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 | F | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 28 | 28 | 28 | 21 | 0 | 52 | 0 | 34 | 0 |
N.S. |
1 | 1.00 | 1.00 | 0.75 | 0.00 | 1.86 | 0.00 | 1.21 | 0.00 |
time (sec) |
N/A | 0.205 | 0.027 | 0.957 | 0.000 | 0.249 | 0.000 | 0.288 | 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 | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 |
N.S. |
1 | 1.00 | 1.00 | 1.50 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
time (sec) |
N/A | 0.106 | 0.002 | 0.056 | 0.279 | 0.241 | 0.039 | 0.277 | 0.050 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 14 | 14 | 14 | 11 | 25 | 25 | 8 | 25 | 10 |
N.S. |
1 | 1.00 | 1.00 | 0.79 | 1.79 | 1.79 | 0.57 | 1.79 | 0.71 |
time (sec) |
N/A | 0.131 | 0.012 | 0.402 | 0.192 | 0.265 | 0.550 | 0.284 | 0.077 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | C | A | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 252 | 267 | 20 | 15 | 0 | 9 | 31 | 0 | 231 |
N.S. |
1 | 1.06 | 0.08 | 0.06 | 0.00 | 0.04 | 0.12 | 0.00 | 0.92 |
time (sec) |
N/A | 0.317 | 10.019 | 0.158 | 0.000 | 0.069 | 0.401 | 0.000 | 0.245 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | C | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 16 | 16 | 16 | 13 | 29 | 29 | 31 | 30 | 180 |
N.S. |
1 | 1.00 | 1.00 | 0.81 | 1.81 | 1.81 | 1.94 | 1.88 | 11.25 |
time (sec) |
N/A | 0.140 | 0.020 | 0.291 | 0.197 | 0.239 | 0.574 | 0.316 | 0.115 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | B | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 76 | 76 | 78 | 148 | 0 | 72 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.03 | 1.95 | 0.00 | 0.95 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.192 | 3.950 | 1.160 | 0.000 | 0.300 | 0.000 | 0.000 | 0.000 |
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 | 13 | 13 | 26 | 17 | 144 | 16 | 10 | 16 | 11 |
N.S. |
1 | 1.00 | 2.00 | 1.31 | 11.08 | 1.23 | 0.77 | 1.23 | 0.85 |
time (sec) |
N/A | 0.222 | 0.036 | 0.097 | 0.204 | 0.250 | 0.069 | 0.290 | 0.059 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | B | B | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 42 | 42 | 42 | 43 | 142 | 85 | 0 | 0 | 32 |
N.S. |
1 | 1.00 | 1.00 | 1.02 | 3.38 | 2.02 | 0.00 | 0.00 | 0.76 |
time (sec) |
N/A | 0.193 | 0.010 | 0.078 | 0.307 | 0.265 | 0.000 | 0.000 | 0.192 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | C | A | A | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 2 | 2 | 2 | 3 | 13 | 2 | 2 | 2 | 0 |
N.S. |
1 | 1.00 | 1.00 | 1.50 | 6.50 | 1.00 | 1.00 | 1.00 | 0.00 |
time (sec) |
N/A | 0.151 | 0.011 | 0.081 | 0.243 | 0.242 | 0.459 | 0.310 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | A | F(-2) | F | F | B |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 17 | 17 | 17 | 0 | 21 | 0 | 0 | 0 | 14 |
N.S. |
1 | 1.00 | 1.00 | 0.00 | 1.24 | 0.00 | 0.00 | 0.00 | 0.82 |
time (sec) |
N/A | 0.693 | 0.592 | 0.000 | 0.256 | 0.000 | 0.000 | 0.000 | 0.288 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 13 | 13 | 13 | 12 | 11 | 24 | 12 | 12 | 11 |
N.S. |
1 | 1.00 | 1.00 | 0.92 | 0.85 | 1.85 | 0.92 | 0.92 | 0.85 |
time (sec) |
N/A | 0.161 | 0.023 | 0.041 | 0.200 | 0.251 | 0.410 | 0.266 | 0.249 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | A | F | F | B | F | A | B |
verified |
N/A | N/A | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 13 | 0 | 13 | 0 | 0 | 24 | 0 | 11 | 11 |
N.S. |
1 | 0.00 | 1.00 | 0.00 | 0.00 | 1.85 | 0.00 | 0.85 | 0.85 |
time (sec) |
N/A | 0.000 | 0.177 | 0.000 | 0.000 | 0.242 | 0.000 | 0.333 | 0.229 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 23 | 26 | 0 | 19 | 19 | 24 | 19 |
N.S. |
1 | 1.00 | 1.00 | 1.13 | 0.00 | 0.83 | 0.83 | 1.04 | 0.83 |
time (sec) |
N/A | 0.434 | 0.132 | 0.145 | 0.000 | 0.240 | 0.232 | 0.367 | 0.248 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 39 | 39 | 41 | 23 | 0 | 22 | 24 | 22 | 22 |
N.S. |
1 | 1.00 | 1.05 | 0.59 | 0.00 | 0.56 | 0.62 | 0.56 | 0.56 |
time (sec) |
N/A | 0.180 | 0.009 | 0.076 | 0.000 | 0.236 | 0.055 | 0.281 | 0.252 |