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 | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 249 | 257 | 160 | 204 | 1570 | 173 | 0 | 227 | 0 |
N.S. |
1 | 1.03 | 0.64 | 0.82 | 6.31 | 0.69 | 0.00 | 0.91 | 0.00 |
time (sec) |
N/A | 0.752 | 0.471 | 1.853 | 1.070 | 0.320 | 0.000 | 0.338 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 123 | 125 | 112 | 99 | 580 | 119 | 0 | 183 | 0 |
N.S. |
1 | 1.02 | 0.91 | 0.80 | 4.72 | 0.97 | 0.00 | 1.49 | 0.00 |
time (sec) |
N/A | 0.341 | 0.197 | 0.844 | 0.630 | 0.272 | 0.000 | 0.333 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | C | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 98 | 98 | 85 | 81 | 112 | 103 | 88 | 135 | 99 |
N.S. |
1 | 1.00 | 0.87 | 0.83 | 1.14 | 1.05 | 0.90 | 1.38 | 1.01 |
time (sec) |
N/A | 0.268 | 0.115 | 0.357 | 0.239 | 0.302 | 0.218 | 0.333 | 0.080 |
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 | 0.163 | 2.150 | 0.227 | 0.361 | 0.262 | 0.383 | 0.340 | 13.744 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 111 | 111 | 110 | 0 | 0 | 115 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.99 | 0.00 | 0.00 | 1.04 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.296 | 1.719 | 0.000 | 0.000 | 0.287 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 251 | 261 | 164 | 224 | 1570 | 178 | 0 | 229 | 0 |
N.S. |
1 | 1.04 | 0.65 | 0.89 | 6.25 | 0.71 | 0.00 | 0.91 | 0.00 |
time (sec) |
N/A | 0.745 | 0.458 | 1.755 | 1.104 | 0.286 | 0.000 | 0.325 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 124 | 127 | 116 | 110 | 580 | 125 | 0 | 185 | 0 |
N.S. |
1 | 1.02 | 0.94 | 0.89 | 4.68 | 1.01 | 0.00 | 1.49 | 0.00 |
time (sec) |
N/A | 0.355 | 0.206 | 0.855 | 0.618 | 0.293 | 0.000 | 0.373 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | C | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 99 | 99 | 88 | 88 | 112 | 106 | 94 | 137 | 105 |
N.S. |
1 | 1.00 | 0.89 | 0.89 | 1.13 | 1.07 | 0.95 | 1.38 | 1.06 |
time (sec) |
N/A | 0.281 | 0.118 | 0.332 | 0.246 | 0.275 | 0.226 | 0.363 | 0.054 |
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 | 0.165 | 2.046 | 0.251 | 0.366 | 0.266 | 0.398 | 0.338 | 14.376 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 111 | 111 | 114 | 0 | 0 | 123 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.03 | 0.00 | 0.00 | 1.11 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.300 | 2.419 | 0.000 | 0.000 | 0.274 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 82 | 87 | 67 | 59 | 160 | 59 | 0 | 75 | 64 |
N.S. |
1 | 1.06 | 0.82 | 0.72 | 1.95 | 0.72 | 0.00 | 0.91 | 0.78 |
time (sec) |
N/A | 0.363 | 0.095 | 1.862 | 0.440 | 0.260 | 0.000 | 0.433 | 0.116 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | B | C | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 41 | 41 | 39 | 30 | 126 | 32 | 155 | 65 | 32 |
N.S. |
1 | 1.00 | 0.95 | 0.73 | 3.07 | 0.78 | 3.78 | 1.59 | 0.78 |
time (sec) |
N/A | 0.206 | 0.033 | 1.584 | 0.401 | 0.266 | 0.661 | 0.473 | 14.528 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | C | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 24 | 24 | 24 | 20 | 70 | 22 | 29 | 39 | 19 |
N.S. |
1 | 1.00 | 1.00 | 0.83 | 2.92 | 0.92 | 1.21 | 1.62 | 0.79 |
time (sec) |
N/A | 0.173 | 0.019 | 0.662 | 0.320 | 0.274 | 0.200 | 0.398 | 0.043 |
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 | 0.150 | 9.941 | 0.183 | 0.349 | 0.268 | 0.354 | 0.406 | 13.770 |
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 | 0.285 | 10.619 | 0.208 | 0.360 | 0.281 | 0.326 | 0.407 | 14.017 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 248 | 248 | 170 | 191 | 1617 | 178 | 0 | 214 | 0 |
N.S. |
1 | 1.00 | 0.69 | 0.77 | 6.52 | 0.72 | 0.00 | 0.86 | 0.00 |
time (sec) |
N/A | 0.431 | 0.454 | 0.684 | 1.061 | 0.345 | 0.000 | 0.462 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 126 | 126 | 116 | 95 | 610 | 124 | 0 | 172 | 0 |
N.S. |
1 | 1.00 | 0.92 | 0.75 | 4.84 | 0.98 | 0.00 | 1.37 | 0.00 |
time (sec) |
N/A | 0.255 | 0.173 | 0.722 | 0.635 | 0.290 | 0.000 | 0.438 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 100 | 100 | 97 | 72 | 124 | 93 | 83 | 122 | 0 |
N.S. |
1 | 1.00 | 0.97 | 0.72 | 1.24 | 0.93 | 0.83 | 1.22 | 0.00 |
time (sec) |
N/A | 0.222 | 0.070 | 0.590 | 0.325 | 0.270 | 0.466 | 0.444 | 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 | 28 | 19 | 15 | 19 | 19 |
N.S. |
1 | 1.00 | 1.12 | 1.00 | 1.65 | 1.12 | 0.88 | 1.12 | 1.12 |
time (sec) |
N/A | 0.195 | 3.959 | 0.428 | 0.399 | 0.270 | 0.798 | 0.485 | 13.750 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 248 | 248 | 175 | 199 | 1617 | 187 | 0 | 216 | 0 |
N.S. |
1 | 1.00 | 0.71 | 0.80 | 6.52 | 0.75 | 0.00 | 0.87 | 0.00 |
time (sec) |
N/A | 0.429 | 0.399 | 0.663 | 1.055 | 0.269 | 0.000 | 0.418 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 126 | 126 | 122 | 99 | 611 | 133 | 0 | 174 | 0 |
N.S. |
1 | 1.00 | 0.97 | 0.79 | 4.85 | 1.06 | 0.00 | 1.38 | 0.00 |
time (sec) |
N/A | 0.254 | 0.173 | 0.666 | 0.699 | 0.287 | 0.000 | 0.431 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 100 | 100 | 100 | 76 | 124 | 96 | 88 | 124 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.76 | 1.24 | 0.96 | 0.88 | 1.24 | 0.00 |
time (sec) |
N/A | 0.226 | 0.072 | 0.682 | 0.330 | 0.285 | 0.452 | 0.438 | 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 | 28 | 22 | 15 | 20 | 20 |
N.S. |
1 | 1.00 | 1.11 | 1.00 | 1.56 | 1.22 | 0.83 | 1.11 | 1.11 |
time (sec) |
N/A | 0.195 | 4.132 | 0.393 | 0.382 | 0.270 | 0.838 | 0.500 | 13.403 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 85 | 85 | 77 | 64 | 171 | 57 | 0 | 64 | 0 |
N.S. |
1 | 1.00 | 0.91 | 0.75 | 2.01 | 0.67 | 0.00 | 0.75 | 0.00 |
time (sec) |
N/A | 0.231 | 0.104 | 1.428 | 0.430 | 0.275 | 0.000 | 0.503 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | B | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 46 | 46 | 41 | 35 | 136 | 37 | 121 | 54 | 0 |
N.S. |
1 | 1.00 | 0.89 | 0.76 | 2.96 | 0.80 | 2.63 | 1.17 | 0.00 |
time (sec) |
N/A | 0.196 | 0.043 | 1.313 | 0.382 | 0.273 | 0.919 | 0.460 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 27 | 27 | 26 | 20 | 34 | 19 | 22 | 26 | 0 |
N.S. |
1 | 1.00 | 0.96 | 0.74 | 1.26 | 0.70 | 0.81 | 0.96 | 0.00 |
time (sec) |
N/A | 0.167 | 0.011 | 1.264 | 0.304 | 0.266 | 0.374 | 0.593 | 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 | 24 | 15 | 14 | 15 | 15 |
N.S. |
1 | 1.00 | 1.13 | 0.87 | 1.60 | 1.00 | 0.93 | 1.00 | 1.00 |
time (sec) |
N/A | 0.187 | 7.145 | 0.363 | 0.358 | 0.265 | 0.662 | 0.533 | 13.385 |
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 | 26 | 15 | 15 | 15 | 15 |
N.S. |
1 | 1.00 | 1.13 | 0.87 | 1.73 | 1.00 | 1.00 | 1.00 | 1.00 |
time (sec) |
N/A | 0.215 | 7.053 | 0.401 | 0.339 | 0.273 | 0.594 | 0.405 | 13.308 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 285 | 282 | 187 | 396 | 2271 | 230 | 0 | 289 | 0 |
N.S. |
1 | 0.99 | 0.66 | 1.39 | 7.97 | 0.81 | 0.00 | 1.01 | 0.00 |
time (sec) |
N/A | 0.773 | 0.788 | 2.376 | 1.336 | 0.269 | 0.000 | 0.347 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 140 | 134 | 129 | 182 | 695 | 136 | 0 | 199 | 0 |
N.S. |
1 | 0.96 | 0.92 | 1.30 | 4.96 | 0.97 | 0.00 | 1.42 | 0.00 |
time (sec) |
N/A | 0.341 | 0.440 | 0.752 | 0.593 | 0.287 | 0.000 | 0.333 | 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 | 0.168 | 3.237 | 0.429 | 0.385 | 0.269 | 0.345 | 0.351 | 13.456 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 291 | 291 | 215 | 378 | 2360 | 257 | 0 | 292 | 0 |
N.S. |
1 | 1.00 | 0.74 | 1.30 | 8.11 | 0.88 | 0.00 | 1.00 | 0.00 |
time (sec) |
N/A | 0.526 | 0.699 | 1.391 | 1.382 | 0.303 | 0.000 | 0.371 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 150 | 150 | 139 | 170 | 737 | 149 | 0 | 195 | 0 |
N.S. |
1 | 1.00 | 0.93 | 1.13 | 4.91 | 0.99 | 0.00 | 1.30 | 0.00 |
time (sec) |
N/A | 0.285 | 0.326 | 0.776 | 0.649 | 0.280 | 0.000 | 0.353 | 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 | 120 | 23 | 19 | 23 | 23 |
N.S. |
1 | 1.00 | 1.10 | 1.00 | 5.71 | 1.10 | 0.90 | 1.10 | 1.10 |
time (sec) |
N/A | 0.215 | 4.466 | 0.544 | 0.382 | 0.279 | 41.990 | 0.396 | 14.124 |