Detailed conclusion table per each integral is given by 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 just an F.
In this table,the column N.S. in the table below, which stands for normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To help make the table fit, Mathematica was abbreviated to MMA.
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | A | A | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 78 | 78 | 60 | 56 | 0 | 39 | 54 | 63 | 48 |
N.S. |
1 | 1.00 | 0.77 | 0.72 | 0.00 | 0.50 | 0.69 | 0.81 | 0.62 |
time (sec) |
N/A | 0.053 | 0.129 | 0.259 | 0.000 | 3.117 | 0.084 | 0.426 | 0.324 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | F(-2) | A | A | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 29 | 42 | 93 | 0 | 26 | 36 | 69 | 74 |
N.S. |
1 | 1.00 | 1.45 | 3.21 | 0.00 | 0.90 | 1.24 | 2.38 | 2.55 |
time (sec) |
N/A | 0.110 | 0.096 | 0.168 | 0.000 | 3.016 | 0.083 | 0.392 | 0.485 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F(-2) | A | A | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 50 | 50 | 32 | 37 | 0 | 27 | 34 | 51 | 36 |
N.S. |
1 | 1.00 | 0.64 | 0.74 | 0.00 | 0.54 | 0.68 | 1.02 | 0.72 |
time (sec) |
N/A | 0.042 | 0.069 | 0.256 | 0.000 | 2.374 | 0.060 | 0.407 | 0.253 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | F(-2) | A | A | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 19 | 49 | 0 | 14 | 17 | 31 | 40 |
N.S. |
1 | 1.00 | 1.00 | 2.58 | 0.00 | 0.74 | 0.89 | 1.63 | 2.11 |
time (sec) |
N/A | 0.071 | 0.016 | 0.210 | 0.000 | 3.943 | 0.054 | 0.421 | 0.276 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | B | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 18 | 18 | 44 | 34 | 45 | 33 | 0 | 29 | 21 |
N.S. | 1 | 1.00 | 2.44 | 1.89 | 2.50 | 1.83 | 0.00 | 1.61 | 1.17 |
time (sec) |
N/A | 0.072 | 0.040 | 0.214 | 0.278 | 3.696 | 0.000 | 0.408 | 0.250 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 20 | 20 | 17 | 13 | 12 | 36 | 0 | 53 | 12 |
N.S. | 1 | 1.00 | 0.85 | 0.65 | 0.60 | 1.80 | 0.00 | 2.65 | 0.60 |
time (sec) |
N/A | 0.027 | 0.038 | 0.243 | 0.283 | 4.471 | 0.000 | 0.422 | 0.227 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | B | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 22 | 22 | 48 | 68 | 94 | 73 | 0 | 55 | 47 |
N.S. | 1 | 1.00 | 2.18 | 3.09 | 4.27 | 3.32 | 0.00 | 2.50 | 2.14 |
time (sec) |
N/A | 0.107 | 0.160 | 0.250 | 0.280 | 5.672 | 0.000 | 0.448 | 0.303 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 24 | 18 | 13 | 30 | 0 | 13 | 13 |
N.S. | 1 | 1.00 | 1.26 | 0.95 | 0.68 | 1.58 | 0.00 | 0.68 | 0.68 |
time (sec) |
N/A | 0.031 | 0.059 | 0.197 | 0.281 | 2.976 | 0.000 | 0.424 | 0.200 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 40 | 40 | 61 | 116 | 167 | 123 | 0 | 87 | 81 |
N.S. | 1 | 1.00 | 1.52 | 2.90 | 4.18 | 3.08 | 0.00 | 2.18 | 2.02 |
time (sec) |
N/A | 0.129 | 0.649 | 0.262 | 0.298 | 3.498 | 0.000 | 0.404 | 0.460 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 37 | 37 | 26 | 31 | 25 | 48 | 0 | 25 | 27 |
N.S. | 1 | 1.00 | 0.70 | 0.84 | 0.68 | 1.30 | 0.00 | 0.68 | 0.73 |
time (sec) |
N/A | 0.038 | 0.096 | 0.254 | 0.283 | 2.545 | 0.000 | 0.418 | 0.212 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | A | A | A | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 126 | 126 | 179 | 172 | 240 | 178 | 0 | 270 | 279 |
N.S. | 1 | 1.00 | 1.42 | 1.37 | 1.90 | 1.41 | 0.00 | 2.14 | 2.21 |
time (sec) |
N/A | 0.240 | 0.646 | 0.362 | 0.504 | 3.487 | 0.000 | 0.453 | 0.497 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | B | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 123 | 123 | 112 | 170 | 279 | 214 | 0 | 201 | 291 |
N.S. | 1 | 1.00 | 0.91 | 1.38 | 2.27 | 1.74 | 0.00 | 1.63 | 2.37 |
time (sec) |
N/A | 0.156 | 1.196 | 0.394 | 0.528 | 2.901 | 0.000 | 0.447 | 0.671 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | A | A | A | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 73 | 73 | 82 | 98 | 122 | 95 | 0 | 155 | 118 |
N.S. | 1 | 1.00 | 1.12 | 1.34 | 1.67 | 1.30 | 0.00 | 2.12 | 1.62 |
time (sec) |
N/A | 0.121 | 0.340 | 0.271 | 0.534 | 2.766 | 0.000 | 0.429 | 0.366 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 65 | 65 | 61 | 81 | 105 | 144 | 0 | 94 | 93 |
N.S. | 1 | 1.00 | 0.94 | 1.25 | 1.62 | 2.22 | 0.00 | 1.45 | 1.43 |
time (sec) |
N/A | 0.065 | 0.212 | 0.287 | 0.512 | 4.091 | 0.000 | 0.438 | 0.376 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 47 | 47 | 76 | 63 | 98 | 140 | 0 | 90 | 408 |
N.S. | 1 | 1.00 | 1.62 | 1.34 | 2.09 | 2.98 | 0.00 | 1.91 | 8.68 |
time (sec) |
N/A | 0.083 | 0.099 | 0.369 | 0.505 | 3.870 | 0.000 | 0.436 | 0.657 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 29 | 27 | 21 | 20 | 57 | 0 | 21 | 20 |
N.S. | 1 | 1.00 | 0.93 | 0.72 | 0.69 | 1.97 | 0.00 | 0.72 | 0.69 |
time (sec) |
N/A | 0.040 | 0.090 | 0.288 | 0.285 | 3.132 | 0.000 | 0.428 | 0.200 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | B | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 79 | 79 | 192 | 150 | 203 | 166 | 0 | 158 | 346 |
N.S. | 1 | 1.00 | 2.43 | 1.90 | 2.57 | 2.10 | 0.00 | 2.00 | 4.38 |
time (sec) |
N/A | 0.142 | 0.503 | 0.497 | 0.505 | 5.978 | 0.000 | 0.435 | 0.580 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | F | A | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 73 | 73 | 66 | 56 | 56 | 106 | 0 | 60 | 55 |
N.S. | 1 | 1.00 | 0.90 | 0.77 | 0.77 | 1.45 | 0.00 | 0.82 | 0.75 |
time (sec) |
N/A | 0.078 | 0.301 | 0.367 | 0.283 | 2.464 | 0.000 | 0.411 | 0.223 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | B | A | B | B | F | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 25 | 25 | 57 | 37 | 67 | 66 | 0 | 54 | 54 |
N.S. | 1 | 1.00 | 2.28 | 1.48 | 2.68 | 2.64 | 0.00 | 2.16 | 2.16 |
time (sec) |
N/A | 0.061 | 0.058 | 0.276 | 0.524 | 2.869 | 0.000 | 0.441 | 0.322 |