## 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 }

#### 2.1.2 Mathematica

A grade: { 1, 2, 3, 5, 7, 8 }

B grade: { 4, 6 }

#### 2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8 }

#### 2.1.4 Maxima

A grade: { 1, 2, 4, 5, 6, 7, 8 }

#### 2.1.5 FriCAS

A grade: { 1, 3, 5, 6, 7, 8 }

B grade: { 2, 4 }

#### 2.1.6 Sympy

A grade: { 2, 3, 4, 5, 6, 7, 8 }

#### 2.1.7 Giac

A grade: { 1, 3, 4, 5, 6, 7, 8 }

B grade: { 1, 2, 3, 4, 5, 6, 7, 8 }

### 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 Mupad grade A A A A A A B A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 40 40 23 20 23 36 116 23 23 normalized size 1 1.00 0.58 0.50 0.58 0.90 2.90 0.58 0.58 time (sec) N/A 0.007 0.012 0.006 0.498 0.761 1.391 0.885 0.348
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A B A B B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 24 24 24 19 35 40 19 38 18 normalized size 1 1.00 1.00 0.79 1.46 1.67 0.79 1.58 0.75 time (sec) N/A 0.018 0.011 0.005 1.405 0.723 0.147 1.068 0.461
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F(-2) A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 42 42 41 36 0 137 144 61 38 normalized size 1 1.00 0.98 0.86 0.00 3.26 3.43 1.45 0.90 time (sec) N/A 0.042 0.033 0.021 0.000 0.838 4.297 0.826 0.588
 Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A B A A B A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 15 15 34 12 15 23 10 12 11 normalized size 1 1.00 2.27 0.80 1.00 1.53 0.67 0.80 0.73 time (sec) N/A 0.013 0.019 0.052 0.566 0.756 0.263 1.130 0.081
 Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 21 21 39 20 29 21 19 21 11 normalized size 1 1.00 1.86 0.95 1.38 1.00 0.90 1.00 0.52 time (sec) N/A 0.012 0.017 0.057 0.552 0.701 0.273 1.088 0.409
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A B A A A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 12 21 26 11 15 20 8 10 10 normalized size 1 1.75 2.17 0.92 1.25 1.67 0.67 0.83 0.83 time (sec) N/A 0.011 0.033 0.055 0.585 0.668 0.434 0.953 0.359
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 43 43 24 20 23 39 42 49 21 normalized size 1 1.00 0.56 0.47 0.53 0.91 0.98 1.14 0.49 time (sec) N/A 0.040 0.027 0.053 0.980 0.487 0.522 0.936 0.048
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 30 30 31 35 34 38 32 36 28 normalized size 1 1.00 1.03 1.17 1.13 1.27 1.07 1.20 0.93 time (sec) N/A 0.013 0.005 0.021 0.423 0.753 0.178 0.856 0.398

### 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 [.2500]

 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 2 1 1.00 17 0.059

2 A 2 2 1.00 18 0.111

3 A 2 2 1.00 8 0.250

4 A 2 2 1.00 12 0.167

5 A 2 2 1.00 12 0.167

6 A 1 1 1.75 12 0.083

7 A 3 3 1.00 12 0.250

8 A 4 4 1.00 16 0.250