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 | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 144 | 144 | 180 | 205 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.25 | 1.42 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.134 | 0.151 | 0.892 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 115 | 115 | 138 | 186 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.20 | 1.62 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.093 | 0.101 | 0.621 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 82 | 82 | 115 | 163 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.40 | 1.99 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.075 | 0.054 | 0.234 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 76 | 76 | 107 | 156 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.41 | 2.05 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.044 | 0.044 | 0.365 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 71 | 71 | 104 | 168 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.46 | 2.37 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.079 | 0.094 | 0.376 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 107 | 107 | 158 | 168 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.48 | 1.57 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.104 | 0.152 | 0.619 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 124 | 124 | 150 | 283 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.21 | 2.28 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.133 | 0.284 | 0.964 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 180 | 180 | 294 | 264 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.63 | 1.47 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.168 | 0.172 | 1.092 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 155 | 155 | 203 | 281 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.31 | 1.81 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.132 | 0.261 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 136 | 136 | 251 | 228 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.85 | 1.68 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.094 | 0.129 | 0.487 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | B | A | F | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 57 | 57 | 49 | 98 | 136 | 54 | 0 | 100 | -1 |
N.S. | 1 | 1.00 | 0.86 | 1.72 | 2.39 | 0.95 | 0.00 | 1.75 | -0.02 |
time (sec) |
N/A | 0.035 | 0.043 | 0.168 | 0.479 | 2.680 | 0.000 | 0.441 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 132 | 132 | 220 | 228 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.67 | 1.73 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.067 | 0.139 | 0.202 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | B | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 122 | 122 | 152 | 340 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.25 | 2.79 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.123 | 0.292 | 1.216 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F(-1) | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 177 | 177 | 251 | 257 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 1.00 | 1.42 | 1.45 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.134 | 0.254 | 0.632 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 159 | 159 | 217 | 347 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.36 | 2.18 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.188 | 0.428 | 0.635 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 149 | 149 | 153 | 177 | 167 | 127 | 211 | 165 | -1 |
N.S. | 1 | 1.00 | 1.03 | 1.19 | 1.12 | 0.85 | 1.42 | 1.11 | -0.01 |
time (sec) |
N/A | 0.087 | 0.100 | 0.187 | 0.475 | 2.729 | 0.492 | 0.419 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 120 | 120 | 125 | 161 | 146 | 113 | 177 | 141 | -1 |
N.S. | 1 | 1.00 | 1.04 | 1.34 | 1.22 | 0.94 | 1.48 | 1.18 | -0.01 |
time (sec) |
N/A | 0.092 | 0.081 | 0.108 | 0.470 | 2.880 | 0.344 | 0.407 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 122 | 122 | 131 | 172 | 126 | 107 | 158 | 121 | -1 |
N.S. | 1 | 1.00 | 1.07 | 1.41 | 1.03 | 0.88 | 1.30 | 0.99 | -0.01 |
time (sec) |
N/A | 0.062 | 0.074 | 0.158 | 0.487 | 2.473 | 0.245 | 0.441 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 81 | 81 | 91 | 112 | 96 | 87 | 114 | 91 | -1 |
N.S. | 1 | 1.00 | 1.12 | 1.38 | 1.19 | 1.07 | 1.41 | 1.12 | -0.01 |
time (sec) |
N/A | 0.050 | 0.052 | 0.006 | 0.481 | 2.563 | 0.147 | 0.421 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 132 | 132 | 124 | 130 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.94 | 0.98 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.177 | 0.136 | 0.569 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | A | B | B |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 66 | 66 | 80 | 79 | 83 | 160 | 78 | 859 | 71 |
N.S. | 1 | 1.00 | 1.21 | 1.20 | 1.26 | 2.42 | 1.18 | 13.02 | 1.08 |
time (sec) |
N/A | 0.053 | 0.045 | 0.010 | 0.487 | 2.990 | 2.358 | 0.735 | 0.376 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 119 | 119 | 105 | 148 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 0.88 | 1.24 | 0.00 | 0.00 | 0.00 | 0.00 | -0.01 |
time (sec) |
N/A | 0.167 | 0.100 | 1.383 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | B | A | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 85 | 85 | 130 | 119 | 121 | 184 | 170 | 3082 | -1 |
N.S. | 1 | 1.00 | 1.53 | 1.40 | 1.42 | 2.16 | 2.00 | 36.26 | -0.01 |
time (sec) |
N/A | 0.062 | 0.045 | 0.010 | 0.478 | 3.893 | 3.018 | 122.203 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 135 | 135 | 99 | 169 | 160 | 110 | 197 | 160 | -1 |
N.S. | 1 | 1.00 | 0.73 | 1.25 | 1.19 | 0.81 | 1.46 | 1.19 | -0.01 |
time (sec) |
N/A | 0.096 | 0.074 | 0.133 | 0.476 | 3.325 | 0.342 | 0.416 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 205 | 205 | 149 | 270 | 267 | 168 | 326 | 270 | -1 |
N.S. | 1 | 1.00 | 0.73 | 1.32 | 1.30 | 0.82 | 1.59 | 1.32 | -0.00 |
time (sec) |
N/A | 0.174 | 0.098 | 0.133 | 0.469 | 2.342 | 0.719 | 0.448 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | A | A | A | A | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 292 | 292 | 212 | 393 | 400 | 239 | 502 | 408 | -1 |
N.S. | 1 | 1.00 | 0.73 | 1.35 | 1.37 | 0.82 | 1.72 | 1.40 | -0.00 |
time (sec) |
N/A | 0.238 | 0.112 | 0.155 | 0.464 | 2.281 | 1.491 | 0.436 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | C | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 521 | 521 | 811 | 224 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.56 | 0.43 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.559 | 0.772 | 11.921 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | A | C | F | F | F | F | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 727 | 727 | 1065 | 796 | 0 | 0 | 0 | 0 | -1 |
N.S. | 1 | 1.00 | 1.46 | 1.09 | 0.00 | 0.00 | 0.00 | 0.00 | -0.00 |
time (sec) |
N/A | 0.735 | 1.527 | 7.592 | 0.000 | 0.000 | 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 | A |
verified |
N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.05 |
time (sec) |
N/A | 0.012 | 2.099 | 0.603 | 0.000 | 0.000 | 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 | A |
verified |
N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
N.S. |
1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | -0.05 |
time (sec) |
N/A | 0.011 | 1.554 | 0.724 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | F | F(-2) | B | F | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 66 | 66 | 68 | 0 | 0 | 276 | 0 | 75 | -1 |
N.S. |
1 | 1.00 | 1.03 | 0.00 | 0.00 | 4.18 | 0.00 | 1.14 | -0.02 |
time (sec) |
N/A | 0.067 | 0.056 | 0.417 | 0.000 | 2.835 | 0.000 | 0.439 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | F | F(-2) | B | F | A | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 136 | 136 | 120 | 0 | 0 | 580 | 0 | 185 | -1 |
N.S. |
1 | 1.00 | 0.88 | 0.00 | 0.00 | 4.26 | 0.00 | 1.36 | -0.01 |
time (sec) |
N/A | 0.108 | 0.144 | 0.417 | 0.000 | 3.851 | 0.000 | 0.521 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
A | A | C | F | F(-2) | B | F | B | F |
verified |
N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 211 | 211 | 162 | 0 | 0 | 1066 | 0 | 398 | -1 |
N.S. |
1 | 1.00 | 0.77 | 0.00 | 0.00 | 5.05 | 0.00 | 1.89 | -0.00 |
time (sec) |
N/A | 0.581 | 0.220 | 0.405 | 0.000 | 2.958 | 0.000 | 0.648 | 0.000 |