## 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 }

B grade: { }

C grade: { }

F grade: { }

#### 2.1.2 Mathematica

A grade: { 2, 4, 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, 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 }

B grade: { 1, 3, 5, 41, 42 }

C grade: { }

F grade: { }

#### 2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67 }

B grade: { 39, 40, 41, 42, 43, 51 }

C grade: { 17, 18, 19, 20, 21, 22, 23, 24, 55, 56, 57, 58, 59, 60 }

F grade: { 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 68, 69, 70 }

#### 2.1.4 Maxima

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 43, 44, 45, 46, 50, 51, 61, 62, 63, 64, 65, 66, 67 }

B grade: { 39, 40, 41, 42, 47, 48, 49 }

C grade: { }

F grade: { 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, 52, 53, 54, 55, 56, 57, 58, 59, 60, 68, 69, 70 }

#### 2.1.5 FriCAS

A grade: { 2, 4, 6, 8, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67 }

B grade: { 1, 3, 5, 7, 39, 40, 41, 42 }

C grade: { }

F grade: { 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, 55, 56, 57, 58, 59, 60, 68, 69, 70 }

#### 2.1.6 Sympy

A grade: { 1, 43, 44, 45, 46, 51, 52, 53, 54 }

B grade: { }

C grade: { }

F grade: { 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, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 }

#### 2.1.7 Giac

A grade: { 1, 2, 4, 6, 8, 43, 45, 46, 50, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67 }

B grade: { 3, 5, 7, 39, 40, 41, 42, 44, 47, 48, 49, 51 }

C grade: { }

F grade: { 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, 55, 56, 57, 58, 59, 60, 68, 69, 70 }

A grade: { }

B grade: { 1, 2, 3, 4, 5, 6, 7, 8, 12, 20, 43, 51, 61, 62, 63, 64 }

C grade: { }

F grade: { 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 65, 66, 67, 68, 69, 70 }

### 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 B A A B A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 12 12 38 20 19 30 37 15 12 normalized size 1 1.00 3.17 1.67 1.58 2.50 3.08 1.25 1.00 time (sec) N/A 0.004 0.017 0.043 0.320 0.672 1.930 0.167 0.401
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 11 11 11 12 13 19 0 13 11 normalized size 1 1.00 1.00 1.09 1.18 1.73 0.00 1.18 1.00 time (sec) N/A 0.008 0.009 0.821 0.328 0.444 0.000 0.213 0.104
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A B A A B F B B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 34 34 75 40 46 72 0 92 36 normalized size 1 1.00 2.21 1.18 1.35 2.12 0.00 2.71 1.06 time (sec) N/A 0.014 0.013 0.850 0.320 0.781 0.000 0.469 0.114
 Problem 4 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 27 27 35 23 25 45 0 25 21 normalized size 1 1.00 1.30 0.85 0.93 1.67 0.00 0.93 0.78 time (sec) N/A 0.011 0.011 0.870 0.319 0.688 0.000 0.151 0.080
 Problem 5 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A B A A B F B B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 55 55 113 59 71 112 0 138 59 normalized size 1 1.00 2.05 1.07 1.29 2.04 0.00 2.51 1.07 time (sec) N/A 0.026 0.016 0.832 0.328 0.494 0.000 0.186 0.128
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 42 42 56 33 35 64 0 35 38 normalized size 1 1.00 1.33 0.79 0.83 1.52 0.00 0.83 0.90 time (sec) N/A 0.014 0.015 0.787 0.355 0.530 0.000 0.189 0.096
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A B F B B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 76 76 151 78 91 155 0 182 78 normalized size 1 1.00 1.99 1.03 1.20 2.04 0.00 2.39 1.03 time (sec) N/A 0.036 0.014 0.782 0.336 0.757 0.000 0.206 0.155
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 55 55 77 43 45 87 0 45 41 normalized size 1 1.00 1.40 0.78 0.82 1.58 0.00 0.82 0.75 time (sec) N/A 0.017 0.010 0.795 0.403 0.629 0.000 0.350 0.127
 Problem 9 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F(-1) F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 90 90 63 160 0 0 0 0 -1 normalized size 1 1.00 0.70 1.78 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.037 0.182 2.890 0.000 0.610 0.000 0.000 0.000
 Problem 10 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 67 67 50 88 0 0 0 0 -1 normalized size 1 1.00 0.75 1.31 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.026 0.075 2.427 0.000 1.157 0.000 0.000 0.000
 Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 63 63 49 132 0 0 0 0 -1 normalized size 1 1.00 0.78 2.10 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.024 0.051 3.184 0.000 0.564 0.000 0.000 0.000
 Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 41 41 40 69 0 0 0 0 61 normalized size 1 1.00 0.98 1.68 0.00 0.00 0.00 0.00 1.49 time (sec) N/A 0.017 0.032 1.760 0.000 0.838 0.000 0.000 0.121
 Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 41 41 40 91 0 0 0 0 -1 normalized size 1 1.00 0.98 2.22 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.017 0.035 2.648 0.000 0.615 0.000 0.000 0.000
 Problem 14 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 67 67 53 88 0 0 0 0 -1 normalized size 1 1.00 0.79 1.31 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.027 0.053 2.203 0.000 0.687 0.000 0.000 0.000
 Problem 15 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 67 67 60 142 0 0 0 0 -1 normalized size 1 1.00 0.90 2.12 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.026 0.097 2.533 0.000 0.681 0.000 0.000 0.000
 Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 90 90 65 104 0 0 0 0 -1 normalized size 1 1.00 0.72 1.16 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.038 0.150 2.204 0.000 0.518 0.000 0.000 0.000
 Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F(-1) F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 103 103 67 1028 0 0 0 0 -1 normalized size 1 1.00 0.65 9.98 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.053 0.181 1.230 0.000 0.665 0.000 0.000 0.000
 Problem 18 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 75 75 55 313 0 0 0 0 -1 normalized size 1 1.00 0.73 4.17 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.031 0.165 1.035 0.000 0.567 0.000 0.000 0.000
 Problem 19 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 71 71 54 505 0 0 0 0 -1 normalized size 1 1.00 0.76 7.11 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.032 0.107 1.175 0.000 0.462 0.000 0.000 0.000
 Problem 20 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 43 43 42 165 0 0 0 0 63 normalized size 1 1.00 0.98 3.84 0.00 0.00 0.00 0.00 1.47 time (sec) N/A 0.018 0.025 1.223 0.000 0.613 0.000 0.000 0.199
 Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 43 43 42 533 0 0 0 0 -1 normalized size 1 1.00 0.98 12.40 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.018 0.028 1.070 0.000 0.809 0.000 0.000 0.000
 Problem 22 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 77 77 63 186 0 0 0 0 -1 normalized size 1 1.00 0.82 2.42 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.033 0.071 0.842 0.000 0.637 0.000 0.000 0.000
 Problem 23 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 77 77 60 534 0 0 0 0 -1 normalized size 1 1.00 0.78 6.94 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.032 0.136 0.852 0.000 0.606 0.000 0.000 0.000
 Problem 24 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 105 105 70 213 0 0 0 0 -1 normalized size 1 1.00 0.67 2.03 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.049 0.120 0.849 0.000 0.671 0.000 0.000 0.000
 Problem 25 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 51 51 54 0 0 0 0 0 -1 normalized size 1 1.00 1.06 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.022 0.070 0.649 0.000 0.859 0.000 0.000 0.000
 Problem 26 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 51 51 51 0 0 0 0 0 -1 normalized size 1 1.00 1.00 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.022 0.051 0.712 0.000 1.259 0.000 0.000 0.000
 Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 53 53 51 0 0 0 0 0 -1 normalized size 1 1.00 0.96 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.022 0.052 0.951 0.000 0.490 0.000 0.000 0.000
 Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 53 53 51 0 0 0 0 0 -1 normalized size 1 1.00 0.96 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.021 0.045 0.779 0.000 0.584 0.000 0.000 0.000
 Problem 29 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 53 53 51 0 0 0 0 0 -1 normalized size 1 1.00 0.96 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.021 0.046 0.773 0.000 0.586 0.000 0.000 0.000
 Problem 30 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 53 53 68 0 0 0 0 0 -1 normalized size 1 1.00 1.28 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.021 0.113 0.611 0.000 0.605 0.000 0.000 0.000
 Problem 31 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 54 54 57 0 0 0 0 0 -1 normalized size 1 1.00 1.06 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.030 0.045 0.577 0.000 0.665 0.000 0.000 0.000
 Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 54 54 59 0 0 0 0 0 -1 normalized size 1 1.00 1.09 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.027 0.067 0.651 0.000 0.683 0.000 0.000 0.000
 Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 56 56 59 0 0 0 0 0 -1 normalized size 1 1.00 1.05 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.026 0.066 0.753 0.000 0.673 0.000 0.000 0.000
 Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 56 56 59 0 0 0 0 0 -1 normalized size 1 1.00 1.05 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.026 0.065 0.642 0.000 0.850 0.000 0.000 0.000
 Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 56 56 59 0 0 0 0 0 -1 normalized size 1 1.00 1.05 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.027 0.064 0.630 0.000 0.535 0.000 0.000 0.000
 Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 56 56 76 0 0 0 0 0 -1 normalized size 1 1.00 1.36 0.00 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.027 0.150 0.523 0.000 0.614 0.000 0.000 0.000
 Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 69 69 59 0 0 0 0 0 -1 normalized size 1 1.00 0.86 0.00 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.026 0.090 2.652 0.000 0.664 0.000 0.000 0.000
 Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 72 72 65 0 0 0 0 0 -1 normalized size 1 1.00 0.90 0.00 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.030 0.073 2.620 0.000 0.621 0.000 0.000 0.000
 Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B B B F(-1) B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 50 50 92 101 1669 93 0 129 -1 normalized size 1 1.00 1.84 2.02 33.38 1.86 0.00 2.58 -0.02 time (sec) N/A 0.017 0.255 0.447 0.760 0.605 0.000 0.267 0.000
 Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B B B F B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 36 36 72 78 869 69 0 100 -1 normalized size 1 1.00 2.00 2.17 24.14 1.92 0.00 2.78 -0.03 time (sec) N/A 0.012 0.255 0.442 0.601 0.663 0.000 0.505 0.000
 Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A B B B B F B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 22 22 51 52 300 44 0 69 -1 normalized size 1 1.00 2.32 2.36 13.64 2.00 0.00 3.14 -0.05 time (sec) N/A 0.009 0.093 0.463 0.574 0.503 0.000 0.492 0.000
 Problem 42 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A B B B B F B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 5 5 28 31 35 19 0 12 -1 normalized size 1 1.00 5.60 6.20 7.00 3.80 0.00 2.40 -0.20 time (sec) N/A 0.007 0.013 0.608 0.579 0.536 0.000 0.297 0.000
 Problem 43 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B A A A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 12 12 12 26 10 4 12 11 12 normalized size 1 1.00 1.00 2.17 0.83 0.33 1.00 0.92 1.00 time (sec) N/A 0.009 0.007 0.583 0.571 0.646 0.352 0.289 0.170
 Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A A B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 29 29 23 30 11 11 29 44 -1 normalized size 1 1.00 0.79 1.03 0.38 0.38 1.00 1.52 -0.03 time (sec) N/A 0.011 0.016 0.436 0.515 0.566 0.937 0.424 0.000
 Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A A A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 43 43 31 38 17 17 46 61 -1 normalized size 1 1.00 0.72 0.88 0.40 0.40 1.07 1.42 -0.02 time (sec) N/A 0.015 0.026 0.426 0.680 0.579 9.322 0.333 0.000
 Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A A A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 57 57 37 44 23 21 61 78 -1 normalized size 1 1.00 0.65 0.77 0.40 0.37 1.07 1.37 -0.02 time (sec) N/A 0.019 0.040 0.430 0.525 0.667 138.800 0.547 0.000
 Problem 47 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A B A F(-1) B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 84 84 61 102 2183 106 0 164 -1 normalized size 1 1.00 0.73 1.21 25.99 1.26 0.00 1.95 -0.01 time (sec) N/A 0.040 0.322 0.474 4.137 0.615 0.000 0.362 0.000
 Problem 48 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A B A F B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 65 65 51 79 1113 80 0 124 -1 normalized size 1 1.00 0.78 1.22 17.12 1.23 0.00 1.91 -0.02 time (sec) N/A 0.029 0.214 0.437 0.868 0.658 0.000 0.290 0.000
 Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A B A F B F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 46 46 39 53 318 49 0 72 -1 normalized size 1 1.00 0.85 1.15 6.91 1.07 0.00 1.57 -0.02 time (sec) N/A 0.022 0.073 0.462 0.539 0.586 0.000 0.283 0.000
 Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 26 26 30 32 24 64 0 13 -1 normalized size 1 1.00 1.15 1.23 0.92 2.46 0.00 0.50 -0.04 time (sec) N/A 0.016 0.016 0.683 0.789 0.547 0.000 0.561 0.000
 Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A B A A A B B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 14 14 14 27 13 22 17 34 15 normalized size 1 1.00 1.00 1.93 0.93 1.57 1.21 2.43 1.07 time (sec) N/A 0.013 0.006 0.619 0.578 0.835 0.450 0.356 0.220
 Problem 52 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F A A A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 36 36 27 31 0 29 39 43 -1 normalized size 1 1.00 0.75 0.86 0.00 0.81 1.08 1.19 -0.03 time (sec) N/A 0.018 0.018 0.408 0.000 0.534 1.039 0.430 0.000
 Problem 53 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F A A A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 55 55 36 39 0 37 61 56 -1 normalized size 1 1.00 0.65 0.71 0.00 0.67 1.11 1.02 -0.02 time (sec) N/A 0.027 0.032 0.415 0.000 0.598 9.415 0.496 0.000
 Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A F A A A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 74 74 42 45 0 43 82 69 -1 normalized size 1 1.00 0.57 0.61 0.00 0.58 1.11 0.93 -0.01 time (sec) N/A 0.036 0.045 0.454 0.000 0.815 136.156 0.721 0.000
 Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 123 123 58 1313 0 0 0 0 -1 normalized size 1 1.00 0.47 10.67 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.060 0.184 0.767 0.000 0.637 0.000 0.000 0.000
 Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 71 71 46 372 0 0 0 0 -1 normalized size 1 1.00 0.65 5.24 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.038 0.113 0.904 0.000 0.814 0.000 0.000 0.000
 Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 48 48 46 343 0 0 0 0 -1 normalized size 1 1.00 0.96 7.15 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.027 0.040 0.975 0.000 0.594 0.000 0.000 0.000
 Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 50 50 38 125 0 0 0 0 -1 normalized size 1 1.00 0.76 2.50 0.00 0.00 0.00 0.00 -0.02 time (sec) N/A 0.030 0.049 1.089 0.000 0.563 0.000 0.000 0.000
 Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 79 79 52 361 0 0 0 0 -1 normalized size 1 1.00 0.66 4.57 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.038 0.094 0.788 0.000 0.510 0.000 0.000 0.000
 Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A C F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 123 123 63 158 0 0 0 0 -1 normalized size 1 1.00 0.51 1.28 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.061 0.144 0.793 0.000 0.736 0.000 0.000 0.000
 Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F(-1) A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 164 164 59 53 66 118 0 69 603 normalized size 1 1.00 0.36 0.32 0.40 0.72 0.00 0.42 3.68 time (sec) N/A 0.035 0.069 0.764 0.439 0.579 0.000 0.651 5.052
 Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 118 118 47 41 48 88 0 51 121 normalized size 1 1.00 0.40 0.35 0.41 0.75 0.00 0.43 1.03 time (sec) N/A 0.028 0.041 0.513 0.436 0.975 0.000 0.308 2.419
 Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 62 62 33 29 30 52 0 23 44 normalized size 1 1.00 0.53 0.47 0.48 0.84 0.00 0.37 0.71 time (sec) N/A 0.021 0.019 0.424 0.477 0.566 0.000 0.291 0.674
 Problem 64 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 16 16 16 18 9 24 0 9 7 normalized size 1 1.00 1.00 1.12 0.56 1.50 0.00 0.56 0.44 time (sec) N/A 0.015 0.006 0.632 0.584 0.416 0.000 0.249 0.178
 Problem 65 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 36 36 25 24 25 46 0 22 -1 normalized size 1 1.00 0.69 0.67 0.69 1.28 0.00 0.61 -0.03 time (sec) N/A 0.016 0.026 0.742 0.468 0.546 0.000 0.414 0.000
 Problem 66 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 86 86 38 41 58 61 0 42 -1 normalized size 1 1.00 0.44 0.48 0.67 0.71 0.00 0.49 -0.01 time (sec) N/A 0.029 0.044 0.642 0.453 0.665 0.000 0.302 0.000
 Problem 67 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A A A A F A F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 132 132 55 57 88 73 0 49 -1 normalized size 1 1.00 0.42 0.43 0.67 0.55 0.00 0.37 -0.01 time (sec) N/A 0.045 0.099 0.780 0.506 0.470 0.000 0.269 0.000
 Problem 68 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 80 80 71 0 0 0 0 0 -1 normalized size 1 1.00 0.89 0.00 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.050 0.122 3.011 0.000 0.621 0.000 0.000 0.000
 Problem 69 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 82 82 73 0 0 0 0 0 -1 normalized size 1 1.00 0.89 0.00 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.049 0.088 2.232 0.000 0.671 0.000 0.000 0.000
 Problem 70 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac Mupad grade A A A F F F F F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD TBD size 91 91 77 0 0 0 0 0 -1 normalized size 1 1.00 0.85 0.00 0.00 0.00 0.00 0.00 -0.01 time (sec) N/A 0.045 0.142 3.901 0.000 0.667 0.000 0.000 0.000

### 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 [47] had the largest ratio of [.4000]

 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 1.00 6 0.167

2 A 2 2 1.00 8 0.250

3 A 2 2 1.00 8 0.250

4 A 2 1 1.00 8 0.125

5 A 3 2 1.00 8 0.250

6 A 2 1 1.00 8 0.125

7 A 4 2 1.00 8 0.250

8 A 2 1 1.00 8 0.125

9 A 4 3 1.00 10 0.300

10 A 3 3 1.00 10 0.300

11 A 3 3 1.00 10 0.300

12 A 2 2 1.00 10 0.200

13 A 2 2 1.00 10 0.200

14 A 3 3 1.00 10 0.300

15 A 3 3 1.00 10 0.300

16 A 4 3 1.00 10 0.300

17 A 4 3 1.00 12 0.250

18 A 3 3 1.00 12 0.250

19 A 3 3 1.00 12 0.250

20 A 2 2 1.00 12 0.167

21 A 2 2 1.00 12 0.167

22 A 3 3 1.00 12 0.250

23 A 3 3 1.00 12 0.250

24 A 4 3 1.00 12 0.250

25 A 2 2 1.00 10 0.200

26 A 2 2 1.00 10 0.200

27 A 2 2 1.00 10 0.200

28 A 2 2 1.00 10 0.200

29 A 2 2 1.00 10 0.200

30 A 2 2 1.00 10 0.200

31 A 2 2 1.00 12 0.167

32 A 2 2 1.00 12 0.167

33 A 2 2 1.00 12 0.167

34 A 2 2 1.00 12 0.167

35 A 2 2 1.00 12 0.167

36 A 2 2 1.00 12 0.167

37 A 2 2 1.00 8 0.250

38 A 2 2 1.00 10 0.200

39 A 5 3 1.00 8 0.375

40 A 4 3 1.00 8 0.375

41 A 3 3 1.00 8 0.375

42 A 2 2 1.00 8 0.250

43 A 2 2 1.00 8 0.250

44 A 3 3 1.00 8 0.375

45 A 4 3 1.00 8 0.375

46 A 5 3 1.00 8 0.375

47 A 6 4 1.00 10 0.400

48 A 5 4 1.00 10 0.400

49 A 4 4 1.00 10 0.400

50 A 3 3 1.00 10 0.300

51 A 2 2 1.00 10 0.200

52 A 3 3 1.00 10 0.300

53 A 4 3 1.00 10 0.300

54 A 5 3 1.00 10 0.300

55 A 7 4 1.00 10 0.400

56 A 5 4 1.00 10 0.400

57 A 4 4 1.00 10 0.400

58 A 4 4 1.00 10 0.400

59 A 5 4 1.00 10 0.400

60 A 7 4 1.00 10 0.400

61 A 3 2 1.00 10 0.200

62 A 3 2 1.00 10 0.200

63 A 3 2 1.00 10 0.200

64 A 3 3 1.00 10 0.300

65 A 3 3 1.00 10 0.300

66 A 5 3 1.00 10 0.300

67 A 7 3 1.00 10 0.300

68 A 3 3 1.00 12 0.250

69 A 3 3 1.00 14 0.214

70 A 3 3 1.00 21 0.143