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 | A | A | A | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 184 | 185 | 255 | 246 | 252 | 264 | 345 | 309 | 309 | 273 |
N.S. |
1 | 1.01 | 1.39 | 1.34 | 1.37 | 1.43 | 1.88 | 1.68 | 1.68 | 1.48 |
time (sec) |
N/A | 0.386 | 0.306 | 0.530 | 0.113 | 0.113 | 0.525 | 0.528 | 0.186 | 1.424 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | B | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 144 | 139 | 218 | 191 | 186 | 196 | 262 | 223 | 223 | 197 |
N.S. |
1 | 0.97 | 1.51 | 1.33 | 1.29 | 1.36 | 1.82 | 1.55 | 1.55 | 1.37 |
time (sec) |
N/A | 0.354 | 0.277 | 0.306 | 0.164 | 0.111 | 0.435 | 0.238 | 0.189 | 0.636 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 103 | 110 | 163 | 119 | 126 | 133 | 160 | 160 | 149 | 127 |
N.S. |
1 | 1.07 | 1.58 | 1.16 | 1.22 | 1.29 | 1.55 | 1.55 | 1.45 | 1.23 |
time (sec) |
N/A | 0.315 | 0.214 | 0.227 | 0.125 | 0.133 | 0.356 | 0.220 | 0.179 | 0.881 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 76 | 77 | 77 | 69 | 71 | 71 | 87 | 85 | 76 | 67 |
N.S. |
1 | 1.01 | 1.01 | 0.91 | 0.93 | 0.93 | 1.14 | 1.12 | 1.00 | 0.88 |
time (sec) |
N/A | 0.267 | 0.004 | 0.122 | 0.129 | 0.112 | 0.302 | 0.133 | 0.187 | 0.676 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 138 | 138 | 138 | 156 | 0 | 0 | 0 | 0 | 30 | 0 |
N.S. |
1 | 1.00 | 1.00 | 1.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.22 | 0.00 |
time (sec) |
N/A | 0.478 | 0.040 | 0.246 | 0.000 | 0.000 | 0.000 | 0.000 | 0.184 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | C | B | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 98 | 94 | 111 | 110 | 107 | 116 | 658 | 1152 | 135 | 112 |
N.S. |
1 | 0.96 | 1.13 | 1.12 | 1.09 | 1.18 | 6.71 | 11.76 | 1.38 | 1.14 |
time (sec) |
N/A | 0.275 | 0.126 | 0.240 | 0.125 | 0.134 | 1.936 | 0.320 | 0.192 | 4.101 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | B | C | C | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 146 | 157 | 192 | 153 | 214 | 313 | 2866 | 594 | 390 | 591 |
N.S. |
1 | 1.08 | 1.32 | 1.05 | 1.47 | 2.14 | 19.63 | 4.07 | 2.67 | 4.05 |
time (sec) |
N/A | 0.383 | 0.206 | 0.353 | 0.127 | 0.232 | 4.025 | 0.470 | 0.188 | 5.024 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | B | C | C | B | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 206 | 210 | 254 | 204 | 374 | 642 | 9202 | 1156 | 799 | 0 |
N.S. |
1 | 1.02 | 1.23 | 0.99 | 1.82 | 3.12 | 44.67 | 5.61 | 3.88 | 0.00 |
time (sec) |
N/A | 0.476 | 0.402 | 0.507 | 0.126 | 0.725 | 7.997 | 1.882 | 0.194 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 376 | 375 | 472 | 667 | 0 | 0 | 0 | 0 | 564 | 0 |
N.S. |
1 | 1.00 | 1.26 | 1.77 | 0.00 | 0.00 | 0.00 | 0.00 | 1.50 | 0.00 |
time (sec) |
N/A | 0.808 | 0.533 | 0.819 | 0.000 | 0.000 | 0.000 | 0.000 | 0.189 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 270 | 284 | 312 | 503 | 0 | 0 | 0 | 0 | 364 | 0 |
N.S. |
1 | 1.05 | 1.16 | 1.86 | 0.00 | 0.00 | 0.00 | 0.00 | 1.35 | 0.00 |
time (sec) |
N/A | 0.624 | 0.364 | 0.557 | 0.000 | 0.000 | 0.000 | 0.000 | 0.188 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 171 | 184 | 172 | 292 | 0 | 0 | 0 | 0 | 186 | 0 |
N.S. |
1 | 1.08 | 1.01 | 1.71 | 0.00 | 0.00 | 0.00 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.513 | 0.265 | 0.349 | 0.000 | 0.000 | 0.000 | 0.000 | 0.195 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | F | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | N/A | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 223 | 223 | 0 | 1199 | 0 | 0 | 0 | 0 | 55 | 0 |
N.S. |
1 | 1.00 | 0.00 | 5.38 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.00 |
time (sec) |
N/A | 0.309 | 0.000 | 4.974 | 0.000 | 0.000 | 0.000 | 0.000 | 0.195 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 341 | 350 | 300 | 513 | 0 | 0 | 0 | 0 | 1426 | 0 |
N.S. |
1 | 1.03 | 0.88 | 1.50 | 0.00 | 0.00 | 0.00 | 0.00 | 4.18 | 0.00 |
time (sec) |
N/A | 0.683 | 1.869 | 1.993 | 0.000 | 0.000 | 0.000 | 0.000 | 0.199 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | F(-1) | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 496 | 496 | 479 | 729 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.97 | 1.47 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.898 | 3.882 | 3.012 | 0.000 | 0.000 | 0.000 | 0.000 | 0.215 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 652 | 645 | 855 | 3122 | 0 | 0 | 0 | 0 | 1008 | 0 |
N.S. |
1 | 0.99 | 1.31 | 4.79 | 0.00 | 0.00 | 0.00 | 0.00 | 1.55 | 0.00 |
time (sec) |
N/A | 1.512 | 1.148 | 17.915 | 0.000 | 0.000 | 0.000 | 0.000 | 0.189 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 411 | 430 | 621 | 2633 | 0 | 0 | 0 | 0 | 642 | 0 |
N.S. |
1 | 1.05 | 1.51 | 6.41 | 0.00 | 0.00 | 0.00 | 0.00 | 1.56 | 0.00 |
time (sec) |
N/A | 1.015 | 0.663 | 8.147 | 0.000 | 0.000 | 0.000 | 0.000 | 0.199 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | Yes | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 264 | 278 | 342 | 3886 | 0 | 0 | 0 | 0 | 321 | 0 |
N.S. |
1 | 1.05 | 1.30 | 14.72 | 0.00 | 0.00 | 0.00 | 0.00 | 1.22 | 0.00 |
time (sec) |
N/A | 0.806 | 0.555 | 2.636 | 0.000 | 0.000 | 0.000 | 0.000 | 0.195 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | F | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | N/A | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 320 | 320 | 0 | 2398 | 0 | 0 | 0 | 0 | 80 | 0 |
N.S. |
1 | 1.00 | 0.00 | 7.49 | 0.00 | 0.00 | 0.00 | 0.00 | 0.25 | 0.00 |
time (sec) |
N/A | 0.418 | 0.000 | 4.460 | 0.000 | 0.000 | 0.000 | 0.000 | 0.205 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | F | C | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | N/A | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 499 | 502 | 0 | 2398 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.01 | 0.00 | 4.81 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.877 | 0.000 | 5.253 | 0.000 | 0.000 | 0.000 | 0.000 | 0.235 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | F | C | F(-1) | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | N/A | No | TBD | TBD | TBD | TBD | TBD | TBD |
size | 936 | 921 | 0 | 40034 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 0.98 | 0.00 | 42.77 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 1.544 | 0.000 | 312.132 | 0.000 | 0.000 | 0.000 | 0.000 | 0.350 | 0.000 |
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 | 206 | 256 | 252 | 303 | 323 | 1013 | 403 | 304 | 332 | 419 |
N.S. |
1 | 1.24 | 1.22 | 1.47 | 1.57 | 4.92 | 1.96 | 1.48 | 1.61 | 2.03 |
time (sec) |
N/A | 0.563 | 1.879 | 0.898 | 0.120 | 0.165 | 12.452 | 0.936 | 0.200 | 3.846 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | A | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 154 | 200 | 153 | 146 | 168 | 147 | 206 | 184 | 169 | 203 |
N.S. |
1 | 1.30 | 0.99 | 0.95 | 1.09 | 0.95 | 1.34 | 1.19 | 1.10 | 1.32 |
time (sec) |
N/A | 0.437 | 0.070 | 0.272 | 0.118 | 0.117 | 6.811 | 0.371 | 0.206 | 2.970 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 501 | 469 | 326 | 138 | 0 | 0 | 0 | 0 | 32 | 0 |
N.S. |
1 | 0.94 | 0.65 | 0.28 | 0.00 | 0.00 | 0.00 | 0.00 | 0.06 | 0.00 |
time (sec) |
N/A | 1.120 | 14.615 | 0.385 | 0.000 | 0.000 | 0.000 | 0.000 | 0.202 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | C | F(-1) | F | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 269 | 338 | 321 | 297 | 287 | 2478078 | 0 | 0 | 784 | 883 |
N.S. |
1 | 1.26 | 1.19 | 1.10 | 1.07 | 9212.19 | 0.00 | 0.00 | 2.91 | 3.28 |
time (sec) |
N/A | 0.779 | 0.448 | 0.491 | 0.114 | 84.490 | 0.000 | 0.000 | 0.318 | 0.934 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | B | F | F | F | F | F | F | F(-1) |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 1325 | 1325 | 4850 | 0 | 0 | 0 | 0 | 0 | 216 | 0 |
N.S. |
1 | 1.00 | 3.66 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.16 | 0.00 |
time (sec) |
N/A | 2.508 | 26.494 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.198 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | N/A | N/A | N/A | N/A | N/A | F(-1) | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 20 | 20 | 22 | 20 | 47 | 36 | 0 | 22 | 59 | 22 |
N.S. |
1 | 1.00 | 1.10 | 1.00 | 2.35 | 1.80 | 0.00 | 1.10 | 2.95 | 1.10 |
time (sec) |
N/A | 0.200 | 39.004 | 0.073 | 0.441 | 0.105 | 0.000 | 0.133 | 0.233 | 0.597 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | N/A | N/A | N/A | F(-2) | N/A | F(-1) | N/A | N/A | N/A |
verified |
N/A | N/A | N/A | N/A | TBD | TBD | TBD | TBD | TBD | TBD |
size | 20 | 20 | 22 | 20 | 0 | 47 | 0 | 22 | 977 | 22 |
N.S. |
1 | 1.00 | 1.10 | 1.00 | 0.00 | 2.35 | 0.00 | 1.10 | 48.85 | 1.10 |
time (sec) |
N/A | 0.196 | 103.424 | 0.224 | 0.000 | 0.099 | 0.000 | 2.029 | 4.098 | 0.784 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | B | A | F(-2) | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 275 | 325 | 297 | 503 | 280 | 0 | 151 | 312 | 362 | 988 |
N.S. |
1 | 1.18 | 1.08 | 1.83 | 1.02 | 0.00 | 0.55 | 1.13 | 1.32 | 3.59 |
time (sec) |
N/A | 0.853 | 125.210 | 2.134 | 0.116 | 0.000 | 27.097 | 8.809 | 0.184 | 0.849 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | F(-2) | A | A | B | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 246 | 297 | 310 | 305 | 232 | 0 | 104 | 236 | 237 | 485 |
N.S. |
1 | 1.21 | 1.26 | 1.24 | 0.94 | 0.00 | 0.42 | 0.96 | 0.96 | 1.97 |
time (sec) |
N/A | 0.783 | 0.067 | 0.479 | 0.118 | 0.000 | 12.407 | 0.689 | 0.182 | 0.681 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | C | C | F | F | F(-1) | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 739 | 687 | 522 | 172 | 0 | 0 | 0 | 0 | 32 | 0 |
N.S. |
1 | 0.93 | 0.71 | 0.23 | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.00 |
time (sec) |
N/A | 1.490 | 9.899 | 0.456 | 0.000 | 0.000 | 0.000 | 0.000 | 0.254 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Reduce | Mupad | |
grade |
N/A | A | A | A | A | F(-1) | F(-1) | F | F | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD | TBD |
size | 734 | 913 | 536 | 862 | 464 | 0 | 0 | 0 | 20 | 2105 |
N.S. |
1 | 1.24 | 0.73 | 1.17 | 0.63 | 0.00 | 0.00 | 0.00 | 0.03 | 2.87 |
time (sec) |
N/A | 1.557 | 11.746 | 0.852 | 0.121 | 0.000 | 0.000 | 0.000 | 200.016 | 1.094 |