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 | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 220 | 209 | 185 | 218 | 0 | 1886 | 332 | 310 | 1828 | 430 |
N.S. |
1 | 0.95 | 0.84 | 0.99 | 0.00 | 8.57 | 1.51 | 1.41 | 8.31 | 1.95 |
time (sec) |
N/A | 0.553 | 0.086 | 0.623 | 0.000 | 0.186 | 1.970 | 0.149 | 0.240 | 0.861 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 216 | 200 | 182 | 190 | 0 | 1822 | 298 | 298 | 1796 | 349 |
N.S. |
1 | 0.93 | 0.84 | 0.88 | 0.00 | 8.44 | 1.38 | 1.38 | 8.31 | 1.62 |
time (sec) |
N/A | 0.541 | 0.013 | 0.617 | 0.000 | 0.128 | 1.891 | 0.150 | 0.223 | 0.996 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | B | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 26 | 26 | 26 | 47 | 136 | 136 | 155 | 46 | 135 | 134 |
N.S. |
1 | 1.00 | 1.00 | 1.81 | 5.23 | 5.23 | 5.96 | 1.77 | 5.19 | 5.15 |
time (sec) |
N/A | 0.245 | 0.008 | 0.580 | 0.046 | 0.118 | 1.711 | 0.142 | 0.275 | 0.631 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 207 | 207 | 181 | 216 | 0 | 1780 | 323 | 286 | 1829 | 357 |
N.S. |
1 | 1.00 | 0.87 | 1.04 | 0.00 | 8.60 | 1.56 | 1.38 | 8.84 | 1.72 |
time (sec) |
N/A | 0.553 | 0.009 | 0.340 | 0.000 | 0.182 | 1.647 | 0.159 | 0.245 | 0.613 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | B | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 94 | 77 | 66 | 148 | 258 | 474 | 292 | 153 | 1405 | 249 |
N.S. |
1 | 0.82 | 0.70 | 1.57 | 2.74 | 5.04 | 3.11 | 1.63 | 14.95 | 2.65 |
time (sec) |
N/A | 0.345 | 0.032 | 0.784 | 0.049 | 0.092 | 2.250 | 0.148 | 0.283 | 1.257 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 237 | 228 | 199 | 219 | 0 | 879 | 445 | 295 | 2310 | 485 |
N.S. |
1 | 0.96 | 0.84 | 0.92 | 0.00 | 3.71 | 1.88 | 1.24 | 9.75 | 2.05 |
time (sec) |
N/A | 0.603 | 0.063 | 0.870 | 0.000 | 0.161 | 2.717 | 0.142 | 0.281 | 2.041 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 235 | 225 | 195 | 192 | 0 | 1051 | 500 | 371 | 2805 | 563 |
N.S. |
1 | 0.96 | 0.83 | 0.82 | 0.00 | 4.47 | 2.13 | 1.58 | 11.94 | 2.40 |
time (sec) |
N/A | 0.610 | 0.082 | 0.897 | 0.000 | 0.139 | 4.090 | 0.154 | 0.273 | 2.923 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | B | B | B | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 116 | 95 | 83 | 165 | 474 | 919 | 578 | 262 | 2664 | 507 |
N.S. |
1 | 0.82 | 0.72 | 1.42 | 4.09 | 7.92 | 4.98 | 2.26 | 22.97 | 4.37 |
time (sec) |
N/A | 0.378 | 0.033 | 0.855 | 0.066 | 0.169 | 3.782 | 0.144 | 0.268 | 2.820 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | B | F(-1) | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 30 | 30 | 30 | 29 | 28 | 104 | 0 | 62 | 105 | 88 |
N.S. |
1 | 1.00 | 1.00 | 0.97 | 0.93 | 3.47 | 0.00 | 2.07 | 3.50 | 2.93 |
time (sec) |
N/A | 0.268 | 0.100 | 1.216 | 0.032 | 0.112 | 0.000 | 0.125 | 0.250 | 0.605 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 29 | 80 | 22 | 21 | 108 | 126 | 25 | 117 | 107 |
N.S. |
1 | 1.26 | 3.48 | 0.96 | 0.91 | 4.70 | 5.48 | 1.09 | 5.09 | 4.65 |
time (sec) |
N/A | 0.281 | 0.014 | 0.629 | 0.025 | 0.072 | 0.035 | 0.124 | 0.242 | 0.067 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 172 | 22 | 21 | 250 | 299 | 21 | 286 | 229 |
N.S. |
1 | 1.00 | 7.48 | 0.96 | 0.91 | 10.87 | 13.00 | 0.91 | 12.43 | 9.96 |
time (sec) |
N/A | 0.257 | 0.027 | 0.625 | 0.026 | 0.086 | 0.061 | 0.124 | 0.209 | 0.506 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 308 | 22 | 21 | 432 | 541 | 21 | 523 | 376 |
N.S. |
1 | 1.00 | 13.39 | 0.96 | 0.91 | 18.78 | 23.52 | 0.91 | 22.74 | 16.35 |
time (sec) |
N/A | 0.255 | 0.056 | 0.642 | 0.033 | 0.105 | 0.078 | 0.116 | 0.208 | 0.289 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 22 | 22 | 22 | 21 | 20 | 54 | 56 | 20 | 54 | 54 |
N.S. |
1 | 1.00 | 1.00 | 0.95 | 0.91 | 2.45 | 2.55 | 0.91 | 2.45 | 2.45 |
time (sec) |
N/A | 0.255 | 0.008 | 0.556 | 0.031 | 0.095 | 0.174 | 0.131 | 0.247 | 0.102 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 23 | 22 | 21 | 66 | 73 | 21 | 55 | 55 |
N.S. |
1 | 1.00 | 1.00 | 0.96 | 0.91 | 2.87 | 3.17 | 0.91 | 2.39 | 2.39 |
time (sec) |
N/A | 0.256 | 0.009 | 0.582 | 0.025 | 0.083 | 0.847 | 0.130 | 0.223 | 0.583 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | B | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 23 | 23 | 23 | 22 | 21 | 172 | 197 | 21 | 173 | 171 |
N.S. |
1 | 1.00 | 1.00 | 0.96 | 0.91 | 7.48 | 8.57 | 0.91 | 7.52 | 7.43 |
time (sec) |
N/A | 0.255 | 0.011 | 0.620 | 0.026 | 0.161 | 2.291 | 0.147 | 0.237 | 1.555 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | C | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 234 | 227 | 132 | 108 | 0 | 6315 | 238 | 0 | 460 | 374 |
N.S. |
1 | 0.97 | 0.56 | 0.46 | 0.00 | 26.99 | 1.02 | 0.00 | 1.97 | 1.60 |
time (sec) |
N/A | 0.819 | 0.038 | 0.201 | 0.000 | 10.892 | 1.749 | 0.000 | 0.240 | 1.502 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | C | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 210 | 197 | 81 | 74 | 0 | 4759 | 158 | 0 | 327 | 437 |
N.S. |
1 | 0.94 | 0.39 | 0.35 | 0.00 | 22.66 | 0.75 | 0.00 | 1.56 | 2.08 |
time (sec) |
N/A | 0.773 | 0.021 | 0.198 | 0.000 | 0.774 | 0.624 | 0.000 | 0.258 | 0.706 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | C | A | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 180 | 171 | 79 | 72 | 0 | 1950 | 83 | 0 | 232 | 145 |
N.S. |
1 | 0.95 | 0.44 | 0.40 | 0.00 | 10.83 | 0.46 | 0.00 | 1.29 | 0.81 |
time (sec) |
N/A | 0.618 | 0.018 | 0.188 | 0.000 | 0.730 | 0.417 | 0.000 | 0.203 | 0.657 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 140 | 131 | 116 | 71 | 0 | 442 | 26 | 160 | 111 | 144 |
N.S. |
1 | 0.94 | 0.83 | 0.51 | 0.00 | 3.16 | 0.19 | 1.14 | 0.79 | 1.03 |
time (sec) |
N/A | 0.462 | 0.009 | 0.178 | 0.000 | 0.110 | 0.160 | 0.121 | 0.198 | 0.003 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | C | F(-1) | F(-2) | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 224 | 241 | 119 | 105 | 0 | 4370 | 0 | 0 | 1347 | 553 |
N.S. |
1 | 1.08 | 0.53 | 0.47 | 0.00 | 19.51 | 0.00 | 0.00 | 6.01 | 2.47 |
time (sec) |
N/A | 0.886 | 0.036 | 0.250 | 0.000 | 0.780 | 0.000 | 0.000 | 0.223 | 0.537 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | C | F(-1) | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 312 | 315 | 173 | 143 | 0 | 8919 | 0 | 0 | 3657 | 1588 |
N.S. |
1 | 1.01 | 0.55 | 0.46 | 0.00 | 28.59 | 0.00 | 0.00 | 11.72 | 5.09 |
time (sec) |
N/A | 1.200 | 0.064 | 0.275 | 0.000 | 1.146 | 0.000 | 0.000 | 0.276 | 0.719 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | C | F(-1) | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 388 | 380 | 244 | 200 | 0 | 14765 | 0 | 0 | 6603 | 1328 |
N.S. |
1 | 0.98 | 0.63 | 0.52 | 0.00 | 38.05 | 0.00 | 0.00 | 17.02 | 3.42 |
time (sec) |
N/A | 1.413 | 0.104 | 0.312 | 0.000 | 3.838 | 0.000 | 0.000 | 0.305 | 0.884 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 361 | 349 | 1258 | 4787 | 0 | 0 | 0 | 0 | 403 | 0 |
N.S. |
1 | 0.97 | 3.48 | 13.26 | 0.00 | 0.00 | 0.00 | 0.00 | 1.12 | 0.00 |
time (sec) |
N/A | 0.770 | 15.652 | 18.929 | 0.000 | 0.000 | 0.000 | 0.000 | 0.300 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 326 | 318 | 553 | 2153 | 0 | 0 | 0 | 0 | 473 | 0 |
N.S. |
1 | 0.98 | 1.70 | 6.60 | 0.00 | 0.00 | 0.00 | 0.00 | 1.45 | 0.00 |
time (sec) |
N/A | 0.676 | 11.671 | 5.142 | 0.000 | 0.000 | 0.000 | 0.000 | 0.325 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 154 | 152 | 389 | 1528 | 0 | 0 | 0 | 0 | 186 | 0 |
N.S. |
1 | 0.99 | 2.53 | 9.92 | 0.00 | 0.00 | 0.00 | 0.00 | 1.21 | 0.00 |
time (sec) |
N/A | 0.466 | 10.377 | 3.628 | 0.000 | 0.000 | 0.000 | 0.000 | 0.294 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 111 | 111 | 90 | 1036 | 0 | 0 | 0 | 0 | 96 | 56 |
N.S. |
1 | 1.00 | 0.81 | 9.33 | 0.00 | 0.00 | 0.00 | 0.00 | 0.86 | 0.50 |
time (sec) |
N/A | 0.296 | 10.040 | 2.385 | 0.000 | 0.000 | 0.000 | 0.000 | 0.288 | 0.843 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 413 | 436 | 442 | 1582 | 0 | 0 | 0 | 0 | 19 | 0 |
N.S. |
1 | 1.06 | 1.07 | 3.83 | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.00 |
time (sec) |
N/A | 1.154 | 10.534 | 3.066 | 0.000 | 0.000 | 0.000 | 0.000 | 200.040 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F(-1) | F | F | F | F(-1) |
verified |
N/A | Yes | No | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 629 | 658 | 1385 | 5348 | 0 | 0 | 0 | 0 | 19 | 0 |
N.S. |
1 | 1.05 | 2.20 | 8.50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.03 | 0.00 |
time (sec) |
N/A | 1.979 | 13.871 | 13.339 | 0.000 | 0.000 | 0.000 | 0.000 | 200.037 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | F | B | B | F(-1) |
verified |
N/A | No | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 55 | 61 | 64 | 864 | 0 | 379 | 0 | 134 | 131 | 0 |
N.S. |
1 | 1.11 | 1.16 | 15.71 | 0.00 | 6.89 | 0.00 | 2.44 | 2.38 | 0.00 |
time (sec) |
N/A | 0.328 | 0.324 | 10.571 | 0.000 | 0.342 | 0.000 | 0.154 | 0.213 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | F | B | B | F(-1) |
verified |
N/A | No | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 55 | 62 | 64 | 864 | 0 | 379 | 0 | 134 | 131 | 0 |
N.S. |
1 | 1.13 | 1.16 | 15.71 | 0.00 | 6.89 | 0.00 | 2.44 | 2.38 | 0.00 |
time (sec) |
N/A | 0.339 | 0.003 | 2.330 | 0.000 | 0.405 | 0.000 | 0.161 | 0.220 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | F | B | B | F(-1) |
verified |
N/A | No | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 55 | 34 | 64 | 864 | 0 | 379 | 0 | 134 | 131 | 0 |
N.S. |
1 | 0.62 | 1.16 | 15.71 | 0.00 | 6.89 | 0.00 | 2.44 | 2.38 | 0.00 |
time (sec) |
N/A | 0.527 | 0.006 | 2.227 | 0.000 | 0.358 | 0.000 | 0.159 | 0.232 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | B | F | B | B | F(-1) |
verified |
N/A | No | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 56 | 62 | 64 | 864 | 0 | 379 | 0 | 134 | 131 | 0 |
N.S. |
1 | 1.11 | 1.14 | 15.43 | 0.00 | 6.77 | 0.00 | 2.39 | 2.34 | 0.00 |
time (sec) |
N/A | 0.551 | 0.004 | 2.195 | 0.000 | 0.348 | 0.000 | 0.159 | 0.199 | 0.000 |