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 | F | A | A | B | A | A | B | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 28 | 0 | 43 | 52 | 73 | 43 | 37 | 73 | 78 |
N.S. |
1 | 0.00 | 1.54 | 1.86 | 2.61 | 1.54 | 1.32 | 2.61 | 2.79 |
time (sec) |
N/A | 11.651 | 0.122 | 0.078 | 0.596 | 0.364 | 0.217 | 0.604 | 7.543 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | A | A | A | A | F(-1) |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 30 | 0 | 26 | 26 | 27 | 25 | 22 | 25 | -1 |
N.S. |
1 | 0.00 | 0.87 | 0.87 | 0.90 | 0.83 | 0.73 | 0.83 | -0.03 |
time (sec) |
N/A | 6.399 | 0.074 | 0.025 | 0.622 | 0.354 | 0.187 | 0.670 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | C | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 21 | 25 | 25 | 25 | 31 | 24 | 19 | 24 | 19 |
N.S. |
1 | 1.19 | 1.19 | 1.19 | 1.48 | 1.14 | 0.90 | 1.14 | 0.90 |
time (sec) |
N/A | 0.059 | 0.012 | 0.072 | 0.306 | 0.395 | 0.048 | 0.422 | 6.235 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | C | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 19 | 19 | 19 | 26 | 24 | 15 | 32 | 18 |
N.S. |
1 | 1.00 | 1.00 | 1.00 | 1.37 | 1.26 | 0.79 | 1.68 | 0.95 |
time (sec) |
N/A | 0.294 | 0.061 | 0.014 | 0.287 | 0.382 | 0.074 | 0.417 | 6.803 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | A | A | A | A | F |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 32 | 0 | 30 | 33 | 30 | 30 | 27 | 32 | -1 |
N.S. |
1 | 0.00 | 0.94 | 1.03 | 0.94 | 0.94 | 0.84 | 1.00 | -0.03 |
time (sec) |
N/A | 1.451 | 2.226 | 0.060 | 0.308 | 0.384 | 0.132 | 0.405 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | F | A | A | B | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 0 | 26 | 28 | 0 | 26 | 31 | 46 | 20 |
N.S. |
1 | 0.00 | 0.90 | 0.97 | 0.00 | 0.90 | 1.07 | 1.59 | 0.69 |
time (sec) |
N/A | 0.590 | 0.297 | 0.045 | 0.000 | 0.380 | 0.146 | 0.396 | 7.535 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | B | A | B | B | F | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 28 | 0 | 32 | 62 | 26 | 82 | 53 | 0 | 48 |
N.S. |
1 | 0.00 | 1.14 | 2.21 | 0.93 | 2.93 | 1.89 | 0.00 | 1.71 |
time (sec) |
N/A | 48.113 | 0.136 | 0.094 | 0.514 | 0.403 | 0.471 | 0.000 | 8.743 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | B | F | A | A | A | F |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 32 | 0 | 36 | 86 | 0 | 61 | 31 | 62 | -1 |
N.S. |
1 | 0.00 | 1.12 | 2.69 | 0.00 | 1.91 | 0.97 | 1.94 | -0.03 |
time (sec) |
N/A | 4.659 | 0.125 | 0.271 | 0.000 | 0.382 | 0.172 | 0.440 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 28 | 27 | 24 | 28 | 21 | 22 | 27 | 22 | 24 |
N.S. |
1 | 0.96 | 0.86 | 1.00 | 0.75 | 0.79 | 0.96 | 0.79 | 0.86 |
time (sec) |
N/A | 0.465 | 0.089 | 0.200 | 0.304 | 0.369 | 0.073 | 0.417 | 9.360 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 16 | 11 | 11 | 12 | 9 | 9 | 10 | 9 | 9 |
N.S. |
1 | 0.69 | 0.69 | 0.75 | 0.56 | 0.56 | 0.62 | 0.56 | 0.56 |
time (sec) |
N/A | 0.003 | 0.010 | 0.012 | 0.273 | 0.394 | 0.027 | 0.403 | 0.015 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F(-1) | A | A | A | A | A | B | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 28 | 0 | 40 | 37 | 46 | 40 | 36 | 219 | 40 |
N.S. |
1 | 0.00 | 1.43 | 1.32 | 1.64 | 1.43 | 1.29 | 7.82 | 1.43 |
time (sec) |
N/A | 180.002 | 0.193 | 1.057 | 0.419 | 0.368 | 0.193 | 8.937 | 9.270 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 22 | 37 | 21 | 36 | 17 | 17 | 14 | 17 | 18 |
N.S. |
1 | 1.68 | 0.95 | 1.64 | 0.77 | 0.77 | 0.64 | 0.77 | 0.82 |
time (sec) |
N/A | 0.050 | 0.024 | 0.015 | 0.262 | 0.357 | 0.052 | 0.396 | 0.079 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | A | A | A | A | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 18 | 0 | 17 | 18 | 20 | 20 | 15 | 22 | 17 |
N.S. |
1 | 0.00 | 0.94 | 1.00 | 1.11 | 1.11 | 0.83 | 1.22 | 0.94 |
time (sec) |
N/A | 0.234 | 0.038 | 0.793 | 0.303 | 0.379 | 0.043 | 0.402 | 8.556 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 31 | 22 | 29 | 30 | 20 | 20 | 30 | 18 |
N.S. |
1 | 1.35 | 0.96 | 1.26 | 1.30 | 0.87 | 0.87 | 1.30 | 0.78 |
time (sec) |
N/A | 0.020 | 0.025 | 0.019 | 0.284 | 0.348 | 0.070 | 0.406 | 9.301 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 27 | 26 | 25 | 18 | 17 | 25 | 12 | 17 | 17 |
N.S. |
1 | 0.96 | 0.93 | 0.67 | 0.63 | 0.93 | 0.44 | 0.63 | 0.63 |
time (sec) |
N/A | 0.069 | 0.013 | 0.054 | 0.270 | 0.349 | 0.070 | 0.414 | 7.122 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 18 | 17 | 17 | 19 | 14 | 22 | 14 | 14 | 14 |
N.S. |
1 | 0.94 | 0.94 | 1.06 | 0.78 | 1.22 | 0.78 | 0.78 | 0.78 |
time (sec) |
N/A | 0.084 | 0.018 | 0.030 | 0.261 | 0.378 | 0.050 | 0.411 | 0.106 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | B | A | B | B | B | B | B | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 30 | 163 | 35 | 288 | 227 | 91 | 128 | 96 | 33 |
N.S. |
1 | 5.43 | 1.17 | 9.60 | 7.57 | 3.03 | 4.27 | 3.20 | 1.10 |
time (sec) |
N/A | 0.423 | 2.053 | 0.050 | 0.300 | 0.360 | 0.134 | 0.391 | 6.707 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | C | A | B | F(-2) | B | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 25 | 0 | 34 | 621 | 34 | 67 | 0 | 74 | 67 |
N.S. |
1 | 0.00 | 1.36 | 24.84 | 1.36 | 2.68 | 0.00 | 2.96 | 2.68 |
time (sec) |
N/A | 1.683 | 0.051 | 1.082 | 0.528 | 0.362 | 0.000 | 0.563 | 7.219 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 27 | 13 | 29 | 0 | 12 | 15 | 21 | 12 |
N.S. |
1 | 1.42 | 0.68 | 1.53 | 0.00 | 0.63 | 0.79 | 1.11 | 0.63 |
time (sec) |
N/A | 0.240 | 0.025 | 0.054 | 0.000 | 0.372 | 0.100 | 0.419 | 7.287 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | B | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 33 | 21 | 43 | 21 | 49 | 20 | 17 | 22 | 20 |
N.S. |
1 | 0.64 | 1.30 | 0.64 | 1.48 | 0.61 | 0.52 | 0.67 | 0.61 |
time (sec) |
N/A | 0.056 | 0.884 | 0.052 | 0.501 | 0.375 | 0.060 | 0.398 | 8.052 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | B | B | B | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 34 | 46 | 37 | 115 | 141 | 143 | 53 | 59 | 133 |
N.S. |
1 | 1.35 | 1.09 | 3.38 | 4.15 | 4.21 | 1.56 | 1.74 | 3.91 |
time (sec) |
N/A | 1.183 | 0.122 | 0.343 | 0.521 | 0.370 | 0.179 | 0.439 | 6.982 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | B | A | A | F | A | A | F | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 99 | 31 | 37 | 0 | 31 | 26 | 0 | 55 |
N.S. |
1 | 3.41 | 1.07 | 1.28 | 0.00 | 1.07 | 0.90 | 0.00 | 1.90 |
time (sec) |
N/A | 2.888 | 1.137 | 0.177 | 0.000 | 0.388 | 0.217 | 0.000 | 6.565 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | A | A | F(-2) | A | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 32 | 0 | 32 | 37 | 30 | 29 | 0 | 30 | 47 |
N.S. |
1 | 0.00 | 1.00 | 1.16 | 0.94 | 0.91 | 0.00 | 0.94 | 1.47 |
time (sec) |
N/A | 3.047 | 0.082 | 0.165 | 0.504 | 0.377 | 0.000 | 0.419 | 8.348 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 28 | 31 | 23 | 25 | 21 | 30 | 19 | 24 | 24 |
N.S. |
1 | 1.11 | 0.82 | 0.89 | 0.75 | 1.07 | 0.68 | 0.86 | 0.86 |
time (sec) |
N/A | 0.017 | 0.006 | 0.022 | 0.266 | 0.377 | 0.035 | 0.397 | 0.089 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | B | A | A | B | B |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 25 | 0 | 28 | 25 | 108 | 34 | 37 | 62 | 43 |
N.S. |
1 | 0.00 | 1.12 | 1.00 | 4.32 | 1.36 | 1.48 | 2.48 | 1.72 |
time (sec) |
N/A | 29.935 | 0.160 | 0.062 | 4.909 | 0.460 | 0.906 | 0.519 | 7.103 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 11 | 11 | 11 | 10 | 10 | 8 | 10 | 10 |
N.S. |
1 | 0.38 | 0.38 | 0.38 | 0.34 | 0.34 | 0.28 | 0.34 | 0.34 |
time (sec) |
N/A | 0.016 | 0.003 | 0.016 | 0.260 | 0.377 | 0.034 | 0.409 | 0.093 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 15 | 15 | 16 | 15 | 15 | 14 | 15 | 15 |
N.S. |
1 | 0.79 | 0.79 | 0.84 | 0.79 | 0.79 | 0.74 | 0.79 | 0.79 |
time (sec) |
N/A | 0.005 | 0.000 | 0.011 | 0.296 | 0.383 | 0.009 | 0.398 | 7.034 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | F | A | F(-2) | F | F |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 29 | 0 | 28 | 26 | 0 | 32 | 0 | 0 | -1 |
N.S. |
1 | 0.00 | 0.97 | 0.90 | 0.00 | 1.10 | 0.00 | 0.00 | -0.03 |
time (sec) |
N/A | 5.708 | 0.156 | 0.093 | 0.000 | 0.381 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 17 | 17 | 17 | 14 | 13 | 27 | 12 | 13 | 14 |
N.S. |
1 | 1.00 | 1.00 | 0.82 | 0.76 | 1.59 | 0.71 | 0.76 | 0.82 |
time (sec) |
N/A | 0.020 | 0.012 | 0.029 | 0.259 | 0.347 | 0.044 | 0.416 | 0.140 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 20 | 17 | 17 | 15 | 0 | 14 | 15 | 14 | 14 |
N.S. |
1 | 0.85 | 0.85 | 0.75 | 0.00 | 0.70 | 0.75 | 0.70 | 0.70 |
time (sec) |
N/A | 0.207 | 0.092 | 0.037 | 0.000 | 0.374 | 0.061 | 0.405 | 6.192 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | B | B | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 24 | 31 | 24 | 61 | 51 | 58 | 19 | 24 | 24 |
N.S. |
1 | 1.29 | 1.00 | 2.54 | 2.12 | 2.42 | 0.79 | 1.00 | 1.00 |
time (sec) |
N/A | 0.189 | 0.129 | 0.112 | 0.501 | 0.350 | 0.087 | 0.428 | 8.464 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 13 | 18 | 15 | 17 | 16 | 16 | 17 | 16 | 16 |
N.S. |
1 | 1.38 | 1.15 | 1.31 | 1.23 | 1.23 | 1.31 | 1.23 | 1.23 |
time (sec) |
N/A | 0.007 | 0.006 | 0.018 | 0.253 | 0.337 | 0.039 | 0.425 | 0.060 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | B | A | A | B | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 25 | 25 | 109 | 62 | 17 | 26 | 62 | 17 |
N.S. |
1 | 1.09 | 1.09 | 4.74 | 2.70 | 0.74 | 1.13 | 2.70 | 0.74 |
time (sec) |
N/A | 0.027 | 0.007 | 0.032 | 0.284 | 0.362 | 0.042 | 0.402 | 7.065 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | B |
verified | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 24 | 21 | 21 | 21 | 21 | 20 | 19 | 20 | 20 |
N.S. |
1 | 0.88 | 0.88 | 0.88 | 0.88 | 0.83 | 0.79 | 0.83 | 0.83 |
time (sec) |
N/A | 0.154 | 0.064 | 0.066 | 0.310 | 0.352 | 0.269 | 0.444 | 8.152 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | F | A | A | A | A | A | A | F |
verified | N/A | N/A | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 31 | 0 | 23 | 22 | 42 | 42 | 42 | 46 | -1 |
N.S. |
1 | 0.00 | 0.74 | 0.71 | 1.35 | 1.35 | 1.35 | 1.48 | -0.03 |
time (sec) |
N/A | 2.492 | 0.118 | 0.033 | 0.377 | 0.349 | 0.097 | 0.452 | 0.000 |