi specific breakdown of results for each integral
|
|
|
|
|
|
|
# |
grade |
|
|
normalized | antiderivative |
leaf size |
|
|
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
|
|
|
|
|
|
|
|
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|
|
1 |
A |
3 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
2 |
A |
3 |
2 |
1. |
11 |
0.182 |
|
|
|
|
|
|
|
3 |
A |
2 |
1 |
1. |
11 |
0.091 |
|
|
|
|
|
|
|
4 |
A |
1 |
1 |
1. |
2 |
0.5 |
|
|
|
|
|
|
|
5 |
A |
1 |
1 |
1. |
7 |
0.143 |
|
|
|
|
|
|
|
6 |
A |
2 |
2 |
1. |
7 |
0.286 |
|
|
|
|
|
|
|
7 |
A |
1 |
1 |
1. |
11 |
0.091 |
|
|
|
|
|
|
|
8 |
A |
1 |
1 |
1. |
6 |
0.167 |
|
|
|
|
|
|
|
9 |
A |
2 |
2 |
1. |
7 |
0.286 |
|
|
|
|
|
|
|
10 |
A |
2 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
11 |
A |
1 |
1 |
1. |
6 |
0.167 |
|
|
|
|
|
|
|
12 |
A |
3 |
2 |
1. |
6 |
0.333 |
|
|
|
|
|
|
|
13 |
A |
4 |
2 |
1. |
16 |
0.125 |
|
|
|
|
|
|
|
14 |
A |
5 |
3 |
1. |
7 |
0.429 |
|
|
|
|
|
|
|
15 |
A |
3 |
1 |
1. |
14 |
0.071 |
|
|
|
|
|
|
|
16 |
A |
2 |
2 |
1. |
5 |
0.4 |
|
|
|
|
|
|
|
17 |
A |
1 |
1 |
1. |
7 |
0.143 |
|
|
|
|
|
|
|
18 |
A |
1 |
1 |
1. |
11 |
0.091 |
|
|
|
|
|
|
|
19 |
A |
2 |
2 |
1. |
11 |
0.182 |
|
|
|
|
|
|
|
20 |
A |
1 |
1 |
1. |
5 |
0.2 |
|
|
|
|
|
|
|
|
21 | A | 1 | 1 | 1. | 6 | 0.167 |
|
|
|
|
|
|
|
22 |
A |
2 |
2 |
1. |
9 |
0.222 |
|
|
|
|
|
|
|
23 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
24 |
A |
2 |
1 |
1. |
9 |
0.111 |
|
|
|
|
|
|
|
25 |
A |
3 |
3 |
1. |
7 |
0.429 |
|
|
|
|
|
|
|
26 |
A |
2 |
2 |
1. |
15 |
0.133 |
|
|
|
|
|
|
|
27 |
A |
2 |
2 |
1. |
17 |
0.118 |
|
|
|
|
|
|
|
28 |
A |
3 |
3 |
1. |
13 |
0.231 |
|
|
|
|
|
|
|
29 |
A |
1 |
1 |
1. |
5 |
0.2 |
|
|
|
|
|
|
|
30 |
A |
3 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
31 |
A |
7 |
7 |
1. |
11 |
0.636 |
|
|
|
|
|
|
|
32 |
A |
3 |
2 |
1. |
12 |
0.167 |
|
|
|
|
|
|
|
33 |
A |
3 |
3 |
1. |
6 |
0.5 |
|
|
|
|
|
|
|
34 |
A |
2 |
1 |
1. |
9 |
0.111 |
|
|
|
|
|
|
|
35 |
A |
4 |
3 |
1. |
13 |
0.231 |
|
|
|
|
|
|
|
36 |
A |
4 |
4 |
1. |
15 |
0.267 |
|
|
|
|
|
|
|
37 |
A |
3 |
2 |
1. |
15 |
0.133 |
|
|
|
|
|
|
|
38 |
A |
6 |
3 |
1. |
13 |
0.231 |
|
|
|
|
|
|
|
39 |
A |
3 |
2 |
1. |
15 |
0.133 |
|
|
|
|
|
|
|
40 |
A |
5 |
5 |
1. |
29 |
0.172 |
|
|
|
|
|
|
|
|
41 |
A |
2 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
42 |
A |
6 |
6 |
1.16 |
19 |
0.316 |
|
|
|
|
|
|
|
43 |
A |
1 |
1 |
1. |
6 |
0.167 |
|
|
|
|
|
|
|
44 |
A |
2 |
2 |
1. |
5 |
0.4 |
|
|
|
|
|
|
|
45 |
A |
1 |
1 |
1. |
10 |
0.1 |
|
|
|
|
|
|
|
46 |
A |
5 |
3 |
1. |
13 |
0.231 |
|
|
|
|
|
|
|
47 |
A |
1 |
1 |
1. |
5 |
0.2 |
|
|
|
|
|
|
|
48 |
A |
1 |
1 |
1. |
7 |
0.143 |
|
|
|
|
|
|
|
49 |
A |
6 |
6 |
1. |
9 |
0.667 |
|
|
|
|
|
|
|
50 |
A |
10 |
6 |
1.55 |
7 |
0.857 |
|
|
|
|
|
|
|
51 |
A |
1 |
1 |
1. |
27 |
0.037 |
|
|
|
|
|
|
|
52 |
A |
1 |
1 |
1. |
4 |
0.25 |
|
|
|
|
|
|
|
53 |
A |
4 |
3 |
1. |
6 |
0.5 |
|
|
|
|
|
|
|
54 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
55 |
A |
7 |
4 |
1. |
9 |
0.444 |
|
|
|
|
|
|
|
56 |
A |
2 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
57 |
A |
1 |
1 |
1. |
4 |
0.25 |
|
|
|
|
|
|
|
58 |
A |
6 |
4 |
1. |
7 |
0.571 |
|
|
|
|
|
|
|
59 |
A |
4 |
3 |
1. |
11 |
0.273 |
|
|
|
|
|
|
|
60 |
A |
6 |
3 |
1. |
9 |
0.333 |
|
|
|
|
|
|
|
|
61 |
A |
2 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
62 |
A |
3 |
3 |
1. |
6 |
0.5 |
|
|
|
|
|
|
|
63 |
A |
2 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
64 |
A |
2 |
2 |
1. |
6 |
0.333 |
|
|
|
|
|
|
|
65 |
A |
2 |
1 |
1. |
9 |
0.111 |
|
|
|
|
|
|
|
66 |
A |
2 |
1 |
1. |
12 |
0.083 |
|
|
|
|
|
|
|
67 |
A |
2 |
2 |
1. |
20 |
0.1 |
|
|
|
|
|
|
|
68 |
A |
2 |
2 |
1. |
10 |
0.2 |
|
|
|
|
|
|
|
69 |
A |
6 |
6 |
1. |
30 |
0.2 |
|
|
|
|
|
|
|
70 |
A |
5 |
5 |
1. |
39 |
0.128 |
|
|
|
|
|
|
|
71 |
A |
6 |
5 |
1. |
45 |
0.111 |
|
|
|
|
|
|
|
72 |
A |
4 |
4 |
1. |
31 |
0.129 |
|
|
|
|
|
|
|
73 |
A |
3 |
2 |
1. |
15 |
0.133 |
|
|
|
|
|
|
|
74 |
A |
3 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
75 |
A |
3 |
2 |
1. |
11 |
0.182 |
|
|
|
|
|
|
|
76 |
A |
5 |
3 |
1. |
13 |
0.231 |
|
|
|
|
|
|
|
77 |
A |
10 |
6 |
1. |
33 |
0.182 |
|
|
|
|
|
|
|
78 |
A |
1 |
1 |
1. |
7 |
0.143 |
|
|
|
|
|
|
|
79 |
A |
5 |
5 |
1. |
8 |
0.625 |
|
|
|
|
|
|
|
80 |
A |
2 |
2 |
1. |
9 |
0.222 |
|
|
|
|
|
|
|
|
81 | A | 3 | 3 | 1. | 11 | 0.273 |
|
|
|
|
|
|
|
82 |
A |
3 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
83 |
A |
3 |
2 |
1. |
15 |
0.133 |
|
|
|
|
|
|
|
84 |
A |
2 |
2 |
1. |
16 |
0.125 |
|
|
|
|
|
|
|
85 |
A |
1 |
1 |
1. |
6 |
0.167 |
|
|
|
|
|
|
|
86 |
A |
1 |
1 |
1. |
3 |
0.333 |
|
|
|
|
|
|
|
87 |
A |
4 |
2 |
1. |
17 |
0.118 |
|
|
|
|
|
|
|
88 |
A |
2 |
2 |
1. |
4 |
0.5 |
|
|
|
|
|
|
|
89 |
A |
2 |
1 |
1. |
11 |
0.091 |
|
|
|
|
|
|
|
90 |
A |
3 |
3 |
1. |
14 |
0.214 |
|
|
|
|
|
|
|
91 |
A |
2 |
2 |
1. |
7 |
0.286 |
|
|
|
|
|
|
|
92 |
A |
2 |
2 |
1. |
11 |
0.182 |
|
|
|
|
|
|
|
93 |
A |
3 |
2 |
1. |
13 |
0.154 |
|
|
|
|
|
|
|
94 |
A |
1 |
1 |
1. |
8 |
0.125 |
|
|
|
|
|
|
|
95 |
A |
2 |
2 |
1. |
7 |
0.286 |
|
|
|
|
|
|
|
96 |
A |
1 |
1 |
1. |
4 |
0.25 |
|
|
|
|
|
|
|
97 |
A |
2 |
2 |
1. |
5 |
0.4 |
|
|
|
|
|
|
|
98 |
A |
3 |
3 |
1. |
17 |
0.176 |
|
|
|
|
|
|
|
99 |
A |
3 |
3 |
1. |
16 |
0.188 |
|
|
|
|
|
|
|
100 |
A |
3 |
2 |
1. |
15 |
0.133 |
|
|
|
|
|
|
|
|
101 |
A |
3 |
3 |
1. |
15 |
0.2 |
|
|
|
|
|
|
|
102 |
A |
3 |
3 |
1. |
8 |
0.375 |
|
|
|
|
|
|
|
103 |
A |
1 |
1 |
1. |
20 |
0.05 |
|
|
|
|
|
|
|
104 |
A |
1 |
1 |
1. |
25 |
0.04 |
|
|
|
|
|
|
|
105 |
A |
1 |
1 |
1. |
26 |
0.038 |
|
|
|
|
|
|
|
106 |
A |
1 |
1 |
1. |
31 |
0.032 |
|
|
|
|
|
|
|
107 |
A |
1 |
1 |
1. |
24 |
0.042 |
|
|
|
|
|
|
|
108 |
A |
1 |
1 |
1. |
29 |
0.034 |
|
|
|
|
|
|
|
109 |
A |
1 |
1 |
1. |
18 |
0.056 |
|
|
|
|
|
|
|
110 |
A |
1 |
1 |
1. |
23 |
0.043 |
|
|
|
|
|
|
|
111 |
A |
1 |
1 |
1. |
24 |
0.042 |
|
|
|
|
|
|
|
112 |
A |
1 |
1 |
1. |
29 |
0.034 |
|
|
|
|
|
|
|
113 |
A |
1 |
1 |
1. |
27 |
0.037 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|