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 N.S. in the table below, which stands for normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To help make the table fit, Mathematica was abbreviated to MMA.
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | A | F | A | B |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 175 | 175 | 171 | 343 | 0 | 469 | 0 | 212 | 320 |
| N.S. |
1 | 1.00 | 0.98 | 1.96 | 0.00 | 2.68 | 0.00 | 1.21 | 1.83 |
| time (sec) |
N/A | 0.103 | 0.629 | 0.134 | 0.000 | 4.324 | 0.000 | 5.102 | 3.906 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | C | B | F(-2) | F(-1) | F | F(-2) | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 431 | 431 | 396 | 1547 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 0.92 | 3.59 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 0.706 | 0.499 | 0.222 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | C | B | F | F(-1) | F(-1) | F(-1) | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 488 | 488 | 972 | 4691 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 1.99 | 9.61 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 1.892 | 1.662 | 0.179 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | B | F | B | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 564 | 564 | 766 | 1654 | 0 | 2175 | 0 | 1150 | -1 |
| N.S. |
1 | 1.00 | 1.36 | 2.93 | 0.00 | 3.86 | 0.00 | 2.04 | -0.00 |
| time (sec) |
N/A | 0.606 | 5.080 | 0.149 | 0.000 | 4.584 | 0.000 | 6.533 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | A | F | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 236 | 236 | 290 | 499 | 0 | 849 | 0 | 417 | -1 |
| N.S. |
1 | 1.00 | 1.23 | 2.11 | 0.00 | 3.60 | 0.00 | 1.77 | -0.00 |
| time (sec) |
N/A | 0.144 | 1.216 | 0.136 | 0.000 | 5.340 | 0.000 | 7.859 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | C | B | F(-2) | F(-1) | F(-1) | F(-2) | F |
| verified |
N/A | NO | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 679 | 678 | 1150 | 2860 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 1.69 | 4.21 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 7.726 | 1.261 | 0.210 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | C | B | F | F(-1) | F(-1) | F(-2) | F |
| verified |
N/A | NO | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 704 | 704 | 2416 | 7856 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 3.43 | 11.16 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 7.553 | 3.794 | 0.155 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | B | B | F | F(-1) | F(-1) | F(-1) | F |
| verified |
N/A | NO | NO | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 671 | 669 | 1621 | 16309 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 2.42 | 24.31 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 11.091 | 16.304 | 0.161 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | A | F | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 717 | 717 | 615 | 2207 | 0 | 1569 | 0 | 824 | -1 |
| N.S. |
1 | 1.00 | 0.86 | 3.08 | 0.00 | 2.19 | 0.00 | 1.15 | -0.00 |
| time (sec) |
N/A | 1.712 | 2.787 | 0.148 | 0.000 | 3.698 | 0.000 | 3.672 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | A | F | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 316 | 316 | 251 | 714 | 0 | 629 | 0 | 304 | -1 |
| N.S. |
1 | 1.00 | 0.79 | 2.26 | 0.00 | 1.99 | 0.00 | 0.96 | -0.00 |
| time (sec) |
N/A | 0.400 | 0.856 | 0.148 | 0.000 | 4.092 | 0.000 | 6.713 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | A | F(-2) | A | F | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 116 | 116 | 99 | 188 | 0 | 231 | 0 | 98 | -1 |
| N.S. |
1 | 1.00 | 0.85 | 1.62 | 0.00 | 1.99 | 0.00 | 0.84 | -0.01 |
| time (sec) |
N/A | 0.061 | 0.441 | 0.131 | 0.000 | 3.586 | 0.000 | 5.546 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | C | B | F(-2) | B | F | F(-1) | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 374 | 374 | 211 | 761 | 0 | 11127 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 0.56 | 2.03 | 0.00 | 29.75 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 0.362 | 0.393 | 0.156 | 0.000 | 15.622 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F | F(-1) | F(-1) | F(-1) | F |
| verified |
N/A | NO | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 789 | 787 | 1369 | 2108 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 1.74 | 2.67 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 7.679 | 12.951 | 0.161 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | B | F | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 649 | 649 | 768 | 2299 | 0 | 3109 | 0 | 1099 | -1 |
| N.S. |
1 | 1.00 | 1.18 | 3.54 | 0.00 | 4.79 | 0.00 | 1.69 | -0.00 |
| time (sec) |
N/A | 1.274 | 4.238 | 0.160 | 0.000 | 4.188 | 0.000 | 3.938 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | B | F | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 309 | 309 | 291 | 749 | 0 | 1277 | 0 | 407 | -1 |
| N.S. |
1 | 1.00 | 0.94 | 2.42 | 0.00 | 4.13 | 0.00 | 1.32 | -0.00 |
| time (sec) |
N/A | 0.270 | 1.503 | 0.160 | 0.000 | 3.188 | 0.000 | 6.399 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | A | F(-2) | B | F | A | B |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 111 | 111 | 114 | 201 | 0 | 435 | 0 | 122 | 143 |
| N.S. |
1 | 1.00 | 1.03 | 1.81 | 0.00 | 3.92 | 0.00 | 1.10 | 1.29 |
| time (sec) |
N/A | 0.051 | 0.551 | 0.131 | 0.000 | 3.008 | 0.000 | 6.441 | 3.734 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | C | B | F(-2) | F(-1) | F(-1) | F(-1) | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 666 | 666 | 692 | 1906 | 0 | 0 | 0 | 0 | -1 |
| N.S. |
1 | 1.00 | 1.04 | 2.86 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 1.147 | 1.057 | 0.164 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | B | F(-1) | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 891 | 891 | 872 | 3251 | 0 | 4005 | 0 | 1401 | -1 |
| N.S. |
1 | 1.00 | 0.98 | 3.65 | 0.00 | 4.49 | 0.00 | 1.57 | -0.00 |
| time (sec) |
N/A | 1.103 | 10.899 | 0.204 | 0.000 | 9.649 | 0.000 | 1.971 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | A | F(-1) | A | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 444 | 444 | 392 | 1129 | 0 | 1595 | 0 | 587 | -1 |
| N.S. |
1 | 1.00 | 0.88 | 2.54 | 0.00 | 3.59 | 0.00 | 1.32 | -0.00 |
| time (sec) |
N/A | 0.276 | 2.361 | 0.142 | 0.000 | 7.990 | 0.000 | 4.523 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F(-2) | B | F(-1) | A | B |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 131 | 131 | 147 | 351 | 0 | 291 | 0 | 240 | 175 |
| N.S. |
1 | 1.00 | 1.12 | 2.68 | 0.00 | 2.22 | 0.00 | 1.83 | 1.34 |
| time (sec) |
N/A | 0.055 | 0.882 | 0.134 | 0.000 | 6.657 | 0.000 | 3.737 | 3.708 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | B | F | B | F | B | F |
| verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 51 | 51 | 53 | 144 | 0 | 154 | 0 | 205 | -1 |
| N.S. | 1 | 1.00 | 1.04 | 2.82 | 0.00 | 3.02 | 0.00 | 4.02 | -0.02 |
| time (sec) |
N/A | 0.040 | 0.158 | 0.386 | 0.000 | 3.339 | 0.000 | 4.957 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | A | F | F | F | F | F |
| verified |
N/A | NO | NO | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 1432 | 1432 | 670 | 906 | 0 | 0 | 0 | 0 | -1 |
| N.S. | 1 | 1.00 | 0.47 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 3.900 | 3.920 | 0.292 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
| grade |
A | A | A | A | F | F | F | F | F |
| verified |
N/A | Yes | NO | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 652 | 652 | 390 | 420 | 0 | 0 | 0 | 0 | -1 |
| N.S. | 1 | 1.00 | 0.60 | 0.64 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
| time (sec) |
N/A | 0.500 | 2.246 | 0.393 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |