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 | N/A | N/A | N/A | N/A | N/A | F(-1) | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 40 | 40 | 42 | 36 | 39 | 38 | 0 | 38 | 39 |
N.S. |
1 | 1.00 | 1.05 | 0.90 | 0.98 | 0.95 | 0.00 | 0.95 | 0.98 |
time (sec) |
N/A | 0.226 | 0.282 | 2.844 | 1.008 | 0.290 | 0.000 | 0.951 | 0.740 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | F | B | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | Yes | TBD | TBD | TBD | TBD | TBD |
size | 279 | 285 | 0 | 681 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.02 | 0.00 | 2.44 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.763 | 0.000 | 6.016 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | F | A | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | Yes | TBD | TBD | TBD | TBD | TBD |
size | 207 | 213 | 0 | 394 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.03 | 0.00 | 1.90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.600 | 0.000 | 2.973 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | F | A | F | F | F(-1) | F | F(-1) |
verified |
N/A | Yes | N/A | Yes | TBD | TBD | TBD | TBD | TBD |
size | 141 | 143 | 0 | 171 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.01 | 0.00 | 1.21 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.450 | 0.000 | 0.955 | 0.000 | 0.000 | 0.000 | 0.000 | 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 | 40 | 40 | 42 | 36 | 39 | 48 | 61 | 38 | 39 |
N.S. |
1 | 1.00 | 1.05 | 0.90 | 0.98 | 1.20 | 1.52 | 0.95 | 0.98 |
time (sec) |
N/A | 0.227 | 0.471 | 1.430 | 0.406 | 0.254 | 133.802 | 0.446 | 0.482 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | F(-1) | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD |
size | 40 | 40 | 42 | 36 | 293 | 91 | 0 | 38 | 39 |
N.S. |
1 | 1.00 | 1.05 | 0.90 | 7.32 | 2.28 | 0.00 | 0.95 | 0.98 |
time (sec) |
N/A | 0.222 | 4.677 | 1.422 | 1.886 | 0.245 | 0.000 | 0.541 | 1.305 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | F | A | F | F(-2) | F | F | F(-1) |
verified |
N/A | No | N/A | Yes | TBD | TBD | TBD | TBD | TBD |
size | 84 | 86 | 0 | 102 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.02 | 0.00 | 1.21 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.359 | 0.000 | 1.068 | 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 | A | A | A | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 81 | 74 | 50 | 0 | 0 | 55 | 105 | 82 | 0 |
N.S. |
1 | 0.91 | 0.62 | 0.00 | 0.00 | 0.68 | 1.30 | 1.01 | 0.00 |
time (sec) |
N/A | 0.269 | 0.201 | 0.000 | 0.000 | 0.256 | 0.583 | 0.289 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | A | A | A | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 82 | 76 | 50 | 0 | 0 | 46 | 85 | 69 | 0 |
N.S. |
1 | 0.93 | 0.61 | 0.00 | 0.00 | 0.56 | 1.04 | 0.84 | 0.00 |
time (sec) |
N/A | 0.261 | 0.158 | 0.000 | 0.000 | 0.267 | 0.351 | 0.284 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | A | A | A | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 41 | 42 | 30 | 0 | 0 | 36 | 58 | 44 | 0 |
N.S. |
1 | 1.02 | 0.73 | 0.00 | 0.00 | 0.88 | 1.41 | 1.07 | 0.00 |
time (sec) |
N/A | 0.244 | 0.051 | 0.000 | 0.000 | 0.249 | 0.219 | 0.283 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | A | A | A | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 39 | 42 | 32 | 0 | 0 | 28 | 37 | 31 | 0 |
N.S. |
1 | 1.08 | 0.82 | 0.00 | 0.00 | 0.72 | 0.95 | 0.79 | 0.00 |
time (sec) |
N/A | 0.193 | 0.048 | 0.000 | 0.000 | 0.256 | 0.110 | 0.264 | 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 | 45 | 45 | 79 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.76 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.251 | 0.072 | 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 | 87 | 88 | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.01 | 0.63 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.299 | 0.081 | 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 | B | A | F | B | F | B | B |
verified |
N/A | No | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 48 | 49 | 528 | 45 | 0 | 140 | 0 | 95 | 43 |
N.S. |
1 | 1.02 | 11.00 | 0.94 | 0.00 | 2.92 | 0.00 | 1.98 | 0.90 |
time (sec) |
N/A | 0.250 | 0.926 | 0.629 | 0.000 | 0.273 | 0.000 | 0.290 | 0.694 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F(-2) | F(-2) | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 26 | 26 | 56 | 21 | 0 | 0 | 0 | 37 | 0 |
N.S. |
1 | 1.00 | 2.15 | 0.81 | 0.00 | 0.00 | 0.00 | 1.42 | 0.00 |
time (sec) |
N/A | 0.286 | 0.112 | 1.616 | 0.000 | 0.000 | 0.000 | 0.297 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | F | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 5 | 5 | 5 | 6 | 0 | 0 | 0 | 5 | 0 |
N.S. |
1 | 1.00 | 1.00 | 1.20 | 0.00 | 0.00 | 0.00 | 1.00 | 0.00 |
time (sec) |
N/A | 0.249 | 0.056 | 0.776 | 0.000 | 0.000 | 0.000 | 0.278 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F(-2) | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 39 | 0 | 0 | 42 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 0.00 | 1.08 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.269 | 0.067 | 0.000 | 0.000 | 0.265 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F(-2) | A | F | F | B |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 31 | 31 | 25 | 0 | 0 | 27 | 0 | 0 | 25 |
N.S. |
1 | 1.00 | 0.81 | 0.00 | 0.00 | 0.87 | 0.00 | 0.00 | 0.81 |
time (sec) |
N/A | 0.262 | 0.040 | 0.000 | 0.000 | 0.234 | 0.000 | 0.000 | 0.340 |