## Chapter 2detailed summary tables of results

2.1.1 Rubi
2.1.2 Mathematica
2.1.3 Maple
2.1.4 Maxima
2.1.5 FriCAS
2.1.6 Sympy
2.1.7 Giac

### 2.1 List of integrals sorted by grade for each CAS

#### 2.1.1 Rubi

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 }

#### 2.1.2 Mathematica

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 }

#### 2.1.3 Maple

A grade: { 1, 2, 3, 4, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 30, 32, 33 }

B grade: { 28, 29, 31 }

C grade: { 11, 12, 13, 25, 26 }

F grade: { 5, 10 }

#### 2.1.4 Maxima

A grade: { 4, 9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 27, 30, 33

B grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 28, 29, 31, 32 }

C grade: { 24, 25, 26 }

F grade: { 5, 10 }

#### 2.1.5 FriCAS

A grade: { 3, 4, 8, 9, 11, 12, 13, 14, 15, 18, 19, 22, 23, 25, 26, 27, 30, 33 }

B grade: { 1, 2, 5, 6, 7, 10, 16, 17, 20, 21, 24, 28, 29, 31, 32 }

#### 2.1.6 Sympy

A grade: { 4, 9, 14, 15, 19, 23, 27, 30 }

F grade: { 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33 }

#### 2.1.7 Giac

A grade: { 1, 2, 3, 4, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33 }

C grade: { 11, 12, 13, 25, 26 }

F grade: { 5, 10 }

### 2.2 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 grade A A A A B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 225 225 149 252 1019 1076 0 220 normalized size 1 1. 0.66 1.12 4.53 4.78 0. 0.98 time (sec) N/A 0.136 0.286 0.051 1.607 1.866 0. 1.333
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 111 111 130 124 825 892 0 163 normalized size 1 1. 1.17 1.12 7.43 8.04 0. 1.47 time (sec) N/A 0.048 0.143 0.034 1.589 1.85 0. 1.234
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B A F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 91 91 105 83 626 302 0 107 normalized size 1 1. 1.15 0.91 6.88 3.32 0. 1.18 time (sec) N/A 0.034 0.063 0.03 1.511 1.771 0. 1.353
 Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 17 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.01 0.534 0.028 0. 0. 0. 0.
 Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 107 107 132 0 0 872 0 0 normalized size 1 1. 1.23 0. 0. 8.15 0. 0. time (sec) N/A 0.086 7.678 0.116 0. 1.903 0. 0.
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 227 227 152 244 1126 1065 0 225 normalized size 1 1. 0.67 1.07 4.96 4.69 0. 0.99 time (sec) N/A 0.142 0.29 0.044 1.613 2.129 0. 1.229
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 112 112 134 120 910 882 0 166 normalized size 1 1. 1.2 1.07 8.12 7.88 0. 1.48 time (sec) N/A 0.053 0.146 0.035 1.526 2.166 0. 1.274
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B A F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 91 91 109 79 690 297 0 109 normalized size 1 1. 1.2 0.87 7.58 3.26 0. 1.2 time (sec) N/A 0.034 0.066 0.032 1.443 2.089 0. 1.254
 Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 18 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.011 1.176 0.029 0. 0. 0. 0.
 Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 108 108 136 0 0 861 0 0 normalized size 1 1. 1.26 0. 0. 7.97 0. 0. time (sec) N/A 0.089 9.151 0.111 0. 2.099 0. 0.
 Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A C B A F C veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 66 66 72 75 247 180 0 72 normalized size 1 1. 1.09 1.14 3.74 2.73 0. 1.09 time (sec) N/A 0.057 0.159 0.033 1.437 2.495 0. 1.218
 Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A C B A F C veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 52 52 76 49 166 155 0 58 normalized size 1 1. 1.46 0.94 3.19 2.98 0. 1.12 time (sec) N/A 0.025 0.074 0.028 1.485 2.515 0. 1.257
 Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A C B A F C veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 39 39 22 25 127 61 0 28 normalized size 1 1. 0.56 0.64 3.26 1.56 0. 0.72 time (sec) N/A 0.016 0.023 0.027 1.393 2.482 0. 1.233
 Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 15 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.01 7.306 0.026 0. 0. 0. 0.
 Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 68 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.04 9.458 0.036 0. 0. 0. 0.
 Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 268 268 176 281 397 1904 0 246 normalized size 1 1. 0.66 1.05 1.48 7.1 0. 0.92 time (sec) N/A 0.235 0.744 0.065 1.319 2.18 0. 1.277
 Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 136 136 155 141 270 1650 0 192 normalized size 1 1. 1.14 1.04 1.99 12.13 0. 1.41 time (sec) N/A 0.093 0.393 0.049 1.268 2.103 0. 1.271
 Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 110 110 140 94 130 358 0 127 normalized size 1 1. 1.27 0.85 1.18 3.25 0. 1.15 time (sec) N/A 0.066 0.143 0.043 1.568 2.12 0. 1.346
 Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 32 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.033 32.532 0.056 0. 0. 0. 0.
 Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 268 268 181 273 435 1887 0 251 normalized size 1 1. 0.68 1.02 1.62 7.04 0. 0.94 time (sec) N/A 0.235 0.723 0.063 1.394 2.181 0. 1.315
 Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 136 136 161 137 292 1634 0 194 normalized size 1 1. 1.18 1.01 2.15 12.01 0. 1.43 time (sec) N/A 0.093 0.388 0.051 1.294 2.155 0. 1.286
 Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 110 110 144 90 130 352 0 130 normalized size 1 1. 1.31 0.82 1.18 3.2 0. 1.18 time (sec) N/A 0.066 0.137 0.043 1.56 2.111 0. 1.328
 Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 32 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.033 34.525 0.055 0. 0. 0. 0.
 Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A C B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 68 68 99 77 153 859 0 82 normalized size 1 1. 1.46 1.13 2.25 12.63 0. 1.21 time (sec) N/A 0.099 0.202 0.045 1.897 2.133 0. 1.215
 Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A C C A F C veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 75 75 88 75 163 263 0 95 normalized size 1 1. 1.17 1. 2.17 3.51 0. 1.27 time (sec) N/A 0.054 0.205 0.046 1.802 2.277 0. 1.239
 Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A C C A F C veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 56 56 48 49 61 132 0 57 normalized size 1 1. 0.86 0.88 1.09 2.36 0. 1.02 time (sec) N/A 0.034 0.072 0.039 1.529 2.157 0. 1.243
 Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 30 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.032 10.7 0.054 0. 0. 0. 0.
 Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 261 261 194 493 724 1508 0 522 normalized size 1 1. 0.74 1.89 2.77 5.78 0. 2. time (sec) N/A 0.172 0.528 0.056 1.508 2.235 0. 1.274
 Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B F B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 128 128 146 211 343 1033 0 282 normalized size 1 1. 1.14 1.65 2.68 8.07 0. 2.2 time (sec) N/A 0.06 0.232 0.036 1.372 2.248 0. 1.234
 Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 21 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.015 2.462 0.079 0. 0. 0. 0.
 Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 311 311 240 558 814 2700 0 608 normalized size 1 1. 0.77 1.79 2.62 8.68 0. 1.95 time (sec) N/A 0.386 1.337 0.08 2.108 2.106 0. 1.307
 Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B B F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 160 160 177 241 406 1922 0 335 normalized size 1 1. 1.11 1.51 2.54 12.01 0. 2.09 time (sec) N/A 0.146 0.639 0.057 1.931 1.921 0. 1.333
 Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A A A F(-1) A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 43 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.047 8.142 0.116 0. 0. 0. 0.

### 2.3 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 [11] had the largest ratio of [ 0.5385 ]

 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 7 1. 15 0.467

2 A 6 5 1. 13 0.385

3 A 5 4 1. 11 0.364

4 A 0 0 0. 0 0.

5 A 7 5 1. 33 0.152

6 A 12 7 1. 16 0.438

7 A 6 5 1. 14 0.357

8 A 5 4 1. 12 0.333

9 A 0 0 0. 0 0.

10 A 7 5 1. 35 0.143

11 A 12 7 1. 13 0.538

12 A 6 5 1. 11 0.454

13 A 5 4 1. 9 0.444

14 A 0 0 0. 0 0.

15 A 0 0 0. 0 0.

16 A 14 8 1. 17 0.471

17 A 8 6 1. 15 0.4

18 A 7 5 1. 13 0.385

19 A 0 0 0. 0 0.

20 A 14 8 1. 18 0.444

21 A 8 6 1. 16 0.375

22 A 7 5 1. 14 0.357

23 A 0 0 0. 0 0.

24 A 14 8 1. 15 0.533

25 A 8 6 1. 13 0.462

26 A 7 5 1. 11 0.454

27 A 0 0 0. 0 0.

28 A 12 7 1. 19 0.368

29 A 6 5 1. 17 0.294

30 A 0 0 0. 0 0.

31 A 14 8 1. 21 0.381

32 A 8 6 1. 19 0.316

33 A 0 0 0. 0 0.