## 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, 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 }

#### 2.1.2 Mathematica

A grade: { 1, 2, 3, 4, 5, 6, 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, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63 }

B grade: { 7, 36, 44, 45, 49, 50, 61 }

#### 2.1.3 Maple

A grade: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 41, 42, 43, 46, 47, 48, 51, 52, 53, 57, 58, 62, 63 }

B grade: { 39, 40, 44, 45, 49, 50, 54, 55, 56, 59, 60, 61 }

F grade: { 16, 17, 36, 37, 38 }

#### 2.1.4 Maxima

A grade: { 4, 5, 9, 10, 14, 15, 21, 22, 23, 27, 28, 32, 33, 41, 42, 43, 47, 48, 52, 53

B grade: { 1, 2, 3, 6, 7, 8, 11, 12, 13, 39, 40, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61 }

F grade: { 16, 17, 18, 19, 20, 24, 25, 26, 29, 30, 31, 34, 35, 36, 37, 38, 57, 58, 62, 63 }

#### 2.1.5 FriCAS

A grade: { 4, 5, 8, 9, 10, 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, 42, 43, 46, 47, 48, 51, 52, 53, 57, 58, 62, 63 }

B grade: { 3, 7, 13, 41, 56, 61 }

C grade: { 1, 2, 6, 11, 12, 39, 40, 44, 45, 49, 50, 54, 55, 59, 60 }

#### 2.1.6 Sympy

A grade: { 4, 5, 8, 9, 10, 14, 15, 18, 19, 20, 24, 25, 26, 29, 30, 31, 35, 42, 43, 47, 48, 52, 53, 57, 58, 62 }

F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 21, 22, 23, 27, 28, 32, 33, 34, 36, 37, 38, 39, 40, 41, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61, 63 }

#### 2.1.7 Giac

A grade: { 4, 5, 9, 10, 14, 15, 18, 19, 20, 21, 24, 25, 26, 27, 29, 30, 31, 34, 35, 42, 43, 47, 48, 52, 53, 57, 58, 62, 63 }

B grade: { 8, 22, 23, 28, 32, 33 }

F grade: { 1, 2, 3, 6, 7, 11, 12, 13, 16, 17, 36, 37, 38, 39, 40, 41, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61 }

### 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 C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 106 106 106 125 325 802 0 0 normalized size 1 1. 1. 1.18 3.07 7.57 0. 0. time (sec) N/A 0.155 0.064 0.087 1.694 1.611 0. 0.
 Problem 2 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 77 77 77 103 219 562 0 0 normalized size 1 1. 1. 1.34 2.84 7.3 0. 0. time (sec) N/A 0.134 0.008 0.035 1.672 1.621 0. 0.
 Problem 3 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 54 54 54 78 124 332 0 0 normalized size 1 1. 1. 1.44 2.3 6.15 0. 0. time (sec) N/A 0.081 0.005 0.033 1.709 1.621 0. 0.
 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 12 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.015 1.731 0.084 0. 0. 0. 0.
 Problem 5 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 12 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.016 2.509 0.066 0. 0. 0. 0.
 Problem 6 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 98 98 115 133 864 614 0 0 normalized size 1 1. 1.17 1.36 8.82 6.27 0. 0. time (sec) N/A 0.167 0.619 0.054 1.783 1.566 0. 0.
 Problem 7 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B A B B F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 73 73 160 108 348 387 0 0 normalized size 1 1. 2.19 1.48 4.77 5.3 0. 0. time (sec) N/A 0.112 5.41 0.046 1.782 1.629 0. 0.
 Problem 8 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B A A B veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 30 30 43 29 289 96 41 246 normalized size 1 1. 1.43 0.97 9.63 3.2 1.37 8.2 time (sec) N/A 0.023 0.161 0.072 1.466 1.55 0.314 1.525
 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 14 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.028 2.083 0.069 0. 0. 0. 0.
 Problem 10 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 14 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.03 2.417 0.099 0. 0. 0. 0.
 Problem 11 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B C F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 205 205 359 251 1624 938 0 0 normalized size 1 1. 1.75 1.22 7.92 4.58 0. 0. time (sec) N/A 0.292 6.513 0.069 2.192 1.695 0. 0.
 Problem 12 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 128 128 172 180 999 657 0 0 normalized size 1 1. 1.34 1.41 7.8 5.13 0. 0. time (sec) N/A 0.18 2.484 0.058 1.986 1.754 0. 0.
 Problem 13 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B B F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 90 90 171 122 524 396 0 0 normalized size 1 1. 1.9 1.36 5.82 4.4 0. 0. time (sec) N/A 0.103 4.31 0.051 1.876 1.607 0. 0.
 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 14 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.027 4.573 0.552 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 14 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.028 3.164 0.722 0. 0. 0. 0.
 Problem 16 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F A F F(-2) veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 18 18 18 0 0 42 0 0 normalized size 1 1. 1. 0. 0. 2.33 0. 0. time (sec) N/A 0.123 0.885 0.218 0. 1.611 0. 0.
 Problem 17 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A F A F F A F F veriﬁed N/A N/A Yes TBD TBD TBD TBD TBD size 17 0 17 0 0 35 0 0 normalized size 1 0. 1. 0. 0. 2.06 0. 0. time (sec) N/A 0.035 0.56 0.253 0. 1.558 0. 0.
 Problem 18 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 189 189 278 159 0 370 372 261 normalized size 1 1. 1.47 0.84 0. 1.96 1.97 1.38 time (sec) N/A 0.199 0.627 0.194 0. 1.514 0.929 1.195
 Problem 19 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 137 137 178 97 0 248 238 166 normalized size 1 1. 1.3 0.71 0. 1.81 1.74 1.21 time (sec) N/A 0.123 0.31 0.162 0. 1.534 0.683 1.197
 Problem 20 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 84 84 96 139 0 146 128 88 normalized size 1 1. 1.14 1.65 0. 1.74 1.52 1.05 time (sec) N/A 0.054 0.358 0.145 0. 1.531 0.406 1.228
 Problem 21 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 161 161 166 65 149 119 0 192 normalized size 1 1. 1.03 0.4 0.93 0.74 0. 1.19 time (sec) N/A 0.277 0.311 0.15 1.191 1.596 0. 1.178
 Problem 22 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 168 168 224 96 162 212 0 797 normalized size 1 1. 1.33 0.57 0.96 1.26 0. 4.74 time (sec) N/A 0.24 0.767 0.171 1.313 1.801 0. 1.442
 Problem 23 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 227 227 285 216 211 300 0 732 normalized size 1 1. 1.26 0.95 0.93 1.32 0. 3.22 time (sec) N/A 0.322 1.046 0.217 1.569 1.638 0. 1.207
 Problem 24 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 270 270 473 272 0 639 666 517 normalized size 1 1. 1.75 1.01 0. 2.37 2.47 1.91 time (sec) N/A 0.293 1.246 0.298 0. 1.624 1.861 1.193
 Problem 25 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 202 202 282 162 0 406 420 306 normalized size 1 1. 1.4 0.8 0. 2.01 2.08 1.51 time (sec) N/A 0.208 0.668 0.252 0. 1.588 1.254 1.185
 Problem 26 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 151 151 130 82 0 220 228 144 normalized size 1 1. 0.86 0.54 0. 1.46 1.51 0.95 time (sec) N/A 0.142 0.463 0.22 0. 1.589 0.765 1.161
 Problem 27 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) A veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 305 305 211 114 257 203 0 567 normalized size 1 1. 0.69 0.37 0.84 0.67 0. 1.86 time (sec) N/A 0.772 0.422 0.234 1.424 1.564 0. 1.19
 Problem 28 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 436 436 467 175 284 362 0 1551 normalized size 1 1. 1.07 0.4 0.65 0.83 0. 3.56 time (sec) N/A 0.739 1.375 0.277 1.508 1.629 0. 1.993
 Problem 29 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 396 396 667 385 0 941 947 774 normalized size 1 1. 1.68 0.97 0. 2.38 2.39 1.95 time (sec) N/A 0.404 2.384 0.386 0. 1.634 3.432 1.24
 Problem 30 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 294 294 405 227 0 581 590 447 normalized size 1 1. 1.38 0.77 0. 1.98 2.01 1.52 time (sec) N/A 0.265 1.463 0.321 0. 1.591 2.218 1.279
 Problem 31 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 209 209 205 114 0 309 313 204 normalized size 1 1. 0.98 0.55 0. 1.48 1.5 0.98 time (sec) N/A 0.264 0.672 0.275 0. 1.573 1.294 1.18
 Problem 32 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 449 449 336 163 365 284 0 1142 normalized size 1 1. 0.75 0.36 0.81 0.63 0. 2.54 time (sec) N/A 1.783 0.654 0.305 1.377 1.662 0. 1.258
 Problem 33 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A A F(-2) B veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 712 712 833 787 397 509 0 2302 normalized size 1 1. 1.17 1.11 0.56 0.71 0. 3.23 time (sec) N/A 1.732 2.805 0.142 1.789 1.621 0. 2.356
 Problem 34 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A F(-2) A F(-1) A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 25 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.05 22.711 0.147 0. 0. 0. 0.
 Problem 35 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A F(-2) A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 23 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.028 13.886 0.185 0. 0. 0. 0.
 Problem 36 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B F F A F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 98 98 205 0 0 207 0 0 normalized size 1 1. 2.09 0. 0. 2.11 0. 0. time (sec) N/A 0.123 1.227 0.197 0. 1.632 0. 0.
 Problem 37 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F A F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 171 171 192 0 0 354 0 0 normalized size 1 1. 1.12 0. 0. 2.07 0. 0. time (sec) N/A 0.184 38.086 0.118 0. 1.672 0. 0.
 Problem 38 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A F F A F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 251 251 269 0 0 504 0 0 normalized size 1 1. 1.07 0. 0. 2.01 0. 0. time (sec) N/A 0.242 55.015 0.136 0. 1.743 0. 0.
 Problem 39 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 152 152 255 481 898 1257 0 0 normalized size 1 1. 1.68 3.16 5.91 8.27 0. 0. time (sec) N/A 0.251 0.327 0.143 1.803 1.737 0. 0.
 Problem 40 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 115 115 191 295 498 814 0 0 normalized size 1 1. 1.66 2.57 4.33 7.08 0. 0. time (sec) N/A 0.208 0.115 0.074 1.682 1.683 0. 0.
 Problem 41 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A A B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 84 84 87 143 176 420 0 0 normalized size 1 1. 1.04 1.7 2.1 5. 0. 0. time (sec) N/A 0.119 0.014 0.061 1.64 1.63 0. 0.
 Problem 42 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 20 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.028 1.922 0.184 0. 0. 0. 0.
 Problem 43 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 20 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.027 5.718 0.201 0. 0. 0. 0.
 Problem 44 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B B B C F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 300 300 1326 919 3401 1767 0 0 normalized size 1 1. 4.42 3.06 11.34 5.89 0. 0. time (sec) N/A 0.522 7.856 0.168 4.924 1.815 0. 0.
 Problem 45 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B B B C F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 229 229 649 542 1713 1081 0 0 normalized size 1 1. 2.83 2.37 7.48 4.72 0. 0. time (sec) N/A 0.412 7.096 0.109 2.399 1.778 0. 0.
 Problem 46 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B A F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 136 136 200 238 714 540 0 0 normalized size 1 1. 1.47 1.75 5.25 3.97 0. 0. time (sec) N/A 0.19 2.159 0.099 1.766 1.684 0. 0.
 Problem 47 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 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.053 18.984 1.105 0. 0. 0. 0.
 Problem 48 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 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.051 14.364 1.779 0. 0. 0. 0.
 Problem 49 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B B B C F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 612 612 2572 1882 9189 2549 0 0 normalized size 1 1. 4.2 3.08 15.01 4.17 0. 0. time (sec) N/A 0.982 8.338 0.243 51.562 2.062 0. 0.
 Problem 50 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B B B C F F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 436 436 1846 1090 4578 1540 0 0 normalized size 1 1. 4.23 2.5 10.5 3.53 0. 0. time (sec) N/A 0.72 7.621 0.181 10.178 1.829 0. 0.
 Problem 51 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A A B A F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 277 277 277 493 1782 752 0 0 normalized size 1 1. 1. 1.78 6.43 2.71 0. 0. time (sec) N/A 0.338 3.533 0.136 3.144 1.744 0. 0.
 Problem 52 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 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.053 13.592 1.775 0. 0. 0. 0.
 Problem 53 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 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.05 18.698 3.512 0. 0. 0. 0.
 Problem 54 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 243 243 297 1450 1319 2782 0 0 normalized size 1 1. 1.22 5.97 5.43 11.45 0. 0. time (sec) N/A 0.334 1.949 0.284 2.835 1.962 0. 0.
 Problem 55 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B C F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 181 181 236 922 961 1993 0 0 normalized size 1 1. 1.3 5.09 5.31 11.01 0. 0. time (sec) N/A 0.272 1.412 0.179 2.356 1.852 0. 0.
 Problem 56 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B B F F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 125 125 177 462 539 1273 0 0 normalized size 1 1. 1.42 3.7 4.31 10.18 0. 0. time (sec) N/A 0.159 1.468 0.204 2.07 1.877 0. 0.
 Problem 57 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A F(-1) A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.062 1.689 0.685 0. 0. 0. 0.
 Problem 58 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A F(-1) A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.058 4.303 1.657 0. 0. 0. 0.
 Problem 59 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B C F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 848 848 1673 3488 6329 5532 0 0 normalized size 1 1. 1.97 4.11 7.46 6.52 0. 0. time (sec) N/A 2.004 10.051 0.411 13.902 2.501 0. 0.
 Problem 60 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A A B B C F(-2) F veriﬁed N/A Yes Yes TBD TBD TBD TBD TBD size 654 654 714 2160 3464 3513 0 0 normalized size 1 1. 1.09 3.3 5.3 5.37 0. 0. time (sec) N/A 1.537 7.912 0.32 5.494 2.136 0. 0.
 Problem 61 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade A A B B B B F(-2) F veriﬁed N/A Yes NO TBD TBD TBD TBD TBD size 214 214 745 999 1570 1952 0 0 normalized size 1 1. 3.48 4.67 7.34 9.12 0. 0. time (sec) N/A 0.275 6.87 0.279 3.376 2.018 0. 0.
 Problem 62 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A F(-1) A A A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.061 15.847 5.47 0. 0. 0. 0.
 Problem 63 Optimal Rubi Mathematica Maple Maxima Fricas Sympy Giac grade N/A A A A F(-1) A F(-1) A veriﬁed N/A N/A N/A TBD TBD TBD TBD TBD size 22 0 0 0 0 0 0 0 normalized size 1 0. 0. 0. 0. 0. 0. 0. time (sec) N/A 0.059 16.207 10.503 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 [ 0.8333 ]

 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 6 6 1. 10 0.6

2 A 5 5 1. 10 0.5

3 A 4 4 1. 8 0.5

4 A 0 0 0. 0 0.

5 A 0 0 0. 0 0.

6 A 7 7 1. 12 0.583

7 A 6 6 1. 12 0.5

8 A 3 3 1. 10 0.3

9 A 0 0 0. 0 0.

10 A 0 0 0. 0 0.

11 A 13 10 1. 12 0.833

12 A 9 8 1. 12 0.667

13 A 7 7 1. 10 0.7

14 A 0 0 0. 0 0.

15 A 0 0 0. 0 0.

16 A 2 1 1. 45 0.022

17 F 0 0 N/A 0 N/A

18 A 5 3 1. 23 0.13

19 A 4 3 1. 23 0.13

20 A 3 3 1. 21 0.143

21 A 7 4 1. 23 0.174

22 A 7 4 1. 23 0.174

23 A 8 5 1. 23 0.217

24 A 10 3 1. 23 0.13

25 A 8 3 1. 23 0.13

26 A 7 3 1. 21 0.143

27 A 21 5 1. 23 0.217

28 A 24 7 1. 23 0.304

29 A 14 3 1. 23 0.13

30 A 11 3 1. 23 0.13

31 A 11 3 1. 21 0.143

32 A 53 7 1. 23 0.304

33 A 60 9 1. 23 0.391

34 A 0 0 0. 0 0.

35 A 0 0 0. 0 0.

36 A 2 2 1. 23 0.087

37 A 4 2 1. 23 0.087

38 A 5 2 1. 23 0.087

39 A 8 7 1. 18 0.389

40 A 7 6 1. 18 0.333

41 A 6 5 1. 16 0.312

42 A 0 0 0. 0 0.

43 A 0 0 0. 0 0.

44 A 15 9 1. 20 0.45

45 A 13 10 1. 20 0.5

46 A 9 7 1. 18 0.389

47 A 0 0 0. 0 0.

48 A 0 0 0. 0 0.

49 A 28 11 1. 20 0.55

50 A 22 11 1. 20 0.55

51 A 16 9 1. 18 0.5

52 A 0 0 0. 0 0.

53 A 0 0 0. 0 0.

54 A 6 6 1. 20 0.3

55 A 5 5 1. 20 0.25

56 A 4 4 1. 18 0.222

57 A 0 0 0. 0 0.

58 A 0 0 0. 0 0.

59 A 21 9 1. 20 0.45

60 A 18 10 1. 20 0.5

61 A 5 5 1. 18 0.278

62 A 0 0 0. 0 0.

63 A 0 0 0. 0 0.