## 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, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 }

#### 2.1.2 Mathematica

A grade: { 1, 2, 3, 6, 10, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 }

B grade: { 5, 9, 25 }

C grade: { 4, 7, 8, 11, 12, 13, 23, 38, 40, 42, 44, 46 }

#### 2.1.3 Maple

A grade: { 1, 2, 3, 14, 15, 16, 17, 19, 20, 21, 22, 27, 28, 29, 34, 36, 47, 49, 50, 51, 57, 61, 70 }

B grade: { 5, 6, 30, 31, 32, 33, 35, 39, 41, 43, 45, 52, 53, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68 }

C grade: { 38, 40, 42, 44, 46 }

F grade: { 4, 7, 8, 9, 10, 11, 12, 13, 18, 23, 24, 25, 26, 37, 48, 59, 69, 71 }

#### 2.1.4 Maxima

A grade: { 14, 15, 16, 17, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 34, 36, 39, 43, 47, 49, 50, 51, 52, 53, 55, 57, 61, 63, 66, 68, 70, 71

B grade: { 33, 35, 41, 45, 54, 56, 58, 60, 62, 64 }

F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 23, 24, 25, 26, 37, 38, 40, 42, 44, 46, 48, 59, 65, 67, 69 }

#### 2.1.5 FriCAS

A grade: { 14, 15, 16, 17, 19, 20, 21, 22, 27, 28, 29, 30, 32, 34, 36, 39, 47, 49, 50, 51, 53, 57, 58, 60, 61, 62, 63, 64, 66, 68 }

B grade: { 1, 2, 3, 5, 6, 31, 33, 35, 41, 43, 45, 52, 54, 55, 56, 65, 67, 70, 71 }

F grade: { 4, 7, 8, 9, 10, 11, 12, 13, 18, 23, 24, 25, 26, 37, 38, 40, 42, 44, 46, 48, 59, 69 }

#### 2.1.6 Sympy

A grade: { 22, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 39, 41, 43, 45, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 68 }

C grade: { 38, 40, 42, 44, 46 }

F grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 34, 47, 55, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71 }

#### 2.1.7 Giac

A grade: { 27, 28, 29, 30, 31, 35, 39, 41, 43, 47, 60, 61, 62, 63, 64, 66, 68 }

B grade: { 33, 34, 36, 45, 65, 67 }

F 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, 32, 37, 38, 40, 42, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 69, 70, 71 }

### 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 F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 147 147 149 227 0 729 0 0 normalized size 1 1. 1.01 1.54 0. 4.96 0. 0. time (sec) N/A 0.152 0.281 0.249 0. 3.166 0. 0.
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 110 110 129 170 0 686 0 0 normalized size 1 1. 1.17 1.55 0. 6.24 0. 0. time (sec) N/A 0.095 0.176 0.219 0. 3.319 0. 0.
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 75 75 110 97 0 643 0 0 normalized size 1 1. 1.47 1.29 0. 8.57 0. 0. time (sec) N/A 0.061 0.085 0.234 0. 3.16 0. 0.
 Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 162 162 427 0 0 0 0 0 normalized size 1 1. 2.64 0. 0. 0. 0. 0. time (sec) N/A 0.287 0.379 0.25 0. 0. 0. 0.
 Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B B F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 63 63 141 154 0 792 0 0 normalized size 1 1. 2.24 2.44 0. 12.57 0. 0. time (sec) N/A 0.102 0.163 0.233 0. 2.798 0. 0.
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 114 114 220 453 0 1045 0 0 normalized size 1 1. 1.93 3.97 0. 9.17 0. 0. time (sec) N/A 0.21 0.456 0.231 0. 2.972 0. 0.
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 501 501 1429 0 0 0 0 0 normalized size 1 1. 2.85 0. 0. 0. 0. 0. time (sec) N/A 0.887 12.661 0.425 0. 0. 0. 0.
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 351 351 864 0 0 0 0 0 normalized size 1 1. 2.46 0. 0. 0. 0. 0. time (sec) N/A 0.506 9.142 0.403 0. 0. 0. 0.
 Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 194 194 401 0 0 0 0 0 normalized size 1 1. 2.07 0. 0. 0. 0. 0. time (sec) N/A 0.273 1.388 0.222 0. 0. 0. 0.
 Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 85 85 160 0 0 0 0 0 normalized size 1 1. 1.88 0. 0. 0. 0. 0. time (sec) N/A 0.083 0.249 0.218 0. 0. 0. 0.
 Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 475 475 1008 0 0 0 0 0 normalized size 1 1. 2.12 0. 0. 0. 0. 0. time (sec) N/A 1.073 2.783 0.611 0. 0. 0. 0.
 Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 448 448 2061 0 0 0 0 0 normalized size 1 1. 4.6 0. 0. 0. 0. 0. time (sec) N/A 1.11 12.864 0.543 0. 0. 0. 0.
 Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F F(-1) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 1024 1024 8348 0 0 0 0 0 normalized size 1 1. 8.15 0. 0. 0. 0. 0. time (sec) N/A 2.296 14.164 0.53 0. 0. 0. 0.
 Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 114 114 47 43 78 142 0 0 normalized size 1 1. 0.41 0.38 0.68 1.25 0. 0. time (sec) N/A 0.032 0.033 0.128 1.025 2.583 0. 0.
 Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 89 89 42 38 62 128 0 0 normalized size 1 1. 0.47 0.43 0.7 1.44 0. 0. time (sec) N/A 0.025 0.03 0.118 0.996 2.632 0. 0.
 Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 64 64 35 31 46 113 0 0 normalized size 1 1. 0.55 0.48 0.72 1.77 0. 0. time (sec) N/A 0.017 0.024 0.119 1.001 2.563 0. 0.
 Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 31 31 24 24 24 89 0 0 normalized size 1 1. 0.77 0.77 0.77 2.87 0. 0. time (sec) N/A 0.008 0.009 0.113 1.004 2.574 0. 0.
 Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 46 46 45 0 0 0 0 0 normalized size 1 1. 0.98 0. 0. 0. 0. 0. time (sec) N/A 0.097 0.035 0.17 0. 0. 0. 0.
 Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 63 63 42 46 88 109 0 0 normalized size 1 1. 0.67 0.73 1.4 1.73 0. 0. time (sec) N/A 0.024 0.023 0.112 0.998 2.692 0. 0.
 Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 90 90 47 57 124 132 0 0 normalized size 1 1. 0.52 0.63 1.38 1.47 0. 0. time (sec) N/A 0.028 0.034 0.112 1.019 2.506 0. 0.
 Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 115 115 52 67 157 150 0 0 normalized size 1 1. 0.45 0.58 1.37 1.3 0. 0. time (sec) N/A 0.034 0.045 0.117 0.991 2.589 0. 0.
 Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 16 16 18 29 22 57 14 0 normalized size 1 1. 1.12 1.81 1.38 3.56 0.88 0. time (sec) N/A 0.006 0.003 0.099 1. 2.585 0.147 0.
 Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C F F F(-2) F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 61 61 64 0 0 0 0 0 normalized size 1 1. 1.05 0. 0. 0. 0. 0. time (sec) N/A 0.108 0.086 0.378 0. 0. 0. 0.
 Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 54 54 49 0 0 0 0 0 normalized size 1 1. 0.91 0. 0. 0. 0. 0. time (sec) N/A 0.101 0.04 0.279 0. 0. 0. 0.
 Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B F F F(-2) F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 77 77 236 0 0 0 0 0 normalized size 1 1. 3.06 0. 0. 0. 0. 0. time (sec) N/A 0.088 0.569 0.434 0. 0. 0. 0.
 Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F(-2) F A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 52 52 54 0 0 0 51 0 normalized size 1 1. 1.04 0. 0. 0. 0.98 0. time (sec) N/A 0.04 0.045 0.255 0. 0. 8.634 0.
 Problem 27 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 54 54 49 53 68 115 63 108 normalized size 1 1. 0.91 0.98 1.26 2.13 1.17 2. time (sec) N/A 0.029 0.049 0.21 1.008 2.565 3.266 1.153
 Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 75 75 76 109 144 171 73 93 normalized size 1 1. 1.01 1.45 1.92 2.28 0.97 1.24 time (sec) N/A 0.044 0.055 0.177 0.973 2.642 5.507 1.164
 Problem 29 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 31 31 38 43 34 90 41 61 normalized size 1 1. 1.23 1.39 1.1 2.9 1.32 1.97 time (sec) N/A 0.022 0.038 0.171 1.021 2.535 2.457 1.148
 Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A A A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 47 47 47 85 105 140 29 70 normalized size 1 1. 1. 1.81 2.23 2.98 0.62 1.49 time (sec) N/A 0.026 0.03 0.194 0.967 2.57 3.453 1.171
 Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A B A A veriﬁed N/A NO NO TBD TBD TBD TBD TBD size 24 31 35 113 86 203 48 82 normalized size 1 1.29 1.46 4.71 3.58 8.46 2. 3.42 time (sec) N/A 0.015 0.015 0.169 1.002 2.776 2.018 1.161
 Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 38 38 42 107 73 142 41 0 normalized size 1 1. 1.11 2.82 1.92 3.74 1.08 0. time (sec) N/A 0.035 0.028 0.18 1.001 3.016 4.265 0.
 Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B A B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 40 40 43 145 116 232 36 111 normalized size 1 1. 1.08 3.62 2.9 5.8 0.9 2.78 time (sec) N/A 0.029 0.027 0.188 1.008 2.949 4.075 1.135
 Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 31 31 37 42 34 103 0 93 normalized size 1 1. 1.19 1.35 1.1 3.32 0. 3. time (sec) N/A 0.022 0.034 0.179 1.033 2.41 0. 1.193
 Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B A A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 65 65 53 173 174 250 83 139 normalized size 1 1. 0.82 2.66 2.68 3.85 1.28 2.14 time (sec) N/A 0.043 0.049 0.183 1.022 2.586 5.326 1.152
 Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 51 51 46 52 55 124 65 167 normalized size 1 1. 0.9 1.02 1.08 2.43 1.27 3.27 time (sec) N/A 0.038 0.042 0.197 0.989 2.439 3.043 1.255
 Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F(-2) F A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 59 59 55 0 0 0 66 0 normalized size 1 1. 0.93 0. 0. 0. 1.12 0. time (sec) N/A 0.042 0.061 0.199 0. 0. 19.537 0.
 Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C C F F C F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 202 202 112 150 0 0 48 0 normalized size 1 1. 0.55 0.74 0. 0. 0.24 0. time (sec) N/A 0.125 0.249 0.211 0. 0. 3.352 0.
 Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A A A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 52 52 53 94 109 149 36 74 normalized size 1 1. 1.02 1.81 2.1 2.87 0.69 1.42 time (sec) N/A 0.038 0.054 0.29 1.032 2.595 4.294 1.133
 Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C C F F C F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 86 86 113 104 0 0 41 0 normalized size 1 1. 1.31 1.21 0. 0. 0.48 0. time (sec) N/A 0.053 0.215 0.187 0. 0. 3.279 0.
 Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B A A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 40 40 42 116 96 205 58 82 normalized size 1 1. 1.05 2.9 2.4 5.12 1.45 2.05 time (sec) N/A 0.036 0.031 0.285 0.989 2.548 5.24 1.132
 Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C C F F C F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 165 165 96 146 0 0 42 0 normalized size 1 1. 0.58 0.88 0. 0. 0.25 0. time (sec) N/A 0.08 0.141 0.18 0. 0. 0.908 0.
 Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A B A A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 46 46 22 86 73 163 54 80 normalized size 1 1. 0.48 1.87 1.59 3.54 1.17 1.74 time (sec) N/A 0.033 0.045 0.272 1.008 2.56 7.134 1.229
 Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C C F F C F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 91 91 96 111 0 0 42 0 normalized size 1 1. 1.05 1.22 0. 0. 0.46 0. time (sec) N/A 0.047 0.16 0.18 0. 0. 2.604 0.
 Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B A B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 42 42 24 114 124 227 39 103 normalized size 1 1. 0.57 2.71 2.95 5.4 0.93 2.45 time (sec) N/A 0.039 0.044 0.287 1.072 2.52 4.863 1.125
 Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A C C F F C F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 181 181 114 171 0 0 44 0 normalized size 1 1. 0.63 0.94 0. 0. 0.24 0. time (sec) N/A 0.095 0.193 0.215 0. 0. 3.465 0.
 Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 31 31 39 42 34 105 0 34 normalized size 1 1. 1.26 1.35 1.1 3.39 0. 1.1 time (sec) N/A 0.022 0.053 0.273 1.037 2.519 0. 1.209
 Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F(-2) F A F(-2) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 64 64 57 0 0 0 71 0 normalized size 1 1. 0.89 0. 0. 0. 1.11 0. time (sec) N/A 0.322 0.065 0.185 0. 0. 14.121 0.
 Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 85 85 84 117 158 193 82 0 normalized size 1 1. 0.99 1.38 1.86 2.27 0.96 0. time (sec) N/A 0.254 0.063 0.174 1.035 2.562 6.686 0.
 Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A F(-2) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 38 38 44 61 43 109 51 0 normalized size 1 1. 1.16 1.61 1.13 2.87 1.34 0. time (sec) N/A 0.225 0.047 0.183 0.992 2.536 3.489 0.
 Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 52 52 57 90 120 159 36 0 normalized size 1 1. 1.1 1.73 2.31 3.06 0.69 0. time (sec) N/A 0.24 0.038 0.171 0.989 2.525 4.371 0.
 Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A B A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 43 43 44 120 101 235 63 0 normalized size 1 1. 1.02 2.79 2.35 5.47 1.47 0. time (sec) N/A 0.158 0.047 0.179 0.964 2.498 4.363 0.
 Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 47 47 52 112 80 165 49 0 normalized size 1 1. 1.11 2.38 1.7 3.51 1.04 0. time (sec) N/A 0.072 0.035 0.128 0.965 2.532 4.22 0.
 Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 38 38 39 150 126 252 34 0 normalized size 1 1. 1.03 3.95 3.32 6.63 0.89 0. time (sec) N/A 0.214 0.057 0.177 1.016 2.595 4.739 0.
 Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A B F(-2) F(-2) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 34 54 46 63 38 127 0 0 normalized size 1 1.59 1.35 1.85 1.12 3.74 0. 0. time (sec) N/A 0.204 0.042 0.19 1. 2.479 0. 0.
 Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 73 73 73 176 188 269 92 0 normalized size 1 1. 1. 2.41 2.58 3.68 1.26 0. time (sec) N/A 0.234 0.049 0.21 1.01 2.529 6.106 0.
 Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A A F(-2) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 58 58 54 73 70 146 76 0 normalized size 1 1. 0.93 1.26 1.21 2.52 1.31 0. time (sec) N/A 0.233 0.054 0.184 1.001 2.681 3.767 0.
 Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B A A F(-2) veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 96 96 74 209 243 296 114 0 normalized size 1 1. 0.77 2.18 2.53 3.08 1.19 0. time (sec) N/A 0.253 0.072 0.187 1.002 2.788 8.605 0.
 Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 85 85 88 0 0 0 0 0 normalized size 1 1. 1.04 0. 0. 0. 0. 0. time (sec) N/A 0.104 0.243 0.382 0. 0. 0. 0.
 Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B A F A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 92 92 85 165 216 209 0 120 normalized size 1 1. 0.92 1.79 2.35 2.27 0. 1.3 time (sec) N/A 0.108 0.208 0.201 1.537 3.023 0. 1.204
 Problem 61 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 72 72 64 120 66 127 0 101 normalized size 1 1. 0.89 1.67 0.92 1.76 0. 1.4 time (sec) N/A 0.098 0.127 0.143 1.027 2.981 0. 1.148
 Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B A F A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 59 59 54 133 142 162 0 82 normalized size 1 1. 0.92 2.25 2.41 2.75 0. 1.39 time (sec) N/A 0.09 0.131 0.15 1.498 2.612 0. 1.138
 Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A F A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 36 36 35 89 42 89 0 61 normalized size 1 1. 0.97 2.47 1.17 2.47 0. 1.69 time (sec) N/A 0.07 0.067 0.151 1.031 2.607 0. 1.131
 Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B A F A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 27 27 38 85 80 89 0 46 normalized size 1 1. 1.41 3.15 2.96 3.3 0. 1.7 time (sec) N/A 0.059 0.049 0.14 1.556 2.552 0. 1.158
 Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 33 33 37 172 0 194 0 95 normalized size 1 1. 1.12 5.21 0. 5.88 0. 2.88 time (sec) N/A 0.043 0.033 0.15 0. 2.677 0. 1.159
 Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A F(-2) A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 30 30 30 154 46 97 0 58 normalized size 1 1. 1. 5.13 1.53 3.23 0. 1.93 time (sec) N/A 0.071 0.101 0.24 1.528 2.587 0. 1.154
 Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B F B F B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 60 60 58 210 0 293 0 154 normalized size 1 1. 0.97 3.5 0. 4.88 0. 2.57 time (sec) N/A 0.097 0.109 0.195 0. 2.613 0. 1.144
 Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B A A A A veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 61 61 54 193 76 155 75 111 normalized size 1 1. 0.89 3.16 1.25 2.54 1.23 1.82 time (sec) N/A 0.095 0.147 0.196 1.589 2.528 5.461 1.166
 Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F F F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 61 61 52 0 0 0 0 0 normalized size 1 1. 0.85 0. 0. 0. 0. 0. time (sec) N/A 0.098 0.058 0.509 0. 0. 0. 0.
 Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A B F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 46 46 74 63 77 570 0 0 normalized size 1 1. 1.61 1.37 1.67 12.39 0. 0. time (sec) N/A 0.058 0.136 0.25 1.011 2.814 0. 0.
 Problem 71 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F A B F(-1) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 46 46 74 0 81 969 0 0 normalized size 1 1. 1.61 0. 1.76 21.07 0. 0. time (sec) N/A 0.07 0.174 0.349 1.031 3.012 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 [42] had the largest ratio of [ 0.875 ]

 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 8 7 1. 10 0.7

2 A 7 6 1. 10 0.6

3 A 6 6 1. 8 0.75

4 A 14 8 1. 10 0.8

5 A 6 6 1. 10 0.6

6 A 8 8 1. 10 0.8

7 A 20 9 1. 20 0.45

8 A 17 9 1. 20 0.45

9 A 11 8 1. 18 0.444

10 A 8 6 1. 12 0.5

11 A 17 9 1. 20 0.45

12 A 12 8 1. 20 0.4

13 A 23 11 1. 20 0.55

14 A 4 3 1. 10 0.3

15 A 4 3 1. 10 0.3

16 A 4 3 1. 8 0.375

17 A 3 3 1. 6 0.5

18 A 7 6 1. 10 0.6

19 A 5 5 1. 10 0.5

20 A 6 5 1. 10 0.5

21 A 7 5 1. 10 0.5

22 A 3 3 1. 4 0.75

23 A 7 6 1. 10 0.6

24 A 7 6 1. 10 0.6

25 A 7 7 1. 10 0.7

26 A 4 4 1. 10 0.4

27 A 4 4 1. 10 0.4

28 A 7 7 1. 10 0.7

29 A 3 3 1. 10 0.3

30 A 6 6 1. 8 0.75

31 A 5 5 1.29 6 0.833

32 A 6 6 1. 10 0.6

33 A 5 5 1. 10 0.5

34 A 3 3 1. 10 0.3

35 A 6 6 1. 10 0.6

36 A 5 4 1. 10 0.4

37 A 4 4 1. 12 0.333

38 A 8 8 1. 12 0.667

39 A 6 6 1. 12 0.5

40 A 5 5 1. 12 0.417

41 A 6 6 1. 10 0.6

42 A 7 7 1. 8 0.875

43 A 6 6 1. 12 0.5

44 A 5 5 1. 12 0.417

45 A 6 6 1. 12 0.5

46 A 7 7 1. 12 0.583

47 A 3 3 1. 12 0.25

48 A 5 4 1. 12 0.333

49 A 8 7 1. 12 0.583

50 A 4 3 1. 12 0.25

51 A 7 6 1. 12 0.5

52 A 6 5 1. 10 0.5

53 A 7 6 1. 8 0.75

54 A 6 5 1. 12 0.417

55 A 4 3 1.59 12 0.25

56 A 7 6 1. 12 0.5

57 A 6 4 1. 12 0.333

58 A 8 6 1. 12 0.5

59 A 4 3 1. 23 0.13

60 A 9 7 1. 21 0.333

61 A 6 5 1. 21 0.238

62 A 7 7 1. 21 0.333

63 A 3 3 1. 21 0.143

64 A 5 5 1. 19 0.263

65 A 7 7 1. 18 0.389

66 A 4 4 1. 21 0.19

67 A 7 6 1. 21 0.286

68 A 7 5 1. 21 0.238

69 A 8 8 1. 19 0.421

70 A 6 6 1. 12 0.5

71 A 6 6 1. 14 0.429