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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 C A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 22 22 77 21 27 34 61 24 15
normalized size 1 1. 3.5 0.95 1.23 1.55 2.77 1.09 0.68
time (sec) N/A 0.03 0.088 0.156 1.573 0.205 10.787 0.23 0.55
Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 32 32 37 50 0 109 0 198 26
normalized size 1 1. 1.16 1.56 0. 3.41 0. 6.19 0.81
time (sec) N/A 0.062 0.034 0.037 0. 0.215 0. 0.236 2.904
Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 25 25 16 16 31 27 14 20 20
normalized size 1 1. 0.64 0.64 1.24 1.08 0.56 0.8 0.8
time (sec) N/A 0.025 0.019 0.01 1.414 0.22 0.624 0.21 0.534
Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 58 58 52 42 151 62 0 58 148
normalized size 1 1. 0.9 0.72 2.6 1.07 0. 1. 2.55
time (sec) N/A 0.135 0.071 0.024 1.633 0.23 0. 0.229 8.681
Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F(-1) A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 50 50 26 42 76 90 0 43 48
normalized size 1 1. 0.52 0.84 1.52 1.8 0. 0.86 0.96
time (sec) N/A 0.047 0.051 0.098 1.446 0.207 0. 0.228 0.703
Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 334 334 204 36 0 1369 26 335 537
normalized size 1 1. 0.61 0.11 0. 4.1 0.08 1. 1.61
time (sec) N/A 0.745 0.011 0.038 0. 0.23 2.057 0.235 27.133
Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F B C A F(-2) F F F
verified N/A N/A Yes TBD TBD TBD TBD TBD TBD
size 291 0 816 486 497 0 0 0 0
normalized size 1 0. 2.8 1.67 1.71 0. 0. 0. 0.
time (sec) N/A 0.081 2.141 0.291 1.676 0. 0. 0. 0.
Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A F B C A F(-2) F F A
verified N/A N/A Yes TBD TBD TBD TBD TBD TBD
size 308 0 654 513 513 0 0 0 280
normalized size 1 0. 2.12 1.67 1.67 0. 0. 0. 0.91
time (sec) N/A 0.073 0.602 0.015 1.696 0. 0. 0. 23.677
Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B F F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 84 84 347 0 0 116 0 0 76
normalized size 1 1. 4.13 0. 0. 1.38 0. 0. 0.9
time (sec) N/A 0.095 4.668 0.032 0. 0.204 0. 0. 6.26
Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A A A F F(-2) F
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 41 41 147 34 69 93 0 0 0
normalized size 1 1. 3.59 0.83 1.68 2.27 0. 0. 0.
time (sec) N/A 0.347 0.352 0.007 1.59 0.211 0. 0. 0.
Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B A A F F(-2) A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 73 73 136 175 85 112 0 0 80
normalized size 1 1. 1.86 2.4 1.16 1.53 0. 0. 1.1
time (sec) N/A 0.204 0.165 0.19 1.585 0.234 0. 0. 7.001
Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A A A F F(-2) F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 73 73 151 60 104 119 0 0 0
normalized size 1 1. 2.07 0.82 1.42 1.63 0. 0. 0.
time (sec) N/A 0.499 0.228 0.013 1.579 0.212 0. 0. 0.
Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B C F F(-1) F F(-2) A
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 365 365 2177 109 0 0 0 0 700
normalized size 1 1. 5.96 0.3 0. 0. 0. 0. 1.92
time (sec) N/A 1.785 6.38 0.04 0. 0. 0. 0. 168.401
Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B C F F(-1) F F F(-1)
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 337 337 2581 105 0 0 0 0 0
normalized size 1 1. 7.66 0.31 0. 0. 0. 0. 0.
time (sec) N/A 1.303 6.416 0.02 0. 0. 0. 0. 0.
Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F F A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 77 77 0 0 0 76 0 0 70
normalized size 1 1. 0. 0. 0. 0.99 0. 0. 0.91
time (sec) N/A 0.126 0.142 0.032 0. 0.485 0. 0. 3.137
Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F A F(-1) F(-1) A
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 118 118 0 0 0 105 0 0 105
normalized size 1 1. 0. 0. 0. 0.89 0. 0. 0.89
time (sec) N/A 0.341 0.332 0.032 0. 0.852 0. 0. 4.348
Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F F(-1) F F(-1) A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 83 83 97 298 0 0 0 0 71
normalized size 1 1. 1.17 3.59 0. 0. 0. 0. 0.86
time (sec) N/A 0.227 0.058 0.022 0. 0. 0. 0. 11.899
Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F F F(-1) F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 96 96 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.198 0.078 0.075 0. 0. 0. 0. 0.
Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B B A A F A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 25 25 112 49 51 46 0 101 24
normalized size 1 1. 4.48 1.96 2.04 1.84 0. 4.04 0.96
time (sec) N/A 0.171 0.085 0.031 1.611 0.214 0. 0.237 21.508
Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A A A F A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 25 25 126 33 50 74 0 100 26
normalized size 1 1. 5.04 1.32 2. 2.96 0. 4. 1.04
time (sec) N/A 0.129 0.108 0.197 1.567 0.233 0. 0.236 10.8
Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A B C A F(-1) A F(-1) A F(-1)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 108 786 478 95 0 370 0 140 0
normalized size 1 7.28 4.43 0.88 0. 3.43 0. 1.3 0.
time (sec) N/A 2.333 6.585 0.096 0. 0.3 0. 0.214 0.
Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A A A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 29 29 56 27 54 62 66 41 22
normalized size 1 1. 1.93 0.93 1.86 2.14 2.28 1.41 0.76
time (sec) N/A 0.033 0.041 0.062 1.475 0.231 1.468 0.205 1.069
Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A A A A F A A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 26 26 26 20 59 92 0 28 24
normalized size 1 1. 1. 0.77 2.27 3.54 0. 1.08 0.92
time (sec) N/A 0.024 0.071 0.04 1.608 0.213 0. 0.201 44.336
Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B B F A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 110 147 444 364 0 1497 0 0 190
normalized size 1 1.34 4.04 3.31 0. 13.61 0. 0. 1.73
time (sec) N/A 1.124 0.491 0.23 0. 0.267 0. 0. 46.073
Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A F F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 40 40 84 75 0 0 0 0 0
normalized size 1 1. 2.1 1.88 0. 0. 0. 0. 0.
time (sec) N/A 0.131 2.266 0.346 0. 0. 0. 0. 0.
Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A B F A F A F(-1)
verified N/A NO N/A TBD TBD TBD TBD TBD TBD
size 185 349 0 392 0 512 0 406 0
normalized size 1 1.89 0. 2.12 0. 2.77 0. 2.19 0.
time (sec) N/A 1.886 0.41 0.191 0. 0.264 0. 0.289 0.
Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F F F F F
verified N/A Yes N/A TBD TBD TBD TBD TBD TBD
size 102 102 0 83 0 0 0 0 0
normalized size 1 1. 0. 0.81 0. 0. 0. 0. 0.
time (sec) N/A 0.239 0.134 0.073 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 N/A TBD TBD TBD TBD TBD TBD
size 159 159 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.34 7.893 3.404 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 N/A TBD TBD TBD TBD TBD TBD
size 395 395 0 0 0 0 0 0 0
normalized size 1 1. 0. 0. 0. 0. 0. 0. 0.
time (sec) N/A 0.907 8.51 4.602 0. 0. 0. 0. 0.
Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F(-2) F F F F(-2)
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 981 981 868 730 0 0 0 0 0
normalized size 1 1. 0.88 0.74 0. 0. 0. 0. 0.
time (sec) N/A 2.28 1.054 0.114 0. 0. 0. 0. 0.
Problem 31 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 Yes TBD TBD TBD TBD TBD TBD
size 555 555 1283 0 0 0 0 0 517
normalized size 1 1. 2.31 0. 0. 0. 0. 0. 0.93
time (sec) N/A 1.201 3.092 0.019 0. 0. 0. 0. 62.607
Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A C F F F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 313 313 346 252 0 0 0 0 250
normalized size 1 1. 1.11 0.81 0. 0. 0. 0. 0.8
time (sec) N/A 0.637 0.216 0.027 0. 0. 0. 0. 34.03
Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A A A F F A
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 80 80 262 113 113 216 0 0 65
normalized size 1 1. 3.28 1.41 1.41 2.7 0. 0. 0.81
time (sec) N/A 0.135 0.393 0.161 1.607 0.243 0. 0. 9.568
Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A A F A F F F F
verified N/A Yes Yes TBD TBD TBD TBD TBD TBD
size 57 57 44 0 42 0 0 0 0
normalized size 1 1. 0.77 0. 0.74 0. 0. 0. 0.
time (sec) N/A 0.117 0.103 0.834 1.711 0. 0. 0. 0.
Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Rubi in Sympy
grade A A B A F(-1) F F F(-2) F
verified N/A Yes NO TBD TBD TBD TBD TBD TBD
size 121 121 405 171 0 0 0 0 0
normalized size 1 1. 3.35 1.41 0. 0. 0. 0. 0.
time (sec) N/A 0.179 3.285 0.539 0. 0. 0. 0. 0.

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 [29] had the largest ratio of [ 1.154 ]

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 1 1 1. 12 0.083
2 A 4 3 1. 19 0.158
3 A 2 2 1. 6 0.333
4 A 5 5 1. 10 0.5
5 A 3 2 1. 7 0.286
6 A 22 9 1. 8 1.125
7 F 0 0 N/A 0 N/A
8 F 0 0 N/A 0 N/A
9 A 4 3 1. 19 0.158
10 A 6 3 1. 25 0.12
11 A 5 2 1. 19 0.105
12 A 6 3 1. 21 0.143
13 A 20 8 1. 28 0.286
14 A 22 9 1. 21 0.429
15 A 2 1 1. 27 0.037
16 A 3 2 1. 36 0.056
17 A 7 5 1. 17 0.294
18 A 7 5 1. 17 0.294
19 A 6 6 1. 25 0.24
20 A 7 7 1. 14 0.5
21 B 45 7 7.28 9 0.778
22 A 3 3 1. 8 0.375
23 A 2 2 1. 10 0.2
24 A 13 8 1.34 16 0.5
25 A 5 5 1. 11 0.454
26 A 31 12 1.89 16 0.75
27 A 12 10 1. 16 0.625
28 A 13 12 1. 12 1.
29 A 28 15 1. 13 1.154
30 A 44 10 1. 18 0.556
31 A 35 16 1. 18 0.889
32 A 21 7 1. 14 0.5
33 A 7 4 1. 5 0.8
34 A 5 5 1. 8 0.625
35 A 10 7 1. 14 0.5

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