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 | 16 | 16 | 11 | 10 | 18 | 20 | 22 | 19 | 8 |
normalized size | 1 | 1. | 0.69 | 0.62 | 1.12 | 1.25 | 1.38 | 1.19 | 0.5 |
time (sec) |
N/A |
0.023 |
0.014 |
0.016 |
1.521 |
0.241 |
0.328 |
0.203 |
1.003 |
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Problem 2 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
A |
B |
A |
F(-2) |
A |
F |
F |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 38 | 46 | 13 | 0 | 65 | 0 | 0 | 0 |
normalized size | 1 | 2. | 2.42 | 0.68 | 0. | 3.42 | 0. | 0. | 0. |
time (sec) |
N/A |
6. |
0.047 |
0.146 |
0. |
0.24 |
0. |
0. |
0. |
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Problem 3 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
B |
A |
A |
A |
A |
A |
A |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 42 | 19 | 35 | 72 | 58 | 32 | 49 | 0 |
normalized size | 1 | 2.21 | 1. | 1.84 | 3.79 | 3.05 | 1.68 | 2.58 | 0. |
time (sec) |
N/A |
1.01 |
0.059 |
0.105 |
1.595 |
0.247 |
2.922 |
0.222 |
0. |
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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 |
F |
A |
F(-1) |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 19 | 31 | 19 | 26 | 53 | 55 | 0 | 35 | 0 |
normalized size | 1 | 1.63 | 1. | 1.37 | 2.79 | 2.89 | 0. | 1.84 | 0. |
time (sec) |
N/A |
3.373 |
0.047 |
0.149 |
1.544 |
0.236 |
0. |
0.214 |
0. |
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Problem 5 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
F |
A |
A |
F(-1) |
A |
F(-1) |
F |
F(-1) |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 43 | 0 | 61 | 17 | 0 | 42 | 0 | 0 | 0 |
normalized size | 1 | 0. | 1.42 | 0.4 | 0. | 0.98 | 0. | 0. | 0. |
time (sec) |
N/A |
1.344 |
0.092 |
0.085 |
0. |
0.232 |
0. |
0. |
0. |
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Problem 6 |
Optimal |
Rubi |
Mathematica |
Maple |
Maxima |
Fricas |
Sympy |
Giac |
Rubi in Sympy |
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grade |
A |
F |
B |
A |
F(-1) |
A |
F(-1) |
F |
F(-1) |
verified |
N/A |
N/A |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 25 | 0 | 63 | 19 | 0 | 50 | 0 | 0 | 0 |
normalized size | 1 | 0. | 2.52 | 0.76 | 0. | 2. | 0. | 0. | 0. |
time (sec) |
N/A |
1.304 |
0.103 |
0.086 |
0. |
0.264 |
0. |
0. |
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 |
B |
A |
A |
A |
A |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 14 | 14 | 79 | 12 | 20 | 18 | 27 | 34 | 14 |
normalized size | 1 | 1. | 5.64 | 0.86 | 1.43 | 1.29 | 1.93 | 2.43 | 1. |
time (sec) |
N/A |
0.018 |
0.016 |
0.019 |
1.478 |
0.235 |
0.427 |
0.202 |
1.16 |
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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 | 34 | 34 | 15 | 13 | 16 | 36 | 60 | 61 | 15 |
normalized size | 1 | 1. | 0.44 | 0.38 | 0.47 | 1.06 | 1.76 | 1.79 | 0.44 |
time (sec) |
N/A |
0.033 |
0.016 |
0.019 |
1.508 |
0.233 |
1.557 |
0.2 |
0.926 |
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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 |
F(-2) |
A |
F(-1) |
A |
A |
verified |
N/A |
Yes |
Yes |
TBD |
TBD |
TBD |
TBD |
TBD |
TBD |
size | 54 | 54 | 50 | 53 | 0 | 1 | 0 | 97 | 39 |
normalized size | 1 | 1. | 0.93 | 0.98 | 0. | 0.02 | 0. | 1.8 | 0.72 |
time (sec) |
N/A |
0.119 |
0.112 |
0.072 |
0. |
0.263 |
0. |
0.206 |
3.799 |
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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 [2] had the largest ratio of [ 0.4444
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| Table 1: Rubi specific breakdown of results for each integral
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# |
grade |
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normalized | antiderivative |
leaf size |
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\(\frac{\text{number of rules}}{\text{integrand leaf size}}\) |
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1 |
A |
2 |
2 |
1. |
10 |
0.2 |
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2 |
A |
43 |
12 |
2. |
27 |
0.444 |
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3 |
B |
25 |
7 |
2.21 |
31 |
0.226 |
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4 |
A |
32 |
9 |
1.63 |
35 |
0.257 |
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5 |
F |
0 |
0 |
N/A |
0 |
N/A |
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6 |
F |
0 |
0 |
N/A |
0 |
N/A |
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7 |
A |
2 |
2 |
1. |
10 |
0.2 |
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8 |
A |
3 |
3 |
1. |
10 |
0.3 |
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9 |
A |
3 |
3 |
1. |
12 |
0.25 |
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