Detailed conclusion table per each integral is given by the 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 as F.
In this table, the column N.S.
means normalized size and is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To make the table
fit the page, the name Mathematica
was abbreviated to MMA
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 71 | 71 | 71 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.08 | 0.00 |
time (sec) |
N/A | 0.301 | 0.102 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.165 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 9 | 9 | 9 | 9 | 9 | 8 | 7 | 10 | 9 | 8 |
N.S. |
1 | 1.00 | 1.00 | 1.00 | 1.00 | 0.89 | 0.78 | 1.11 | 1.00 | 0.89 |
time (sec) |
N/A | 0.305 | 0.121 | 0.093 | 0.116 | 0.083 | 0.227 | 0.138 | 0.150 | 24.250 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | A | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 35 | 35 | 28 | 26 | 0 | 28 | 65 | 0 | 26 | 25 |
N.S. |
1 | 1.00 | 0.80 | 0.74 | 0.00 | 0.80 | 1.86 | 0.00 | 0.74 | 0.71 |
time (sec) |
N/A | 0.438 | 0.426 | 0.227 | 0.000 | 0.073 | 0.714 | 0.000 | 0.156 | 0.112 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | A | B | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 58 | 60 | 38 | 36 | 0 | 46 | 155 | 0 | 41 | 53 |
N.S. |
1 | 1.03 | 0.66 | 0.62 | 0.00 | 0.79 | 2.67 | 0.00 | 0.71 | 0.91 |
time (sec) |
N/A | 0.583 | 0.613 | 0.660 | 0.000 | 0.071 | 1.813 | 0.000 | 0.145 | 0.140 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | A | F | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 27 | 27 | 25 | 23 | 0 | 30 | 0 | 0 | 29 | 25 |
N.S. |
1 | 1.00 | 0.93 | 0.85 | 0.00 | 1.11 | 0.00 | 0.00 | 1.07 | 0.93 |
time (sec) |
N/A | 0.336 | 0.156 | 0.070 | 0.000 | 0.078 | 0.000 | 0.000 | 0.170 | 0.138 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | A | F | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 55 | 55 | 51 | 35 | 0 | 50 | 0 | 0 | 46 | 42 |
N.S. |
1 | 1.00 | 0.93 | 0.64 | 0.00 | 0.91 | 0.00 | 0.00 | 0.84 | 0.76 |
time (sec) |
N/A | 0.485 | 0.278 | 0.066 | 0.000 | 0.078 | 0.000 | 0.000 | 0.157 | 24.197 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | A | F(-1) | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 82 | 90 | 95 | 45 | 0 | 68 | 0 | 0 | 61 | 58 |
N.S. |
1 | 1.10 | 1.16 | 0.55 | 0.00 | 0.83 | 0.00 | 0.00 | 0.74 | 0.71 |
time (sec) |
N/A | 0.663 | 0.344 | 0.072 | 0.000 | 0.083 | 0.000 | 0.000 | 0.177 | 0.160 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 147 | 147 | 89 | 0 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.00 | 0.61 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.18 | 0.00 |
time (sec) |
N/A | 0.732 | 0.250 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.166 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 147 | 147 | 89 | 0 | 0 | 0 | 0 | 0 | 26 | 0 |
N.S. |
1 | 1.00 | 0.61 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.18 | 0.00 |
time (sec) |
N/A | 0.757 | 0.272 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.176 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 207 | 207 | 89 | 0 | 0 | 0 | 0 | 0 | 47 | 0 |
N.S. |
1 | 1.00 | 0.43 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.23 | 0.00 |
time (sec) |
N/A | 1.011 | 0.260 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.163 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 207 | 207 | 89 | 0 | 0 | 0 | 0 | 0 | 47 | 0 |
N.S. |
1 | 1.00 | 0.43 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.23 | 0.00 |
time (sec) |
N/A | 1.037 | 0.274 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.174 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 272 | 269 | 100 | 0 | 0 | 0 | 0 | 0 | 66 | 0 |
N.S. |
1 | 0.99 | 0.37 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.24 | 0.00 |
time (sec) |
N/A | 1.362 | 0.286 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.183 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | C | C | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 46 | 0 | 100 | 59 | 37 | 77 | 17 | 0 |
N.S. |
1 | 0.00 | 46.00 | 0.00 | 100.00 | 59.00 | 37.00 | 77.00 | 17.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.068 | 0.000 | 0.043 | 0.067 | 0.240 | 0.138 | 0.164 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | C | C | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 34 | 0 | 43 | 25 | 20 | 21 | 17 | 0 |
N.S. |
1 | 0.00 | 34.00 | 0.00 | 43.00 | 25.00 | 20.00 | 21.00 | 17.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.028 | 0.000 | 0.038 | 0.078 | 0.106 | 0.135 | 0.162 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | C | C | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 34 | 0 | 43 | 25 | 20 | 21 | 19 | 0 |
N.S. |
1 | 0.00 | 34.00 | 0.00 | 43.00 | 25.00 | 20.00 | 21.00 | 19.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.027 | 0.000 | 0.043 | 0.069 | 0.105 | 0.118 | 0.150 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | C | C | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 46 | 0 | 100 | 59 | 37 | 77 | 19 | 0 |
N.S. |
1 | 0.00 | 46.00 | 0.00 | 100.00 | 59.00 | 37.00 | 77.00 | 19.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.059 | 0.000 | 0.047 | 0.068 | 0.241 | 0.138 | 0.174 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | F | F | F | F(-1) |
verified |
N/A | N/A | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 82 | 0 | 96 | 125 | 0 | 0 | 17 | 0 |
N.S. |
1 | 0.00 | 82.00 | 0.00 | 96.00 | 125.00 | 0.00 | 0.00 | 17.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.160 | 0.000 | 0.072 | 0.088 | 0.000 | 0.000 | 0.158 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | F | F | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 65 | 0 | 61 | 79 | 0 | 0 | 16 | 0 |
N.S. |
1 | 0.00 | 65.00 | 0.00 | 61.00 | 79.00 | 0.00 | 0.00 | 16.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.119 | 0.000 | 0.056 | 0.078 | 0.000 | 0.000 | 0.154 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | F | F | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 65 | 0 | 61 | 79 | 0 | 0 | 18 | 0 |
N.S. |
1 | 0.00 | 65.00 | 0.00 | 61.00 | 79.00 | 0.00 | 0.00 | 18.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.114 | 0.000 | 0.201 | 0.075 | 0.000 | 0.000 | 0.155 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | F | C | F | C | C | F | F | F | F(-1) |
verified |
N/A | N/A | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1 | 0 | 82 | 0 | 96 | 79 | 0 | 0 | 19 | 0 |
N.S. |
1 | 0.00 | 82.00 | 0.00 | 96.00 | 79.00 | 0.00 | 0.00 | 19.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.162 | 0.000 | 0.194 | 0.079 | 0.000 | 0.000 | 0.174 | 0.000 |