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 | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 101 | 101 | 91 | 97 | 81 | 130 | 0 | 91 | 0 |
N.S. |
1 | 1.00 | 0.90 | 0.96 | 0.80 | 1.29 | 0.00 | 0.90 | 0.00 |
time (sec) |
N/A | 0.341 | 0.114 | 0.289 | 0.197 | 0.268 | 0.000 | 0.261 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | C | C | A | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 65 | 65 | 51 | 52 | 45 | 72 | 0 | 45 | 0 |
N.S. |
1 | 1.00 | 0.78 | 0.80 | 0.69 | 1.11 | 0.00 | 0.69 | 0.00 |
time (sec) |
N/A | 0.263 | 0.052 | 0.190 | 0.199 | 0.256 | 0.000 | 0.274 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 65 | 65 | 71 | 48 | 47 | 76 | 0 | 49 | 0 |
N.S. |
1 | 1.00 | 1.09 | 0.74 | 0.72 | 1.17 | 0.00 | 0.75 | 0.00 |
time (sec) |
N/A | 0.284 | 0.070 | 0.369 | 0.189 | 0.267 | 0.000 | 0.268 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 115 | 115 | 122 | 105 | 89 | 165 | 0 | 101 | 0 |
N.S. |
1 | 1.00 | 1.06 | 0.91 | 0.77 | 1.43 | 0.00 | 0.88 | 0.00 |
time (sec) |
N/A | 0.378 | 0.250 | 0.386 | 0.198 | 0.263 | 0.000 | 0.276 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 110 | 110 | 102 | 100 | 90 | 211 | 0 | 106 | 0 |
N.S. |
1 | 1.00 | 0.93 | 0.91 | 0.82 | 1.92 | 0.00 | 0.96 | 0.00 |
time (sec) |
N/A | 0.355 | 0.092 | 0.126 | 0.202 | 0.276 | 0.000 | 0.281 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 148 | 148 | 149 | 126 | 127 | 278 | 0 | 355 | 0 |
N.S. |
1 | 1.00 | 1.01 | 0.85 | 0.86 | 1.88 | 0.00 | 2.40 | 0.00 |
time (sec) |
N/A | 0.424 | 0.474 | 0.394 | 0.288 | 0.265 | 0.000 | 0.306 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 239 | 239 | 286 | 207 | 200 | 443 | 0 | 223 | 0 |
N.S. |
1 | 1.00 | 1.20 | 0.87 | 0.84 | 1.85 | 0.00 | 0.93 | 0.00 |
time (sec) |
N/A | 0.551 | 0.293 | 1.074 | 0.293 | 0.294 | 0.000 | 0.281 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 115 | 115 | 123 | 126 | 102 | 251 | 0 | 132 | 0 |
N.S. |
1 | 1.00 | 1.07 | 1.10 | 0.89 | 2.18 | 0.00 | 1.15 | 0.00 |
time (sec) |
N/A | 0.434 | 0.206 | 0.146 | 0.207 | 0.275 | 0.000 | 0.268 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | C | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 161 | 161 | 220 | 158 | 143 | 334 | 0 | 387 | 0 |
N.S. |
1 | 1.00 | 1.37 | 0.98 | 0.89 | 2.07 | 0.00 | 2.40 | 0.00 |
time (sec) |
N/A | 0.519 | 0.454 | 0.502 | 0.286 | 0.263 | 0.000 | 0.293 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 257 | 257 | 353 | 265 | 228 | 539 | 0 | 281 | 0 |
N.S. |
1 | 1.00 | 1.37 | 1.03 | 0.89 | 2.10 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.764 | 0.564 | 1.339 | 0.297 | 0.264 | 0.000 | 0.288 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 133 | 133 | 104 | 117 | 105 | 216 | 0 | 132 | 0 |
N.S. |
1 | 1.00 | 0.78 | 0.88 | 0.79 | 1.62 | 0.00 | 0.99 | 0.00 |
time (sec) |
N/A | 0.400 | 0.110 | 0.114 | 0.204 | 0.277 | 0.000 | 0.256 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | A | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 161 | 161 | 131 | 139 | 131 | 242 | 0 | 150 | 0 |
N.S. |
1 | 1.00 | 0.81 | 0.86 | 0.81 | 1.50 | 0.00 | 0.93 | 0.00 |
time (sec) |
N/A | 0.434 | 0.171 | 0.331 | 0.206 | 0.260 | 0.000 | 0.272 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 271 | 271 | 214 | 234 | 211 | 426 | 0 | 264 | 0 |
N.S. |
1 | 1.00 | 0.79 | 0.86 | 0.78 | 1.57 | 0.00 | 0.97 | 0.00 |
time (sec) |
N/A | 0.599 | 0.335 | 0.973 | 0.212 | 0.276 | 0.000 | 0.279 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 81 | 81 | 75 | 70 | 69 | 145 | 0 | 75 | 0 |
N.S. |
1 | 1.00 | 0.93 | 0.86 | 0.85 | 1.79 | 0.00 | 0.93 | 0.00 |
time (sec) |
N/A | 0.366 | 0.253 | 0.106 | 0.198 | 0.268 | 0.000 | 0.268 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 128 | 128 | 179 | 101 | 100 | 254 | 0 | 107 | 0 |
N.S. |
1 | 1.00 | 1.40 | 0.79 | 0.78 | 1.98 | 0.00 | 0.84 | 0.00 |
time (sec) |
N/A | 0.424 | 0.389 | 0.319 | 0.207 | 0.272 | 0.000 | 0.278 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 171 | 171 | 270 | 144 | 143 | 491 | 0 | 155 | 0 |
N.S. |
1 | 1.00 | 1.58 | 0.84 | 0.84 | 2.87 | 0.00 | 0.91 | 0.00 |
time (sec) |
N/A | 0.532 | 0.907 | 0.954 | 0.203 | 0.280 | 0.000 | 0.283 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 140 | 140 | 165 | 147 | 127 | 321 | 0 | 172 | 0 |
N.S. |
1 | 1.00 | 1.18 | 1.05 | 0.91 | 2.29 | 0.00 | 1.23 | 0.00 |
time (sec) |
N/A | 0.518 | 0.475 | 0.145 | 0.207 | 0.274 | 0.000 | 0.279 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 183 | 183 | 258 | 177 | 161 | 420 | 0 | 198 | 0 |
N.S. |
1 | 1.00 | 1.41 | 0.97 | 0.88 | 2.30 | 0.00 | 1.08 | 0.00 |
time (sec) |
N/A | 0.604 | 1.078 | 0.438 | 0.206 | 0.292 | 0.000 | 0.275 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 300 | 300 | 478 | 302 | 263 | 847 | 0 | 352 | 0 |
N.S. |
1 | 1.00 | 1.59 | 1.01 | 0.88 | 2.82 | 0.00 | 1.17 | 0.00 |
time (sec) |
N/A | 0.843 | 4.445 | 1.354 | 0.211 | 0.292 | 0.000 | 0.290 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 153 | 153 | 134 | 162 | 129 | 262 | 0 | 167 | 0 |
N.S. |
1 | 1.00 | 0.88 | 1.06 | 0.84 | 1.71 | 0.00 | 1.09 | 0.00 |
time (sec) |
N/A | 0.554 | 0.201 | 0.122 | 0.205 | 0.276 | 0.000 | 0.285 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 219 | 219 | 183 | 210 | 185 | 341 | 0 | 223 | 0 |
N.S. |
1 | 1.00 | 0.84 | 0.96 | 0.84 | 1.56 | 0.00 | 1.02 | 0.00 |
time (sec) |
N/A | 0.576 | 0.360 | 0.358 | 0.201 | 0.271 | 0.000 | 0.287 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 315 | 315 | 262 | 326 | 263 | 526 | 0 | 339 | 0 |
N.S. |
1 | 1.00 | 0.83 | 1.03 | 0.83 | 1.67 | 0.00 | 1.08 | 0.00 |
time (sec) |
N/A | 0.744 | 0.675 | 1.081 | 0.207 | 0.265 | 0.000 | 0.282 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 154 | 154 | 185 | 160 | 139 | 324 | 0 | 181 | 0 |
N.S. |
1 | 1.00 | 1.20 | 1.04 | 0.90 | 2.10 | 0.00 | 1.18 | 0.00 |
time (sec) |
N/A | 0.545 | 0.482 | 0.137 | 0.208 | 0.273 | 0.000 | 0.276 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 225 | 225 | 257 | 217 | 199 | 466 | 0 | 239 | 0 |
N.S. |
1 | 1.00 | 1.14 | 0.96 | 0.88 | 2.07 | 0.00 | 1.06 | 0.00 |
time (sec) |
N/A | 0.642 | 1.582 | 0.398 | 0.212 | 0.273 | 0.000 | 0.295 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 323 | 323 | 501 | 326 | 287 | 851 | 0 | 369 | 0 |
N.S. |
1 | 1.00 | 1.55 | 1.01 | 0.89 | 2.63 | 0.00 | 1.14 | 0.00 |
time (sec) |
N/A | 0.842 | 4.937 | 1.274 | 0.212 | 0.276 | 0.000 | 0.297 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F | A | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
size | 161 | 161 | 251 | 186 | 151 | 362 | 0 | 207 | 0 |
N.S. |
1 | 1.00 | 1.56 | 1.16 | 0.94 | 2.25 | 0.00 | 1.29 | 0.00 |
time (sec) |
N/A | 0.635 | 1.073 | 0.164 | 0.211 | 0.270 | 0.000 | 0.293 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | A | B | F(-1) | A | F(-1) |
verified |
N/A | Yes | No | Yes | TBD | TBD | TBD | TBD | TBD |
size | 239 | 239 | 339 | 249 | 215 | 516 | 0 | 271 | 0 |
N.S. |
1 | 1.00 | 1.42 | 1.04 | 0.90 | 2.16 | 0.00 | 1.13 | 0.00 |
time (sec) |
N/A | 0.805 | 4.372 | 0.522 | 0.209 | 0.266 | 0.000 | 0.308 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | A | A | B | F(-1) | A | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 344 | 344 | 2991 | 384 | 315 | 939 | 0 | 427 | 0 |
N.S. |
1 | 1.00 | 8.69 | 1.12 | 0.92 | 2.73 | 0.00 | 1.24 | 0.00 |
time (sec) |
N/A | 1.060 | 6.463 | 1.617 | 0.215 | 0.278 | 0.000 | 0.321 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | B | F | F | F(-2) | F | F | B |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
size | 20 | 20 | 46 | 0 | 0 | 0 | 0 | 0 | 39 |
N.S. |
1 | 1.00 | 2.30 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.95 |
time (sec) |
N/A | 0.198 | 0.246 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.172 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | B | F | F | B |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 24 | 24 | 16 | 0 | 0 | 109 | 0 | 0 | 42 |
N.S. |
1 | 1.00 | 0.67 | 0.00 | 0.00 | 4.54 | 0.00 | 0.00 | 1.75 |
time (sec) |
N/A | 0.199 | 0.033 | 0.000 | 0.000 | 0.258 | 0.000 | 0.000 | 1.736 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | F | F | F(-2) | F(-1) | F | B |
verified |
N/A | Yes | No | N/A | TBD | TBD | TBD | TBD | TBD |
size | 47 | 47 | 64 | 0 | 0 | 0 | 0 | 0 | 110 |
N.S. |
1 | 1.00 | 1.36 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 2.34 |
time (sec) |
N/A | 0.219 | 0.411 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.914 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | F | F | F(-2) | F | F | F(-1) |
verified |
N/A | Yes | Yes | N/A | TBD | TBD | TBD | TBD | TBD |
size | 36 | 36 | 76 | 0 | 0 | 0 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 2.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.236 | 0.112 | 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 | A | A | A | A | A | A | B |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 30 | 30 | 26 | 25 | 36 | 23 | 41 | 36 | 24 |
N.S. |
1 | 1.00 | 0.87 | 0.83 | 1.20 | 0.77 | 1.37 | 1.20 | 0.80 |
time (sec) |
N/A | 0.220 | 0.049 | 0.107 | 0.192 | 0.242 | 0.063 | 0.264 | 0.053 |
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 | 56 | 56 | 51 | 48 | 81 | 54 | 85 | 75 | 48 |
N.S. |
1 | 1.00 | 0.91 | 0.86 | 1.45 | 0.96 | 1.52 | 1.34 | 0.86 |
time (sec) |
N/A | 0.269 | 0.063 | 0.172 | 0.200 | 0.265 | 0.093 | 0.262 | 0.079 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | C | A | F | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 213 | 213 | 166 | 212 | 0 | 316 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.78 | 1.00 | 0.00 | 1.48 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.724 | 0.397 | 0.254 | 0.000 | 0.262 | 0.000 | 0.000 | 0.000 |
Optimal | Rubi | MMA | Maple | Maxima | Fricas | Sympy | Giac | Mupad | |
grade |
N/A | A | A | A | F(-2) | B | F | F | F(-1) |
verified |
N/A | Yes | Yes | Yes | TBD | TBD | TBD | TBD | TBD |
size | 271 | 271 | 220 | 376 | 0 | 671 | 0 | 0 | 0 |
N.S. |
1 | 1.00 | 0.81 | 1.39 | 0.00 | 2.48 | 0.00 | 0.00 | 0.00 |
time (sec) |
N/A | 0.987 | 0.606 | 0.382 | 0.000 | 0.267 | 0.000 | 0.000 | 0.000 |