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2 detailed summary tables of results

 2.1 Detailed conclusion table per each integral for all CAS systems
 2.2 Detailed conclusion table specific for Rubi results

2.1 Detailed conclusion table per each integral for all CAS systems

Detailed conclusion table per each integral is given by 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 just an F.

In this table,the column normalized size is defined as \(\frac{\text{antiderivative leaf size}}{\text{optimal antiderivative leaf size}}\)

Problem 1 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 210 210 172 4972 0 4149 0 1 255
normalized size 1 1. 0.82 23.68 0. 19.76 0. 0. 1.21
time (sec) N/A 0.652 1.516 0.152 0. 0.274 0. 0.231 101.107
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 160 160 129 2410 0 2057 0 1 194
normalized size 1 1. 0.81 15.06 0. 12.86 0. 0.01 1.21
time (sec) N/A 0.424 1.04 0.114 0. 0.256 0. 0.222 70.857
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 108 108 84 891 0 759 0 1 133
normalized size 1 1. 0.78 8.25 0. 7.03 0. 0.01 1.23
time (sec) N/A 0.225 0.247 0.087 0. 0.245 0. 0.216 46.128
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-2) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 66 66 49 262 0 250 0 512 75
normalized size 1 1. 0.74 3.97 0. 3.79 0. 7.76 1.14
time (sec) N/A 0.107 0.06 0.067 0. 0.234 0. 0.219 17.551
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 120 120 115 0 0 0 0 0 141
normalized size 1 1. 0.96 0. 0. 0. 0. 0. 1.18
time (sec) N/A 0.329 0.183 0.068 0. 0. 0. 0. 40.022
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 177 177 195 0 0 0 0 0 148
normalized size 1 1. 1.1 0. 0. 0. 0. 0. 0.84
time (sec) N/A 0.695 0.39 0.077 0. 0. 0. 0. 49.542
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 228 228 1153 0 0 0 0 0 199
normalized size 1 1. 5.06 0. 0. 0. 0. 0. 0.87
time (sec) N/A 0.779 0.99 0.095 0. 0. 0. 0. 58.743
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 318 318 273 11389 0 8961 0 1 0
normalized size 1 1. 0.86 35.81 0. 28.18 0. 0. 0.
time (sec) N/A 0.967 3.46 0.217 0. 0.32 0. 0.254 0.
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 237 237 199 5908 0 4745 0 1 275
normalized size 1 1. 0.84 24.93 0. 20.02 0. 0. 1.16
time (sec) N/A 0.765 2.101 0.158 0. 0.275 0. 0.24 117.925
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 160 160 129 2410 0 1925 0 1 194
normalized size 1 1. 0.81 15.06 0. 12.03 0. 0.01 1.21
time (sec) N/A 0.421 0.802 0.113 0. 0.255 0. 0.226 71.037
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 102 102 78 732 0 711 0 1 119
normalized size 1 1. 0.76 7.18 0. 6.97 0. 0.01 1.17
time (sec) N/A 0.187 0.149 0.084 0. 0.238 0. 0.215 33.399
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 185 185 156 0 0 0 0 0 214
normalized size 1 1. 0.84 0. 0. 0. 0. 0. 1.16
time (sec) N/A 0.554 0.355 0.076 0. 0. 0. 0. 64.808
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 268 268 227 0 0 0 0 0 0
normalized size 1 1. 0.85 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.663 1.887 180. 0. 0. 0. 0. 0.
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 322 322 1924 0 0 0 0 0 298
normalized size 1 1. 5.98 0. 0. 0. 0. 0. 0.93
time (sec) N/A 1.447 1.809 0.097 0. 0. 0. 0. 127.055
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 410 410 358 20937 0 15698 0 1 0
normalized size 1 1. 0.87 51.07 0. 38.29 0. 0. 0.
time (sec) N/A 1.55 2.138 0.313 0. 0.385 0. 0.282 0.
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 310 310 265 11389 0 8852 0 1 0
normalized size 1 1. 0.85 36.74 0. 28.55 0. 0. 0.
time (sec) N/A 1.017 2.951 0.215 0. 0.328 0. 0.251 0.
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 210 210 172 4972 0 3825 0 1 255
normalized size 1 1. 0.82 23.68 0. 18.21 0. 0. 1.21
time (sec) N/A 0.629 1.426 0.149 0. 0.278 0. 0.229 100.451
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 137 137 106 1609 0 1490 0 1 165
normalized size 1 1. 0.77 11.74 0. 10.88 0. 0.01 1.2
time (sec) N/A 0.275 0.227 0.105 0. 0.261 0. 0.22 47.394
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 270 270 212 0 0 0 0 0 286
normalized size 1 1. 0.79 0. 0. 0. 0. 0. 1.06
time (sec) N/A 0.957 0.65 0.076 0. 0. 0. 0. 89.902
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 394 389 345 0 0 0 0 0 0
normalized size 1 0.99 0.88 0. 0. 0. 0. 0. 0.
time (sec) N/A 2.701 4.248 0.088 0. 0. 0. 0. 0.
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 380 380 290 0 0 0 0 0 357
normalized size 1 1. 0.76 0. 0. 0. 0. 0. 0.94
time (sec) N/A 1.516 1.247 0.078 0. 0. 0. 0. 114.282
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 272 272 213 0 0 0 0 0 286
normalized size 1 1. 0.78 0. 0. 0. 0. 0. 1.05
time (sec) N/A 0.998 0.694 0.076 0. 0. 0. 0. 90.017
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 187 187 156 0 0 0 0 0 214
normalized size 1 1. 0.83 0. 0. 0. 0. 0. 1.14
time (sec) N/A 0.607 0.371 0.075 0. 0. 0. 0. 66.298
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 122 122 115 0 0 0 0 0 141
normalized size 1 1. 0.94 0. 0. 0. 0. 0. 1.16
time (sec) N/A 0.321 0.181 0.067 0. 0. 0. 0. 42.45
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 78 78 59 0 0 0 0 0 54
normalized size 1 1. 0.76 0. 0. 0. 0. 0. 0.69
time (sec) N/A 0.119 0.072 0.1 0. 0. 0. 0. 13.506
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 127 127 102 0 0 0 0 0 199
normalized size 1 1. 0.8 0. 0. 0. 0. 0. 1.57
time (sec) N/A 0.379 0.179 0.097 0. 0. 0. 0. 90.228
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 212 212 177 0 0 0 0 0 0
normalized size 1 1. 0.83 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.385 0.519 0.113 0. 0. 0. 0. 0.
Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 407 407 402 0 0 0 0 0 0
normalized size 1 1. 0.99 0. 0. 0. 0. 0. 0.
time (sec) N/A 3.265 2.977 0.132 0. 0. 0. 0. 0.
Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 386 381 365 0 0 0 0 0 0
normalized size 1 0.99 0.95 0. 0. 0. 0. 0. 0.
time (sec) N/A 2.748 3.45 0.088 0. 0. 0. 0. 0.
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 267 267 249 0 0 0 0 0 0
normalized size 1 1. 0.93 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.669 1.753 0.086 0. 0. 0. 0. 0.
Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 178 178 193 0 0 0 0 0 148
normalized size 1 1. 1.08 0. 0. 0. 0. 0. 0.83
time (sec) N/A 0.699 0.408 0.078 0. 0. 0. 0. 46.991
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 107 107 84 0 0 0 0 0 78
normalized size 1 1. 0.79 0. 0. 0. 0. 0. 0.73
time (sec) N/A 0.164 0.178 0.072 0. 0. 0. 0. 17.231
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 211 211 177 0 0 0 0 0 0
normalized size 1 1. 0.84 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.41 0.514 0.112 0. 0. 0. 0. 0.
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 315 315 242 0 0 0 0 0 0
normalized size 1 1. 0.77 0. 0. 0. 0. 0. 0.
time (sec) N/A 2.788 1.325 0.129 0. 0. 0. 0. 0.
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 567 567 2176 0 0 0 0 0 0
normalized size 1 1. 3.84 0. 0. 0. 0. 0. 0.
time (sec) N/A 6.17 5.049 0.156 0. 0. 0. 0. 0.
Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 322 322 1924 0 0 0 0 0 298
normalized size 1 1. 5.98 0. 0. 0. 0. 0. 0.93
time (sec) N/A 1.484 2.004 0.108 0. 0. 0. 0. 120.586
Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 228 228 1153 0 0 0 0 0 199
normalized size 1 1. 5.06 0. 0. 0. 0. 0. 0.87
time (sec) N/A 0.772 1.004 0.082 0. 0. 0. 0. 57.341
Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 112 112 127 0 0 0 0 0 85
normalized size 1 1. 1.13 0. 0. 0. 0. 0. 0.76
time (sec) N/A 0.175 0.272 0.079 0. 0. 0. 0. 16.409
Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 366 366 357 0 0 0 0 0 0
normalized size 1 1. 0.98 0. 0. 0. 0. 0. 0.
time (sec) N/A 3.21 3.013 0.129 0. 0. 0. 0. 0.
Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 482 482 2178 0 0 0 0 0 0
normalized size 1 1. 4.52 0. 0. 0. 0. 0. 0.
time (sec) N/A 5.183 4.516 0.156 0. 0. 0. 0. 0.
Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F(-1) F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 211 211 458 0 0 0 0 0 170
normalized size 1 1. 2.17 0. 0. 0. 0. 0. 0.81
time (sec) N/A 0.639 1.934 0.665 0. 0. 0. 0. 65.955
Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-2) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 271 255 164 0 0 0 0 0 246
normalized size 1 0.94 0.61 0. 0. 0. 0. 0. 0.91
time (sec) N/A 0.807 0.948 0.11 0. 0. 0. 0. 119.87
Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F F F F A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 164 164 438 0 0 0 0 0 133
normalized size 1 1. 2.67 0. 0. 0. 0. 0. 0.81
time (sec) N/A 0.475 1.494 0.132 0. 0. 0. 0. 72.684
Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F F(-1)
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 304 304 438 0 0 0 0 0 0
normalized size 1 1. 1.44 0. 0. 0. 0. 0. 0.
time (sec) N/A 1.505 1.697 0.13 0. 0. 0. 0. 0.
Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 139 139 124 0 0 0 0 0 102
normalized size 1 1. 0.89 0. 0. 0. 0. 0. 0.73
time (sec) N/A 0.34 0.301 0.285 0. 0. 0. 0. 30.591
Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F F(-1) F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 139 167 124 0 0 0 0 0 143
normalized size 1 1.2 0.89 0. 0. 0. 0. 0. 1.03
time (sec) N/A 0.361 0.141 0.276 0. 0. 0. 0. 37.136

2.2 Detailed conclusion table specific for Rubi results

The following table is specific to Rubi. It gives additional statistics for each integral. the column steps is the number of steps used by Rubi to obtain the antiderivative. The rules column is the number of unique rules used. The integrand size column is the leaf size of the integrand. Finally the ratio \(\frac{\text{number of rules}}{\text{integrand size}}\) is given. The larger this ratio is, the harder the integral was to solve. In this test, problem number [44] had the largest ratio of [ 0.1935 ]

Table 1: Rubi specific breakdown of results for each integral
# grade
number of
steps
used
number of
unique
rules
normalized
antiderivative
leaf size
integrand
leaf size
\(\frac{\text{number of rules}}{\text{integrand leaf size}}\)
1 A 12 3 1. 29 0.103
2 A 10 3 1. 29 0.103
3 A 8 3 1. 27 0.111
4 A 6 3 1. 20 0.15
5 A 5 4 1. 29 0.138
6 A 3 3 1. 29 0.103
7 A 3 3 1. 29 0.103
8 A 14 3 1. 31 0.097
9 A 12 3 1. 31 0.097
10 A 10 3 1. 29 0.103
11 A 8 3 1. 22 0.136
12 A 7 4 1. 31 0.129
13 A 6 5 1. 31 0.161
14 A 4 3 1. 31 0.097
15 A 16 3 1. 31 0.097
16 A 14 3 1. 31 0.097
17 A 12 3 1. 29 0.103
18 A 10 3 1. 22 0.136
19 A 9 4 1. 31 0.129
20 A 8 5 0.99 31 0.161
21 A 11 4 1. 31 0.129
22 A 9 4 1. 31 0.129
23 A 7 4 1. 31 0.129
24 A 5 4 1. 29 0.138
25 A 2 2 1. 22 0.091
26 A 4 2 1. 31 0.065
27 A 5 3 1. 31 0.097
28 A 6 3 1. 31 0.097
29 A 8 5 0.99 31 0.161
30 A 6 5 1. 31 0.161
31 A 3 3 1. 29 0.103
32 A 2 2 1. 22 0.091
33 A 5 3 1. 31 0.097
34 A 6 3 1. 31 0.097
35 A 7 3 1. 31 0.097
36 A 4 3 1. 31 0.097
37 A 3 3 1. 29 0.103
38 A 2 2 1. 22 0.091
39 A 6 3 1. 31 0.097
40 A 7 3 1. 31 0.097
41 A 7 3 1. 31 0.097
42 A 4 4 0.94 29 0.138
43 A 6 5 1. 31 0.161
44 A 7 6 1. 31 0.194
45 A 4 4 1. 47 0.085
46 A 4 4 1.2 55 0.073

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