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 | B | B | F(-1) | F(-1) | B | B | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 284 | 275 | 273 | 3286 | 920 | 0 | 0 | 1530 | 6610 | 1183 |
| N.S. |
1 | 0.97 | 0.96 | 11.57 | 3.24 | 0.00 | 0.00 | 5.39 | 23.27 | 4.17 |
| time (sec) |
N/A | 0.618 | 1.125 | 99.204 | 0.105 | 0.000 | 0.000 | 0.561 | 0.306 | 30.990 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | B | B | F(-1) | F(-1) | B | B | B |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 389 | 372 | 366 | 5231 | 1912 | 0 | 0 | 3325 | 7048 | 2570 |
| N.S. |
1 | 0.96 | 0.94 | 13.45 | 4.92 | 0.00 | 0.00 | 8.55 | 18.12 | 6.61 |
| time (sec) |
N/A | 0.845 | 1.196 | 268.477 | 0.187 | 0.000 | 0.000 | 1.404 | 0.311 | 37.329 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | C | B | F | F(-2) | F(-1) | F | F(-1) |
| verified |
N/A | No | No | No | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 570 | 706 | 906 | 8446 | 1671 | 0 | 0 | 0 | 1563 | 0 |
| N.S. |
1 | 1.24 | 1.59 | 14.82 | 2.93 | 0.00 | 0.00 | 0.00 | 2.74 | 0.00 |
| time (sec) |
N/A | 1.370 | 1.829 | 107.758 | 0.598 | 0.000 | 0.000 | 0.000 | 0.288 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | C | B | F | F(-2) | F | F | F(-1) |
| verified |
N/A | No | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 294 | 410 | 472 | 4692 | 903 | 0 | 0 | 0 | 724 | 0 |
| N.S. |
1 | 1.39 | 1.61 | 15.96 | 3.07 | 0.00 | 0.00 | 0.00 | 2.46 | 0.00 |
| time (sec) |
N/A | 0.969 | 1.009 | 27.593 | 0.584 | 0.000 | 0.000 | 0.000 | 0.254 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | C | F | F | F(-2) | F | F | F(-1) |
| verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 142 | 130 | 217 | 2240 | 0 | 0 | 0 | 0 | 217 | 0 |
| N.S. |
1 | 0.92 | 1.53 | 15.77 | 0.00 | 0.00 | 0.00 | 0.00 | 1.53 | 0.00 |
| time (sec) |
N/A | 0.675 | 0.202 | 5.019 | 0.000 | 0.000 | 0.000 | 0.000 | 0.282 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-2) | F | F | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 301 | 367 | 1082 | 0 | 0 | 0 | 0 | 0 | 85 | 0 |
| N.S. |
1 | 1.22 | 3.59 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.28 | 0.00 |
| time (sec) |
N/A | 0.894 | 0.487 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.319 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 208 | 220 | 3460 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.06 | 16.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.648 | 1.058 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.594 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 393 | 517 | 15406 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.32 | 39.20 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.192 | 6.559 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.946 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-2) | F(-1) | F | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 875 | 1020 | 7279 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.17 | 8.32 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.847 | 4.824 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.395 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | B | F | F | F | F(-2) | F | F | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 466 | 568 | 3890 | 0 | 0 | 0 | 0 | 0 | 1305 | 0 |
| N.S. |
1 | 1.22 | 8.35 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 2.80 | 0.00 |
| time (sec) |
N/A | 1.191 | 1.547 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.358 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-2) | F | F | F(-1) |
| verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 203 | 174 | 378 | 0 | 0 | 0 | 0 | 0 | 361 | 0 |
| N.S. |
1 | 0.86 | 1.86 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.78 | 0.00 |
| time (sec) |
N/A | 0.725 | 0.328 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.345 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-2) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 425 | 509 | 0 | 0 | 0 | 0 | 0 | 0 | 125 | 0 |
| N.S. |
1 | 1.20 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.29 | 0.00 |
| time (sec) |
N/A | 1.065 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.554 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 302 | 283 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 0.94 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 0.884 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 3.016 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 629 | 734 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| N.S. |
1 | 1.17 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| time (sec) |
N/A | 1.528 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 22.722 | 0.000 |