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 | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 11 | 17 | 11 | 12 | 54 | 17 | 15 | 16 | 11 |
N.S. |
1 | 1.55 | 1.00 | 1.09 | 4.91 | 1.55 | 1.36 | 1.45 | 1.00 |
time (sec) |
N/A | 0.269 | 0.016 | 0.175 | 0.281 | 0.255 | 92.250 | 0.271 | 16.288 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | C | F(-2) | B | C | B |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 13 | 13 | 13 | 0 | 41 | 0 | 83 | 41 | 9 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 3.15 | 0.00 | 6.38 | 3.15 | 0.69 |
time (sec) |
N/A | 0.170 | 0.019 | 0.000 | 0.286 | 0.000 | 0.243 | 0.272 | 16.432 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | A | B | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 12 | 12 | 12 | 13 | 94 | 22 | 83 | 12 | 35 |
N.S. |
1 | 1.00 | 1.00 | 1.08 | 7.83 | 1.83 | 6.92 | 1.00 | 2.92 |
time (sec) |
N/A | 0.201 | 0.020 | 0.451 | 0.289 | 0.248 | 1.468 | 0.272 | 16.939 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | A | B | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 19 | 86 | 19 | 85 | 82 | 85 | 85 |
N.S. |
1 | 1.00 | 1.00 | 4.53 | 1.00 | 4.47 | 4.32 | 4.47 | 4.47 |
time (sec) |
N/A | 0.233 | 0.051 | 0.264 | 0.195 | 0.233 | 0.466 | 0.289 | 17.036 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | B | A | A | F(-1) | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 86 | 190 | 86 | 67 | 0 | 71 | 100 | 66 | 66 |
N.S. |
1 | 2.21 | 1.00 | 0.78 | 0.00 | 0.83 | 1.16 | 0.77 | 0.77 |
time (sec) |
N/A | 1.243 | 0.028 | 0.306 | 0.000 | 0.362 | 69.501 | 0.272 | 0.197 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 26 | 26 | 26 | 23 | 22 | 22 | 19 | 22 | 22 |
N.S. |
1 | 1.00 | 1.00 | 0.88 | 0.85 | 0.85 | 0.73 | 0.85 | 0.85 |
time (sec) |
N/A | 0.172 | 0.027 | 0.215 | 0.276 | 0.242 | 0.030 | 0.276 | 0.051 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | A | F | F | A | F | A | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 23 | 0 | 25 | 0 | 0 | 12 | 0 | 12 | 0 |
N.S. |
1 | 0.00 | 1.09 | 0.00 | 0.00 | 0.52 | 0.00 | 0.52 | 0.00 |
time (sec) |
N/A | 0.000 | 0.063 | 0.000 | 0.000 | 0.247 | 0.000 | 0.273 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 12 | 12 | 12 | 11 | 10 | 10 | 8 | 10 | 10 |
N.S. |
1 | 1.00 | 1.00 | 0.92 | 0.83 | 0.83 | 0.67 | 0.83 | 0.83 |
time (sec) |
N/A | 0.132 | 0.056 | 0.219 | 0.199 | 0.232 | 0.232 | 0.264 | 0.104 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | C | A | A | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 22 | 34 | 46 | 15 | 27 | 42 | 0 |
N.S. |
1 | 1.00 | 0.96 | 1.48 | 2.00 | 0.65 | 1.17 | 1.83 | 0.00 |
time (sec) |
N/A | 0.175 | 0.041 | 0.178 | 0.301 | 0.246 | 1.344 | 0.295 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | A | A | B | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 9 | 9 | 24 | 8 | 12 | 17 | 7 | 8 | 7 |
N.S. |
1 | 1.00 | 2.67 | 0.89 | 1.33 | 1.89 | 0.78 | 0.89 | 0.78 |
time (sec) |
N/A | 0.174 | 0.027 | 0.167 | 0.206 | 0.258 | 0.099 | 0.267 | 0.097 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 22 | 22 | 22 | 23 | 34 | 24 | 20 | 22 | 18 |
N.S. |
1 | 1.00 | 1.00 | 1.05 | 1.55 | 1.09 | 0.91 | 1.00 | 0.82 |
time (sec) |
N/A | 0.241 | 0.012 | 0.061 | 0.194 | 0.237 | 0.051 | 0.276 | 0.044 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | A | B | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 22 | 22 | 22 | 39 | 34 | 40 | 17 | 31 | 38 |
N.S. |
1 | 1.00 | 1.00 | 1.77 | 1.55 | 1.82 | 0.77 | 1.41 | 1.73 |
time (sec) |
N/A | 0.150 | 0.042 | 0.112 | 0.185 | 0.244 | 0.764 | 0.283 | 0.102 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | A | C | F | B | F | B | B |
verified |
N/A | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 21 | 0 | 21 | 52 | 0 | 84 | 0 | 136 | 15 |
N.S. |
1 | 0.00 | 1.00 | 2.48 | 0.00 | 4.00 | 0.00 | 6.48 | 0.71 |
time (sec) |
N/A | 0.000 | 0.089 | 2.167 | 0.000 | 0.261 | 0.000 | 0.277 | 17.185 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 17 | 17 | 17 | 18 | 17 | 17 | 15 | 21 | 17 |
N.S. |
1 | 1.00 | 1.00 | 1.06 | 1.00 | 1.00 | 0.88 | 1.24 | 1.00 |
time (sec) |
N/A | 0.290 | 0.065 | 0.315 | 0.248 | 0.244 | 0.061 | 0.272 | 17.009 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | A | B | F | F(-1) |
verified |
N/A | No | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 45 | 37 | 107 | 0 | 0 | 37 | 4362 | 0 | 0 |
N.S. |
1 | 0.82 | 2.38 | 0.00 | 0.00 | 0.82 | 96.93 | 0.00 | 0.00 |
time (sec) |
N/A | 0.282 | 0.535 | 0.000 | 0.000 | 0.302 | 3.878 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 54 | 54 | 54 | 45 | 44 | 44 | 49 | 44 | 44 |
N.S. |
1 | 1.00 | 1.00 | 0.83 | 0.81 | 0.81 | 0.91 | 0.81 | 0.81 |
time (sec) |
N/A | 0.221 | 0.005 | 0.198 | 0.214 | 0.230 | 0.023 | 0.272 | 0.111 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | A | C | A | A | F | A | B |
verified |
N/A | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 18 | 0 | 18 | 34 | 23 | 23 | 0 | 23 | 18 |
N.S. |
1 | 0.00 | 1.00 | 1.89 | 1.28 | 1.28 | 0.00 | 1.28 | 1.00 |
time (sec) |
N/A | 0.000 | 0.014 | 0.336 | 0.292 | 0.234 | 0.000 | 0.277 | 16.642 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 61 | 65 | 77 | 0 | 56 | 108 | 0 | 57 | 0 |
N.S. |
1 | 1.07 | 1.26 | 0.00 | 0.92 | 1.77 | 0.00 | 0.93 | 0.00 |
time (sec) |
N/A | 0.269 | 0.284 | 0.000 | 0.277 | 0.249 | 0.000 | 0.290 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | B | A | B | A | A | F(-1) | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 64 | 136 | 64 | 228 | 52 | 55 | 0 | 61 | 66 |
N.S. |
1 | 2.12 | 1.00 | 3.56 | 0.81 | 0.86 | 0.00 | 0.95 | 1.03 |
time (sec) |
N/A | 2.076 | 0.478 | 0.737 | 0.206 | 0.284 | 0.000 | 37.520 | 19.383 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | A | F | F | A | F | F | F(-1) |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 105 | 0 | 105 | 0 | 0 | 148 | 0 | 0 | 0 |
N.S. |
1 | 0.00 | 1.00 | 0.00 | 0.00 | 1.41 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.000 | 0.549 | 0.000 | 0.000 | 0.648 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 25 | 29 | 25 | 18 | 17 | 17 | 15 | 19 | 23 |
N.S. |
1 | 1.16 | 1.00 | 0.72 | 0.68 | 0.68 | 0.60 | 0.76 | 0.92 |
time (sec) |
N/A | 0.182 | 0.006 | 0.055 | 0.202 | 0.236 | 0.080 | 0.335 | 0.105 |