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 | B | A | C | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 135 | 383 | 145 | 737 | 0 | 318 | 0 | 0 | 81 | 0 |
N.S. |
1 | 2.84 | 1.07 | 5.46 | 0.00 | 2.36 | 0.00 | 0.00 | 0.60 | 0.00 |
time (sec) |
N/A | 0.564 | 1.122 | 5.957 | 0.000 | 5.248 | 0.000 | 0.000 | 0.023 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | B | A | C | F | B | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 119 | 357 | 139 | 714 | 0 | 268 | 0 | 0 | 69 | 0 |
N.S. |
1 | 3.00 | 1.17 | 6.00 | 0.00 | 2.25 | 0.00 | 0.00 | 0.58 | 0.00 |
time (sec) |
N/A | 0.525 | 1.088 | 6.401 | 0.000 | 5.265 | 0.000 | 0.000 | 0.022 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 43 | 43 | 43 | 34 | 0 | 0 | 0 | 0 | 133 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.79 | 0.00 | 0.00 | 0.00 | 0.00 | 3.09 | 0.00 |
time (sec) |
N/A | 0.202 | 10.181 | 0.862 | 0.000 | 0.000 | 0.000 | 0.000 | 0.019 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 43 | 43 | 43 | 34 | 0 | 0 | 0 | 0 | 81 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.79 | 0.00 | 0.00 | 0.00 | 0.00 | 1.88 | 0.00 |
time (sec) |
N/A | 0.221 | 10.141 | 0.606 | 0.000 | 0.000 | 0.000 | 0.000 | 0.015 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 43 | 34 | 0 | 0 | 0 | 0 | 81 | 0 |
N.S. |
1 | 1.00 | 1.10 | 0.87 | 0.00 | 0.00 | 0.00 | 0.00 | 2.08 | 0.00 |
time (sec) |
N/A | 0.174 | 10.017 | 0.534 | 0.000 | 0.000 | 0.000 | 0.000 | 0.015 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | B | A | C | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 67 | 70 | 101 | 12 | 105 | 94 | 31 | 0 | 34 | 10 |
N.S. |
1 | 1.04 | 1.51 | 0.18 | 1.57 | 1.40 | 0.46 | 0.00 | 0.51 | 0.15 |
time (sec) |
N/A | 0.154 | 0.082 | 1.276 | 0.287 | 0.250 | 0.550 | 0.000 | 0.008 | 0.457 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | A | A | C | A | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 70 | 76 | 90 | 66 | 73 | 75 | 41 | 74 | 35 | 91 |
N.S. |
1 | 1.09 | 1.29 | 0.94 | 1.04 | 1.07 | 0.59 | 1.06 | 0.50 | 1.30 |
time (sec) |
N/A | 0.177 | 0.050 | 1.934 | 0.290 | 0.249 | 0.576 | 0.295 | 0.009 | 0.449 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | F | F | F | F(-1) | F | F | F | F(-1) |
verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 384 | 375 | 0 | 0 | 0 | 0 | 0 | 0 | 88 | 0 |
N.S. |
1 | 0.98 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.23 | 0.00 |
time (sec) |
N/A | 0.702 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.030 | 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 | 234 | 234 | 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.14 | 0.00 |
time (sec) |
N/A | 0.376 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.055 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | F | C | F | C | F | F | F | F(-1) |
verified |
N/A | Yes | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 199 | 250 | 0 | 618 | 0 | 2298 | 0 | 0 | 648 | 0 |
N.S. |
1 | 1.26 | 0.00 | 3.11 | 0.00 | 11.55 | 0.00 | 0.00 | 3.26 | 0.00 |
time (sec) |
N/A | 0.456 | 0.000 | 26.803 | 0.000 | 1.184 | 0.000 | 0.000 | 0.155 | 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 | 177 | 182 | 0 | 0 | 0 | 0 | 0 | 0 | 69 | 0 |
N.S. |
1 | 1.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.39 | 0.00 |
time (sec) |
N/A | 0.379 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.019 | 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 | 177 | 182 | 0 | 0 | 0 | 0 | 0 | 0 | 69 | 0 |
N.S. |
1 | 1.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.39 | 0.00 |
time (sec) |
N/A | 0.407 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.020 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | A | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 132 | 138 | 204 | 179 | 0 | 191 | 0 | 0 | 19 | 0 |
N.S. |
1 | 1.05 | 1.55 | 1.36 | 0.00 | 1.45 | 0.00 | 0.00 | 0.14 | 0.00 |
time (sec) |
N/A | 0.203 | 0.331 | 1.854 | 0.000 | 0.225 | 0.000 | 0.000 | 0.004 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 250 | 272 | 26 | 0 | 0 | 0 | 0 | 0 | 20 | 0 |
N.S. |
1 | 1.09 | 0.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.08 | 0.00 |
time (sec) |
N/A | 0.410 | 10.012 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.003 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 383 | 655 | 138 | 0 | 0 | 0 | 0 | 0 | 42 | 0 |
N.S. |
1 | 1.71 | 0.36 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.11 | 0.00 |
time (sec) |
N/A | 1.115 | 15.090 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.017 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | A | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 272 | 295 | 283 | 1147 | 0 | 341 | 0 | 0 | 19 | 0 |
N.S. |
1 | 1.08 | 1.04 | 4.22 | 0.00 | 1.25 | 0.00 | 0.00 | 0.07 | 0.00 |
time (sec) |
N/A | 0.335 | 1.905 | 5.600 | 0.000 | 1.540 | 0.000 | 0.000 | 0.003 | 0.000 |