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 | Mupad | |
| grade |
N/A | A | C | C | F | F(-1) | F(-1) | F(-1) | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 1668 | 1668 | 223 | 130 | 0 | 0 | 0 | 0 | 359169 |
| N.S. |
1 | 1.00 | 0.13 | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 215.33 |
| time (sec) |
N/A | 3.717 | 0.739 | 29.016 | 0.000 | 0.000 | 0.000 | 0.000 | 52.377 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 124 | 133 | 261 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.07 | 2.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.233 | 0.303 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 263 | 263 | 525 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 2.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.508 | 1.110 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 404 | 387 | 834 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.96 | 2.06 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.601 | 0.920 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 545 | 511 | 1093 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.94 | 2.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.710 | 1.283 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 283 | 277 | 456 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.98 | 1.61 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.564 | 2.677 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 738 | 738 | 4162 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 5.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.446 | 6.421 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 1194 | 1194 | 6525 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 5.46 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.901 | 6.546 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 1654 | 1654 | 8737 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 5.28 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 2.605 | 6.725 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | A | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 75 | 75 | 84 | 0 | 0 | 137 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 1.12 | 0.00 | 0.00 | 1.83 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.379 | 2.147 | 0.000 | 0.000 | 0.270 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | A | A | A | B | B | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 20 | 20 | 19 | 33 | 35 | 35 | 63 | 66 | 35 |
| N.S. |
1 | 1.00 | 0.95 | 1.65 | 1.75 | 1.75 | 3.15 | 3.30 | 1.75 |
| time (sec) |
N/A | 0.187 | 0.584 | 5.155 | 0.247 | 0.267 | 29.898 | 0.374 | 8.880 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | A | F(-1) | A | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 45 | 45 | 45 | 0 | 0 | 48 | 0 | 39 | 39 |
| N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 1.07 | 0.00 | 0.87 | 0.87 |
| time (sec) |
N/A | 0.292 | 0.715 | 0.000 | 0.000 | 0.249 | 0.000 | 5.158 | 9.081 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | A | C | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 65 | 65 | 64 | 0 | 0 | 69 | 160 | 0 | 0 |
| N.S. |
1 | 1.00 | 0.98 | 0.00 | 0.00 | 1.06 | 2.46 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.460 | 0.673 | 0.000 | 0.000 | 0.256 | 117.269 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | F | F | A | F(-1) | B | B |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 75 | 75 | 84 | 0 | 0 | 109 | 0 | 187 | 80 |
| N.S. |
1 | 1.00 | 1.12 | 0.00 | 0.00 | 1.45 | 0.00 | 2.49 | 1.07 |
| time (sec) |
N/A | 0.322 | 0.078 | 0.000 | 0.000 | 0.266 | 0.000 | 1.001 | 8.877 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | A | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 95 | 95 | 242 | 0 | 0 | 132 | 0 | 0 | 0 |
| N.S. |
1 | 1.00 | 2.55 | 0.00 | 0.00 | 1.39 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.511 | 3.459 | 0.000 | 0.000 | 0.277 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | A | B | B | B | F(-1) | B | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
| size | 29 | 29 | 24 | 90 | 60 | 65 | 0 | 96 | 50 |
| N.S. |
1 | 1.00 | 0.83 | 3.10 | 2.07 | 2.24 | 0.00 | 3.31 | 1.72 |
| time (sec) |
N/A | 0.260 | 1.978 | 102.389 | 0.262 | 0.260 | 0.000 | 0.437 | 8.729 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
| size | 494 | 477 | 5439 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.97 | 11.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.180 | 8.364 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |