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 | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 225 | 244 | 149 | 252 | 755 | 429 | 0 | 163 | 0 |
N.S. |
1 | 1.08 | 0.66 | 1.12 | 3.36 | 1.91 | 0.00 | 0.72 | 0.00 |
time (sec) |
N/A | 0.809 | 0.207 | 0.079 | 0.452 | 0.265 | 0.000 | 0.262 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 111 | 118 | 130 | 124 | 611 | 344 | 0 | 121 | 0 |
N.S. |
1 | 1.06 | 1.17 | 1.12 | 5.50 | 3.10 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.360 | 0.103 | 0.048 | 0.399 | 0.290 | 0.000 | 0.268 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 91 | 91 | 105 | 83 | 464 | 112 | 0 | 79 | 0 |
N.S. |
1 | 1.00 | 1.15 | 0.91 | 5.10 | 1.23 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.286 | 0.051 | 0.033 | 0.366 | 0.264 | 0.000 | 0.271 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 15 | 15 | 17 | 15 | 17 | 17 | 14 | 17 | 17 |
N.S. |
1 | 1.00 | 1.13 | 1.00 | 1.13 | 1.13 | 0.93 | 1.13 | 1.13 |
time (sec) |
N/A | 0.180 | 0.259 | 0.023 | 0.752 | 0.264 | 0.643 | 0.263 | 1.555 |
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 | 107 | 107 | 132 | 0 | 0 | 330 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.23 | 0.00 | 0.00 | 3.08 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.276 | 4.537 | 0.000 | 0.000 | 0.247 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 227 | 246 | 152 | 244 | 834 | 466 | 0 | 167 | 0 |
N.S. |
1 | 1.08 | 0.67 | 1.07 | 3.67 | 2.05 | 0.00 | 0.74 | 0.00 |
time (sec) |
N/A | 0.804 | 0.202 | 0.083 | 0.454 | 0.260 | 0.000 | 0.270 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 112 | 119 | 134 | 120 | 674 | 384 | 0 | 123 | 0 |
N.S. |
1 | 1.06 | 1.20 | 1.07 | 6.02 | 3.43 | 0.00 | 1.10 | 0.00 |
time (sec) |
N/A | 0.358 | 0.103 | 0.050 | 0.390 | 0.261 | 0.000 | 0.266 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 91 | 91 | 109 | 79 | 511 | 116 | 0 | 81 | 0 |
N.S. |
1 | 1.00 | 1.20 | 0.87 | 5.62 | 1.27 | 0.00 | 0.89 | 0.00 |
time (sec) |
N/A | 0.294 | 0.051 | 0.035 | 0.368 | 0.250 | 0.000 | 0.263 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 18 | 16 | 20 | 20 | 14 | 18 | 18 |
N.S. |
1 | 1.00 | 1.12 | 1.00 | 1.25 | 1.25 | 0.88 | 1.12 | 1.12 |
time (sec) |
N/A | 0.179 | 0.446 | 0.023 | 0.727 | 0.250 | 0.657 | 0.271 | 1.557 |
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 | 108 | 108 | 136 | 0 | 0 | 370 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.26 | 0.00 | 0.00 | 3.43 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.275 | 5.784 | 0.000 | 0.000 | 0.266 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 66 | 119 | 72 | 75 | 183 | 129 | 0 | 53 | 0 |
N.S. |
1 | 1.80 | 1.09 | 1.14 | 2.77 | 1.95 | 0.00 | 0.80 | 0.00 |
time (sec) |
N/A | 0.576 | 0.109 | 0.077 | 0.278 | 0.270 | 0.000 | 0.264 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 52 | 57 | 76 | 49 | 123 | 106 | 0 | 43 | 0 |
N.S. |
1 | 1.10 | 1.46 | 0.94 | 2.37 | 2.04 | 0.00 | 0.83 | 0.00 |
time (sec) |
N/A | 0.297 | 0.054 | 0.047 | 0.265 | 0.253 | 0.000 | 0.267 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | B | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 22 | 25 | 94 | 14 | 0 | 21 | 0 |
N.S. |
1 | 1.00 | 0.56 | 0.64 | 2.41 | 0.36 | 0.00 | 0.54 | 0.00 |
time (sec) |
N/A | 0.239 | 0.018 | 0.028 | 0.264 | 0.244 | 0.000 | 0.262 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 13 | 13 | 15 | 11 | 13 | 13 | 12 | 13 | 13 |
N.S. |
1 | 1.00 | 1.15 | 0.85 | 1.00 | 1.00 | 0.92 | 1.00 | 1.00 |
time (sec) |
N/A | 0.169 | 4.956 | 0.034 | 0.641 | 0.249 | 0.652 | 0.251 | 1.551 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 13 | 13 | 15 | 11 | 13 | 13 | 14 | 13 | 13 |
N.S. |
1 | 1.00 | 1.15 | 0.85 | 1.00 | 1.00 | 1.08 | 1.00 | 1.00 |
time (sec) |
N/A | 0.403 | 6.318 | 0.024 | 0.477 | 0.243 | 0.439 | 0.268 | 1.567 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 268 | 268 | 176 | 281 | 294 | 766 | 0 | 182 | 0 |
N.S. |
1 | 1.00 | 0.66 | 1.05 | 1.10 | 2.86 | 0.00 | 0.68 | 0.00 |
time (sec) |
N/A | 0.512 | 0.516 | 0.177 | 0.301 | 0.261 | 0.000 | 0.263 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 136 | 136 | 155 | 141 | 200 | 647 | 0 | 142 | 0 |
N.S. |
1 | 1.00 | 1.14 | 1.04 | 1.47 | 4.76 | 0.00 | 1.04 | 0.00 |
time (sec) |
N/A | 0.297 | 0.274 | 0.136 | 0.274 | 0.259 | 0.000 | 0.265 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 140 | 94 | 96 | 128 | 0 | 94 | 0 |
N.S. |
1 | 1.00 | 1.27 | 0.85 | 0.87 | 1.16 | 0.00 | 0.85 | 0.00 |
time (sec) |
N/A | 0.252 | 0.097 | 0.092 | 0.272 | 0.248 | 0.000 | 0.262 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 17 | 17 | 19 | 17 | 51 | 19 | 15 | 19 | 19 |
N.S. |
1 | 1.00 | 1.12 | 1.00 | 3.00 | 1.12 | 0.88 | 1.12 | 1.12 |
time (sec) |
N/A | 0.213 | 4.715 | 0.052 | 0.349 | 0.275 | 0.841 | 0.283 | 1.570 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 268 | 268 | 181 | 273 | 322 | 843 | 0 | 186 | 0 |
N.S. |
1 | 1.00 | 0.68 | 1.02 | 1.20 | 3.15 | 0.00 | 0.69 | 0.00 |
time (sec) |
N/A | 0.480 | 0.509 | 0.159 | 0.294 | 0.271 | 0.000 | 0.274 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 136 | 136 | 161 | 137 | 216 | 729 | 0 | 144 | 0 |
N.S. |
1 | 1.00 | 1.18 | 1.01 | 1.59 | 5.36 | 0.00 | 1.06 | 0.00 |
time (sec) |
N/A | 0.300 | 0.273 | 0.127 | 0.289 | 0.283 | 0.000 | 0.283 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 144 | 90 | 96 | 132 | 0 | 96 | 0 |
N.S. |
1 | 1.00 | 1.31 | 0.82 | 0.87 | 1.20 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.276 | 0.103 | 0.097 | 0.262 | 0.251 | 0.000 | 0.266 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 20 | 18 | 51 | 22 | 15 | 20 | 20 |
N.S. |
1 | 1.00 | 1.11 | 1.00 | 2.83 | 1.22 | 0.83 | 1.11 | 1.11 |
time (sec) |
N/A | 0.219 | 5.765 | 0.055 | 0.343 | 0.247 | 0.867 | 0.286 | 1.559 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 68 | 68 | 99 | 77 | 113 | 268 | 0 | 61 | 0 |
N.S. |
1 | 1.00 | 1.46 | 1.13 | 1.66 | 3.94 | 0.00 | 0.90 | 0.00 |
time (sec) |
N/A | 0.289 | 0.155 | 0.200 | 0.304 | 0.250 | 0.000 | 0.260 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | C | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 75 | 75 | 88 | 75 | 121 | 306 | 0 | 70 | 0 |
N.S. |
1 | 1.00 | 1.17 | 1.00 | 1.61 | 4.08 | 0.00 | 0.93 | 0.00 |
time (sec) |
N/A | 0.244 | 0.152 | 0.168 | 0.317 | 0.268 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 56 | 56 | 48 | 49 | 45 | 41 | 0 | 42 | 0 |
N.S. |
1 | 1.00 | 0.86 | 0.88 | 0.80 | 0.73 | 0.00 | 0.75 | 0.00 |
time (sec) |
N/A | 0.218 | 0.050 | 0.109 | 0.264 | 0.249 | 0.000 | 0.265 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 15 | 15 | 17 | 13 | 43 | 15 | 14 | 15 | 15 |
N.S. |
1 | 1.00 | 1.13 | 0.87 | 2.87 | 1.00 | 0.93 | 1.00 | 1.00 |
time (sec) |
N/A | 0.202 | 6.504 | 0.069 | 0.322 | 0.248 | 0.766 | 0.263 | 1.558 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 261 | 269 | 194 | 460 | 536 | 633 | 0 | 227 | 0 |
N.S. |
1 | 1.03 | 0.74 | 1.76 | 2.05 | 2.43 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.857 | 0.360 | 0.219 | 0.365 | 0.269 | 0.000 | 0.266 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 128 | 127 | 146 | 200 | 254 | 412 | 0 | 139 | 0 |
N.S. |
1 | 0.99 | 1.14 | 1.56 | 1.98 | 3.22 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.382 | 0.162 | 0.061 | 0.282 | 0.260 | 0.000 | 0.268 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 21 | 19 | 21 | 21 | 17 | 21 | 21 |
N.S. |
1 | 1.00 | 1.11 | 1.00 | 1.11 | 1.11 | 0.89 | 1.11 | 1.11 |
time (sec) |
N/A | 0.185 | 0.892 | 0.023 | 0.676 | 0.249 | 0.538 | 0.269 | 1.573 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 311 | 311 | 240 | 528 | 603 | 1142 | 0 | 263 | 0 |
N.S. |
1 | 1.00 | 0.77 | 1.70 | 1.94 | 3.67 | 0.00 | 0.85 | 0.00 |
time (sec) |
N/A | 0.607 | 0.920 | 0.293 | 0.466 | 0.266 | 0.000 | 0.281 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 160 | 160 | 177 | 231 | 301 | 777 | 0 | 165 | 0 |
N.S. |
1 | 1.00 | 1.11 | 1.44 | 1.88 | 4.86 | 0.00 | 1.03 | 0.00 |
time (sec) |
N/A | 0.357 | 0.440 | 0.162 | 0.373 | 0.273 | 0.000 | 0.281 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 21 | 21 | 23 | 21 | 72 | 23 | 19 | 23 | 23 |
N.S. |
1 | 1.00 | 1.10 | 1.00 | 3.43 | 1.10 | 0.90 | 1.10 | 1.10 |
time (sec) |
N/A | 0.237 | 2.490 | 0.049 | 0.364 | 0.246 | 41.993 | 0.298 | 1.613 |