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 | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 232 | 229 | 114 | 213 | 0 | 20298 | 0 | 3690 | 0 |
N.S. |
1 | 0.99 | 0.49 | 0.92 | 0.00 | 87.49 | 0.00 | 15.91 | 0.00 |
time (sec) |
N/A | 0.395 | 0.412 | 1.731 | 0.000 | 1.779 | 0.000 | 24.122 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 163 | 162 | 82 | 213 | 0 | 10506 | 0 | 1981 | 0 |
N.S. |
1 | 0.99 | 0.50 | 1.31 | 0.00 | 64.45 | 0.00 | 12.15 | 0.00 |
time (sec) |
N/A | 0.324 | 0.091 | 0.785 | 0.000 | 0.746 | 0.000 | 8.262 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 99 | 107 | 60 | 173 | 0 | 4200 | 0 | 0 | 0 |
N.S. |
1 | 1.08 | 0.61 | 1.75 | 0.00 | 42.42 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.280 | 0.060 | 0.496 | 0.000 | 0.414 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 83 | 87 | 49 | 65 | 0 | 3084 | 0 | 0 | 0 |
N.S. |
1 | 1.05 | 0.59 | 0.78 | 0.00 | 37.16 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.278 | 0.043 | 0.477 | 0.000 | 0.371 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 143 | 131 | 69 | 73 | 0 | 7594 | 0 | 0 | 0 |
N.S. |
1 | 0.92 | 0.48 | 0.51 | 0.00 | 53.10 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.312 | 0.214 | 0.738 | 0.000 | 0.589 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 208 | 185 | 117 | 73 | 0 | 15102 | 0 | 0 | 0 |
N.S. |
1 | 0.89 | 0.56 | 0.35 | 0.00 | 72.61 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.373 | 0.295 | 1.687 | 0.000 | 1.294 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 333 | 417 | 252 | 661 | 0 | 15718 | 0 | 0 | 0 |
N.S. |
1 | 1.25 | 0.76 | 1.98 | 0.00 | 47.20 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.556 | 2.861 | 4.485 | 0.000 | 0.790 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | B | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 274 | 344 | 215 | 797 | 0 | 8226 | 0 | 0 | 0 |
N.S. |
1 | 1.26 | 0.78 | 2.91 | 0.00 | 30.02 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.475 | 2.297 | 3.085 | 0.000 | 0.319 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 251 | 258 | 190 | 406 | 0 | 5442 | 0 | 0 | 0 |
N.S. |
1 | 1.03 | 0.76 | 1.62 | 0.00 | 21.68 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.260 | 0.971 | 1.248 | 0.000 | 0.215 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 351 | 412 | 226 | 640 | 0 | 7847 | 0 | 0 | 0 |
N.S. |
1 | 1.17 | 0.64 | 1.82 | 0.00 | 22.36 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.584 | 2.378 | 3.041 | 0.000 | 0.288 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | B | F | B | F | F(-2) | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 385 | 471 | 247 | 923 | 0 | 13823 | 0 | 0 | 0 |
N.S. |
1 | 1.22 | 0.64 | 2.40 | 0.00 | 35.90 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.695 | 2.358 | 4.114 | 0.000 | 0.601 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | F | F | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 122 | 149 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.22 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.389 | 0.000 | 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 | 110 | 110 | 90 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.82 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.283 | 0.176 | 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 | 63 | 63 | 65 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.238 | 0.054 | 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 | 54 | 54 | 54 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.245 | 0.044 | 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 | 94 | 91 | 71 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 0.97 | 0.76 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.265 | 0.274 | 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 | F | F | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 103 | 103 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.302 | 0.000 | 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 | F | F | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 103 | 103 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.299 | 0.000 | 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 | F | F | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 99 | 99 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.284 | 0.000 | 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 | F | F | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 103 | 103 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.287 | 0.000 | 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 | C | F | C | F | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 152 | 152 | 136 | 91 | 0 | 1432 | 0 | 209 | 1129 |
N.S. |
1 | 1.00 | 0.89 | 0.60 | 0.00 | 9.42 | 0.00 | 1.38 | 7.43 |
time (sec) |
N/A | 0.455 | 0.241 | 5.239 | 0.000 | 0.949 | 0.000 | 0.276 | 0.998 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 28 | 28 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.254 | 0.018 | 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 | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 45 | 45 | 45 | 0 | 0 | 1663 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 36.96 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.275 | 0.017 | 0.000 | 0.000 | 1.451 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 29 | 29 | 29 | 24 | 0 | 113 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.83 | 0.00 | 3.90 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.273 | 0.019 | 0.258 | 0.000 | 0.280 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 47 | 45 | 45 | 38 | 0 | 156 | 0 | 0 | 0 |
N.S. |
1 | 0.96 | 0.96 | 0.81 | 0.00 | 3.32 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.275 | 0.017 | 0.212 | 0.000 | 0.264 | 0.000 | 0.000 | 0.000 |