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 | F | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 41 | 41 | 44 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.263 | 0.797 | 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 | C | C | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 184 | 187 | 389 | 1528 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.02 | 2.11 | 8.30 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.384 | 10.356 | 4.993 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | C | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 131 | 134 | 90 | 1036 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.02 | 0.69 | 7.91 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.217 | 20.049 | 2.355 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 54 | 54 | 419 | 47 | 0 | 305 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 7.76 | 0.87 | 0.00 | 5.65 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.348 | 12.408 | 2.702 | 0.000 | 7.879 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 53 | 53 | 416 | 46 | 0 | 304 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 7.85 | 0.87 | 0.00 | 5.74 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.361 | 11.905 | 2.712 | 0.000 | 7.665 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F | B | F | B | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 84 | 83 | 90 | 69 | 0 | 154 | 0 | 164 | 0 |
N.S. |
1 | 0.99 | 1.07 | 0.82 | 0.00 | 1.83 | 0.00 | 1.95 | 0.00 |
time (sec) |
N/A | 0.346 | 0.232 | 1.487 | 0.000 | 0.256 | 0.000 | 0.336 | 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 | 20 | 20 | 17 | 23 | 7 | 22 | 26 | 15 | 13 |
N.S. |
1 | 1.00 | 0.85 | 1.15 | 0.35 | 1.10 | 1.30 | 0.75 | 0.65 |
time (sec) |
N/A | 0.147 | 0.004 | 0.068 | 0.294 | 0.257 | 0.183 | 0.299 | 21.024 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 46 | 60 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.30 | 0.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.181 | 0.026 | 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 | A | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
size | 88 | 88 | 246 | 92 | 0 | 324 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 2.80 | 1.05 | 0.00 | 3.68 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.387 | 0.941 | 3.960 | 0.000 | 2.158 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | A | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 88 | 88 | 213 | 96 | 0 | 331 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 2.42 | 1.09 | 0.00 | 3.76 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.441 | 0.921 | 3.883 | 0.000 | 2.104 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 46 | 46 | 107 | 0 | 0 | 161 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 2.33 | 0.00 | 0.00 | 3.50 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.621 | 5.000 | 0.000 | 0.000 | 5.583 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 46 | 47 | 114 | 0 | 0 | 161 | 0 | 0 | 0 |
N.S. |
1 | 1.02 | 2.48 | 0.00 | 0.00 | 3.50 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.604 | 4.440 | 0.000 | 0.000 | 5.158 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 46 | 46 | 107 | 0 | 0 | 161 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 2.33 | 0.00 | 0.00 | 3.50 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.037 | 0.007 | 0.000 | 0.000 | 5.510 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 46 | 47 | 114 | 0 | 0 | 161 | 0 | 0 | 0 |
N.S. |
1 | 1.02 | 2.48 | 0.00 | 0.00 | 3.50 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.031 | 0.008 | 0.000 | 0.000 | 5.109 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | B | A | B | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 19 | 66 | 27 | 147 | 94 | 96 | 17 | 58 | 132 |
N.S. |
1 | 3.47 | 1.42 | 7.74 | 4.95 | 5.05 | 0.89 | 3.05 | 6.95 |
time (sec) |
N/A | 2.081 | 0.266 | 1.382 | 0.256 | 0.243 | 8.672 | 0.337 | 22.531 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
size | 90 | 90 | 180 | 1673 | 0 | 458 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 2.00 | 18.59 | 0.00 | 5.09 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.208 | 0.356 | 17.656 | 0.000 | 5.029 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | B | A | C | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
size | 103 | 425 | 181 | 692 | 0 | 289 | 0 | 0 | 0 |
N.S. |
1 | 4.13 | 1.76 | 6.72 | 0.00 | 2.81 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.957 | 1.344 | 10.010 | 0.000 | 2.921 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | C | C | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 49 | 175 | 53 | 49 | 0 | 51 | 0 | 0 | 0 |
N.S. |
1 | 3.57 | 1.08 | 1.00 | 0.00 | 1.04 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.381 | 0.171 | 1.869 | 0.000 | 0.288 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 80 | 80 | 383 | 66 | 0 | 472 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 4.79 | 0.82 | 0.00 | 5.90 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.522 | 10.565 | 3.405 | 0.000 | 28.967 | 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 | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 1 | 1 | 1 | 2 | 1 | 1 | 0 | 1 | 1 |
N.S. |
1 | 1.00 | 1.00 | 2.00 | 1.00 | 1.00 | 0.00 | 1.00 | 1.00 |
time (sec) |
N/A | 0.131 | 0.000 | 1.175 | 0.184 | 0.235 | 0.021 | 0.316 | 0.005 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | C | C | F | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 42 | 122 | 129 | 89 | 0 | 73 | 0 | 193 | 549 |
N.S. |
1 | 2.90 | 3.07 | 2.12 | 0.00 | 1.74 | 0.00 | 4.60 | 13.07 |
time (sec) |
N/A | 0.356 | 0.074 | 0.393 | 0.000 | 0.262 | 0.000 | 0.374 | 20.258 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | C | C | C | F | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 42 | 149 | 129 | 89 | 0 | 73 | 0 | 193 | 549 |
N.S. |
1 | 3.55 | 3.07 | 2.12 | 0.00 | 1.74 | 0.00 | 4.60 | 13.07 |
time (sec) |
N/A | 0.559 | 0.070 | 1.071 | 0.000 | 0.261 | 0.000 | 0.347 | 0.506 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | F | A | A | A | A | B | B | B |
verified |
N/A | N/A | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 34 | 0 | 31 | 32 | 16 | 31 | 73 | 81 | 31 |
N.S. |
1 | 0.00 | 0.91 | 0.94 | 0.47 | 0.91 | 2.15 | 2.38 | 0.91 |
time (sec) |
N/A | 0.000 | 0.092 | 1.341 | 0.270 | 0.266 | 17.963 | 0.364 | 19.805 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | F | A | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD |
size | 177 | 177 | 182 | 1597 | 0 | 164 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 1.03 | 9.02 | 0.00 | 0.93 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.296 | 5.604 | 1.541 | 0.000 | 0.353 | 0.000 | 0.000 | 0.000 |
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 | 100 | 139 | 100 | 228 | 0 | 97 | 0 | 0 | 0 |
N.S. |
1 | 1.39 | 1.00 | 2.28 | 0.00 | 0.97 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.254 | 3.632 | 2.072 | 0.000 | 0.337 | 0.000 | 0.000 | 0.000 |