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 | A | F | A | F(-1) | A | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 187 | 75 | 72 | 90 | 0 | 91 | 0 | 87 | 58 | 0 |
| N.S. |
1 | 0.40 | 0.39 | 0.48 | 0.00 | 0.49 | 0.00 | 0.47 | 0.31 | 0.00 |
| time (sec) |
N/A | 0.859 | 0.087 | 0.075 | 0.000 | 0.114 | 0.000 | 0.144 | 0.173 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | A | F | A | F(-1) | A | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 221 | 80 | 76 | 98 | 0 | 99 | 0 | 107 | 66 | 0 |
| N.S. |
1 | 0.36 | 0.34 | 0.44 | 0.00 | 0.45 | 0.00 | 0.48 | 0.30 | 0.00 |
| time (sec) |
N/A | 0.884 | 0.079 | 0.078 | 0.000 | 0.107 | 0.000 | 0.146 | 0.165 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | A | F | A | F(-1) | A | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 263 | 93 | 90 | 106 | 0 | 107 | 0 | 123 | 74 | 0 |
| N.S. |
1 | 0.35 | 0.34 | 0.40 | 0.00 | 0.41 | 0.00 | 0.47 | 0.28 | 0.00 |
| time (sec) |
N/A | 0.887 | 0.093 | 0.076 | 0.000 | 0.107 | 0.000 | 0.123 | 0.188 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | A | A | A | F(-1) | F | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 189 | 114 | 97 | 161 | 206 | 210 | 0 | 0 | 127 | 0 |
| N.S. |
1 | 0.60 | 0.51 | 0.85 | 1.09 | 1.11 | 0.00 | 0.00 | 0.67 | 0.00 |
| time (sec) |
N/A | 0.987 | 0.206 | 0.076 | 0.116 | 0.131 | 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 | F(-1) | F(-2) | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 87 | 64 | 53 | 58 | 50 | 74 | 0 | 0 | 27 | 0 |
| N.S. |
1 | 0.74 | 0.61 | 0.67 | 0.57 | 0.85 | 0.00 | 0.00 | 0.31 | 0.00 |
| time (sec) |
N/A | 0.861 | 0.064 | 0.065 | 0.085 | 0.086 | 0.000 | 0.000 | 0.155 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 58 | 58 | 56 | 0 | 0 | 0 | 0 | 0 | 76 | 0 |
| N.S. |
1 | 1.00 | 0.97 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.31 | 0.00 |
| time (sec) |
N/A | 0.820 | 0.048 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.148 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 149 | 119 | 111 | 0 | 0 | 0 | 0 | 0 | 180 | 0 |
| N.S. |
1 | 0.80 | 0.74 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.21 | 0.00 |
| time (sec) |
N/A | 0.974 | 0.097 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.160 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | A | A | A | F(-1) | F | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 191 | 111 | 125 | 161 | 206 | 209 | 0 | 0 | 127 | 0 |
| N.S. |
1 | 0.58 | 0.65 | 0.84 | 1.08 | 1.09 | 0.00 | 0.00 | 0.66 | 0.00 |
| time (sec) |
N/A | 0.998 | 0.130 | 0.082 | 0.121 | 0.120 | 0.000 | 0.000 | 0.166 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | A | A | A | F(-1) | F(-2) | B | F(-1) |
| verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 88 | 66 | 55 | 60 | 52 | 75 | 0 | 0 | 29 | 0 |
| N.S. |
1 | 0.75 | 0.62 | 0.68 | 0.59 | 0.85 | 0.00 | 0.00 | 0.33 | 0.00 |
| time (sec) |
N/A | 0.868 | 0.059 | 0.064 | 0.089 | 0.103 | 0.000 | 0.000 | 0.160 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 58 | 58 | 56 | 0 | 0 | 0 | 0 | 0 | 75 | 0 |
| N.S. |
1 | 1.00 | 0.97 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.29 | 0.00 |
| time (sec) |
N/A | 0.830 | 0.047 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.150 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F(-1) | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 150 | 121 | 112 | 0 | 0 | 0 | 0 | 0 | 183 | 0 |
| N.S. |
1 | 0.81 | 0.75 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.22 | 0.00 |
| time (sec) |
N/A | 0.946 | 0.083 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.149 | 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 | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 154 | 258 | 123 | 0 | 0 | 0 | 0 | 0 | 63 | 0 |
| N.S. |
1 | 1.68 | 0.80 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.41 | 0.00 |
| time (sec) |
N/A | 0.733 | 0.425 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.154 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 150 | 147 | 94 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| N.S. |
1 | 0.98 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.21 | 0.00 |
| time (sec) |
N/A | 0.568 | 0.423 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.154 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 286 | 290 | 180 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| N.S. |
1 | 1.01 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.14 | 0.00 |
| time (sec) |
N/A | 0.811 | 1.891 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.147 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | A | F | F | F | F | F(-2) | F | F(-1) |
| verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 295 | 239 | 146 | 0 | 0 | 0 | 0 | 0 | 30 | 0 |
| N.S. |
1 | 0.81 | 0.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.10 | 0.00 |
| time (sec) |
N/A | 0.847 | 0.692 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.161 | 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 | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 183 | 157 | 112 | 0 | 0 | 0 | 0 | 0 | 30 | 0 |
| N.S. |
1 | 0.86 | 0.61 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.16 | 0.00 |
| time (sec) |
N/A | 0.729 | 0.651 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.157 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | No | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 121 | 116 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| N.S. |
1 | 0.96 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.33 | 0.00 |
| time (sec) |
N/A | 0.653 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.150 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 76 | 76 | 0 | 0 | 0 | 0 | 0 | 0 | 43 | 0 |
| N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.57 | 0.00 |
| time (sec) |
N/A | 0.591 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.154 | 0.000 |
| Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
| grade |
N/A | A | F | F | F | F | F | F | F | F(-1) |
| verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
| size | 75 | 75 | 0 | 0 | 0 | 0 | 0 | 0 | 41 | 0 |
| N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.55 | 0.00 |
| time (sec) |
N/A | 0.579 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.150 | 0.000 |