## 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 }

#### 2.1.2 Mathematica

A grade: { 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 }

#### 2.1.3 Maple

A grade: { 2, 3, 4, 9, 10, 11, 12 }

B grade: { 1, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20 }

#### 2.1.4 Maxima

A grade: { 9, 10, 11, 12

F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20 }

#### 2.1.5 FriCAS

A grade: { 1, 2, 3, 4, 9, 10, 11, 12 }

B grade: { 5, 6, 7, 13, 14, 15, 16, 17 }

F grade: { 8, 18, 19, 20 }

#### 2.1.6 Sympy

A grade: { 3, 9, 10, 11 }

F grade: { 1, 2, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20 }

#### 2.1.7 Giac

A grade: { 1, 2, 3, 4, 5, 9, 10, 11, 12 }

F grade: { 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20 }

### 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 B F(-2) A F(-1) A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 136 136 239 344 0 1125 0 207 normalized size 1 1. 1.76 2.53 0. 8.27 0. 1.52 time (sec) N/A 0.232 0.534 0.027 0. 2.593 0. 1.144
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A F(-2) A F(-1) A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 76 76 131 141 0 641 0 103 normalized size 1 1. 1.72 1.86 0. 8.43 0. 1.36 time (sec) N/A 0.13 0.257 0.02 0. 1.964 0. 1.138
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A F(-2) A A A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 35 35 39 36 0 317 99 47 normalized size 1 1. 1.11 1.03 0. 9.06 2.83 1.34 time (sec) N/A 0.046 0.033 0.013 0. 1.704 6.011 1.189
 Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A F(-2) A F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 129 129 126 223 0 1100 0 176 normalized size 1 1. 0.98 1.73 0. 8.53 0. 1.36 time (sec) N/A 0.172 0.182 0.036 0. 7.397 0. 1.166
 Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C B F(-2) B F(-1) A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 205 205 392 546 0 4351 0 510 normalized size 1 1. 1.91 2.66 0. 21.22 0. 2.49 time (sec) N/A 0.465 2.185 0.046 0. 66.788 0. 1.15
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 388 386 374 2608 0 10364 0 0 normalized size 1 0.99 0.96 6.72 0. 26.71 0. 0. time (sec) N/A 11.013 0.843 0.086 0. 13.34 0. 0.
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 260 260 238 1157 0 1987 0 0 normalized size 1 1. 0.92 4.45 0. 7.64 0. 0. time (sec) N/A 1.284 0.618 0.044 0. 2.469 0. 0.
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F(-1) F(-1) F F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 326 326 335 2816 0 0 0 0 normalized size 1 1. 1.03 8.64 0. 0. 0. 0. time (sec) N/A 3.34 0.921 0.061 0. 0. 0. 0.
 Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 21 21 19 16 20 68 15 23 normalized size 1 1. 0.9 0.76 0.95 3.24 0.71 1.1 time (sec) N/A 0.024 0.025 0.025 0.948 1.313 0.243 1.783
 Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 23 23 29 16 20 69 15 26 normalized size 1 1. 1.26 0.7 0.87 3. 0.65 1.13 time (sec) N/A 0.028 0.014 0.028 0.952 1.298 0.324 1.404
 Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 19 19 18 18 20 74 26 20 normalized size 1 1. 0.95 0.95 1.05 3.89 1.37 1.05 time (sec) N/A 0.036 0.026 0.022 1.428 1.23 0.508 1.44
 Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A B A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 36 36 34 31 38 139 119 38 normalized size 1 1. 0.94 0.86 1.06 3.86 3.31 1.06 time (sec) N/A 0.033 0.073 0.025 1.443 1.405 1.985 1.476
 Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 326 326 356 3427 0 16717 0 0 normalized size 1 1. 1.09 10.51 0. 51.28 0. 0. time (sec) N/A 4.062 1.059 0.057 0. 29.164 0. 0.
 Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 299 299 309 2503 0 13111 0 0 normalized size 1 1. 1.03 8.37 0. 43.85 0. 0. time (sec) N/A 6.758 0.827 0.05 0. 13.015 0. 0.
 Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 255 255 264 1948 0 9790 0 0 normalized size 1 1. 1.04 7.64 0. 38.39 0. 0. time (sec) N/A 1.262 0.547 0.049 0. 6.426 0. 0.
 Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 230 230 227 1264 0 7050 0 0 normalized size 1 1. 0.99 5.5 0. 30.65 0. 0. time (sec) N/A 0.546 0.571 0.036 0. 3.451 0. 0.
 Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F(-1) F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 223 223 198 1262 0 7070 0 0 normalized size 1 1. 0.89 5.66 0. 31.7 0. 0. time (sec) N/A 0.35 0.398 0.035 0. 3.52 0. 0.
 Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F F(-1) F F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 245 245 281 1957 0 0 0 0 normalized size 1 1. 1.15 7.99 0. 0. 0. 0. time (sec) N/A 0.772 0.641 0.056 0. 0. 0. 0.
 Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F F(-1) F F(-1) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 275 275 348 2530 0 0 0 0 normalized size 1 1. 1.27 9.2 0. 0. 0. 0. time (sec) N/A 1.189 1.162 0.065 0. 0. 0. 0.
 Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F F(-1) F(-1) F(-1) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 334 334 446 3476 0 0 0 0 normalized size 1 1. 1.34 10.41 0. 0. 0. 0. time (sec) N/A 4.674 2.895 0.076 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  had the largest ratio of [ 0.4737 ]

 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 7 6 1. 19 0.316

2 A 7 6 1. 19 0.316

3 A 3 3 1. 17 0.176

4 A 9 8 1. 17 0.471

5 A 10 9 1. 19 0.474

6 A 10 7 0.99 19 0.368

7 A 7 4 1. 19 0.21

8 A 9 5 1. 19 0.263

9 A 4 3 1. 13 0.231

10 A 4 3 1. 15 0.2

11 A 3 3 1. 15 0.2

12 A 4 4 1. 15 0.267

13 A 10 7 1. 19 0.368

14 A 8 5 1. 19 0.263

15 A 7 4 1. 19 0.21

16 A 6 3 1. 17 0.176

17 A 5 3 1. 14 0.214

18 A 8 5 1. 17 0.294

19 A 10 7 1. 19 0.368

20 A 12 8 1. 19 0.421