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2.2.1 Integrals 1 to 52

Detailed conclusion table per each integral is given by the 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 as F.

In this table, the column N.S. means normalized size and is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To make the table fit the page, the name Mathematica was abbreviated to MMA

Problem 1

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A B A A A B A B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

11 13 23 17 11 19 29 12 24

N.S.

1 1.18 2.09 1.55 1.00 1.73 2.64 1.09 2.18

time (sec)

N/A 0.163 0.017 0.308 0.238 0.277 0.089 0.278 0.465

Problem 2

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C C A B A B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

15 15 29 24 18 39 17 35 15

N.S.

1 1.00 1.93 1.60 1.20 2.60 1.13 2.33 1.00

time (sec)

N/A 0.162 0.021 0.025 0.322 0.280 0.083 0.264 0.070

Problem 3

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A A B B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

28 30 34 29 23 47 53 118 78

N.S.

1 1.07 1.21 1.04 0.82 1.68 1.89 4.21 2.79

time (sec)

N/A 0.222 0.104 0.048 0.254 0.259 0.214 0.273 15.231

Problem 4

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B A B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

27 27 33 29 34 84 27 62 23

N.S.

1 1.00 1.22 1.07 1.26 3.11 1.00 2.30 0.85

time (sec)

N/A 0.216 0.017 0.041 0.330 0.261 0.094 0.273 0.073

Problem 5

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B B B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

42 44 53 39 38 83 66 164 182

N.S.

1 1.05 1.26 0.93 0.90 1.98 1.57 3.90 4.33

time (sec)

N/A 0.295 0.029 0.062 0.246 0.271 0.311 0.269 16.317

Problem 6

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B A B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

45 45 33 39 44 123 39 91 36

N.S.

1 1.00 0.73 0.87 0.98 2.73 0.87 2.02 0.80

time (sec)

N/A 0.297 0.024 0.044 0.339 0.270 0.118 0.273 0.088

Problem 7

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B A B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

58 60 56 49 48 126 80 208 340

N.S.

1 1.03 0.97 0.84 0.83 2.17 1.38 3.59 5.86

time (sec)

N/A 0.379 0.372 0.078 0.236 0.282 0.425 0.281 19.374

Problem 8

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B A B B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

57 57 33 49 54 168 51 116 43

N.S.

1 1.00 0.58 0.86 0.95 2.95 0.89 2.04 0.75

time (sec)

N/A 0.366 0.013 0.057 0.321 0.263 0.157 0.279 11.796

Problem 9

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

232 227 175 169 197 362 0 0 91

N.S.

1 0.98 0.75 0.73 0.85 1.56 0.00 0.00 0.39

time (sec)

N/A 0.558 0.501 0.139 0.340 0.276 0.000 0.000 12.901

Problem 10

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

212 197 101 154 185 342 0 0 74

N.S.

1 0.93 0.48 0.73 0.87 1.61 0.00 0.00 0.35

time (sec)

N/A 0.455 0.348 0.032 0.310 0.280 0.000 0.000 12.512

Problem 11

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

210 201 161 149 179 262 0 0 75

N.S.

1 0.96 0.77 0.71 0.85 1.25 0.00 0.00 0.36

time (sec)

N/A 0.471 0.220 0.030 0.336 0.264 0.000 0.000 12.369

Problem 12

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

192 174 71 136 165 232 0 0 50

N.S.

1 0.91 0.37 0.71 0.86 1.21 0.00 0.00 0.26

time (sec)

N/A 0.378 0.110 0.074 0.366 0.270 0.000 0.000 12.022

Problem 13

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

192 180 131 138 165 220 0 0 57

N.S.

1 0.94 0.68 0.72 0.86 1.15 0.00 0.00 0.30

time (sec)

N/A 0.370 0.114 0.057 0.313 0.269 0.000 0.000 12.210

Problem 14

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

212 197 82 157 187 379 0 0 76

N.S.

1 0.93 0.39 0.74 0.88 1.79 0.00 0.00 0.36

time (sec)

N/A 0.465 0.137 0.028 0.310 0.280 0.000 0.000 12.015

Problem 15

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

214 205 86 157 188 376 0 0 77

N.S.

1 0.96 0.40 0.73 0.88 1.76 0.00 0.00 0.36

time (sec)

N/A 0.459 0.280 0.033 0.326 0.298 0.000 0.000 12.585

Problem 16

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

234 225 97 171 205 480 0 0 91

N.S.

1 0.96 0.41 0.73 0.88 2.05 0.00 0.00 0.39

time (sec)

N/A 0.562 0.357 0.032 0.308 0.273 0.000 0.000 12.474

Problem 17

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

242 217 205 214 196 431 0 0 246

N.S.

1 0.90 0.85 0.88 0.81 1.78 0.00 0.00 1.02

time (sec)

N/A 0.483 0.392 0.103 0.327 0.301 0.000 0.000 12.887

Problem 18

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

225 196 185 191 182 416 0 0 260

N.S.

1 0.87 0.82 0.85 0.81 1.85 0.00 0.00 1.16

time (sec)

N/A 0.393 0.228 0.079 0.306 0.295 0.000 0.000 12.595

Problem 19

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

131 112 106 108 102 211 0 0 134

N.S.

1 0.85 0.81 0.82 0.78 1.61 0.00 0.00 1.02

time (sec)

N/A 0.307 0.324 0.050 0.383 0.273 0.000 0.000 12.657

Problem 20

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

131 112 98 108 103 639 0 0 128

N.S.

1 0.85 0.75 0.82 0.79 4.88 0.00 0.00 0.98

time (sec)

N/A 0.317 0.154 0.041 0.344 0.269 0.000 0.000 12.163

Problem 21

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

225 196 189 203 182 384 0 0 231

N.S.

1 0.87 0.84 0.90 0.81 1.71 0.00 0.00 1.03

time (sec)

N/A 0.416 0.228 0.064 0.336 0.258 0.000 0.000 12.443

Problem 22

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B F F B

verified

N/A No Yes Yes TBD TBD TBD TBD TBD

size

244 219 254 215 204 660 0 0 277

N.S.

1 0.90 1.04 0.88 0.84 2.70 0.00 0.00 1.14

time (sec)

N/A 0.495 0.261 0.071 0.315 0.268 0.000 0.000 12.348

Problem 23

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

46 46 46 0 0 0 0 0 0

N.S.

1 1.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.190 0.134 0.000 0.000 0.000 0.000 0.000 0.000

Problem 24

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

51 51 52 0 0 0 0 0 0

N.S.

1 1.00 1.02 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.202 0.129 0.000 0.000 0.000 0.000 0.000 0.000

Problem 25

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A A F A F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

36 27 31 29 30 52 0 31 0

N.S.

1 0.75 0.86 0.81 0.83 1.44 0.00 0.86 0.00

time (sec)

N/A 0.279 0.087 0.101 0.487 0.270 0.000 0.258 0.000

Problem 26

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F A F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

16 16 20 22 20 43 0 20 0

N.S.

1 1.00 1.25 1.38 1.25 2.69 0.00 1.25 0.00

time (sec)

N/A 0.219 0.022 0.033 0.424 0.264 0.000 0.253 0.000

Problem 27

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F A B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

17 17 17 26 12 45 0 19 25

N.S.

1 1.00 1.00 1.53 0.71 2.65 0.00 1.12 1.47

time (sec)

N/A 0.212 0.012 0.031 0.428 0.252 0.000 0.256 11.515

Problem 28

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F A F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

39 28 27 36 22 74 0 46 0

N.S.

1 0.72 0.69 0.92 0.56 1.90 0.00 1.18 0.00

time (sec)

N/A 0.262 0.038 0.030 0.438 0.268 0.000 0.253 0.000

Problem 29

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

200 149 80 189 113 400 0 0 0

N.S.

1 0.74 0.40 0.94 0.56 2.00 0.00 0.00 0.00

time (sec)

N/A 0.510 0.311 0.094 0.418 0.264 0.000 0.000 0.000

Problem 30

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

176 136 122 165 94 378 0 0 0

N.S.

1 0.77 0.69 0.94 0.53 2.15 0.00 0.00 0.00

time (sec)

N/A 0.453 0.135 0.039 0.345 0.266 0.000 0.000 0.000

Problem 31

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

176 136 60 164 94 423 0 0 0

N.S.

1 0.77 0.34 0.93 0.53 2.40 0.00 0.00 0.00

time (sec)

N/A 0.452 0.094 0.037 0.411 0.274 0.000 0.000 0.000

Problem 32

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A C F F F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

212 151 71 185 109 567 0 0 0

N.S.

1 0.71 0.33 0.87 0.51 2.67 0.00 0.00 0.00

time (sec)

N/A 0.501 0.147 0.030 0.372 0.267 0.000 0.000 0.000

Problem 33

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A A F A F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

70 40 30 40 37 110 0 57 0

N.S.

1 0.57 0.43 0.57 0.53 1.57 0.00 0.81 0.00

time (sec)

N/A 0.339 0.028 0.066 0.395 0.265 0.000 0.265 0.000

Problem 34

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C A A B F A F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

32 23 28 27 16 61 0 21 0

N.S.

1 0.72 0.88 0.84 0.50 1.91 0.00 0.66 0.00

time (sec)

N/A 0.234 0.016 0.028 0.391 0.277 0.000 0.267 0.000

Problem 35

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F F(-2) F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

31 21 23 26 13 80 0 0 0

N.S.

1 0.68 0.74 0.84 0.42 2.58 0.00 0.00 0.00

time (sec)

N/A 0.219 0.027 0.034 0.431 0.248 0.000 0.000 0.000

Problem 36

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A B F F(-2) F(-1)

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

77 40 41 42 29 142 0 0 0

N.S.

1 0.52 0.53 0.55 0.38 1.84 0.00 0.00 0.00

time (sec)

N/A 0.332 0.070 0.033 0.429 0.267 0.000 0.000 0.000

Problem 37

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

60 60 60 0 0 0 0 0 0

N.S.

1 1.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.297 0.123 0.000 0.000 0.000 0.000 0.000 0.000

Problem 38

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

62 62 62 0 0 0 0 0 0

N.S.

1 1.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.308 0.115 0.000 0.000 0.000 0.000 0.000 0.000

Problem 39

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C F F F F F F(-1)

verified

N/A Yes No N/A TBD TBD TBD TBD TBD

size

87 87 289 0 0 0 0 0 0

N.S.

1 1.00 3.32 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.338 2.693 0.000 0.000 0.000 0.000 0.000 0.000

Problem 40

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

84 84 83 0 0 0 0 0 0

N.S.

1 1.00 0.99 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.374 0.934 0.000 0.000 0.000 0.000 0.000 0.000

Problem 41

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

64 64 65 0 0 0 0 0 0

N.S.

1 1.00 1.02 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.265 0.163 0.000 0.000 0.000 0.000 0.000 0.000

Problem 42

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

90 90 83 0 0 0 0 0 0

N.S.

1 1.00 0.92 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.477 0.811 0.000 0.000 0.000 0.000 0.000 0.000

Problem 43

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A A F(-1) F B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

76 71 73 90 84 145 0 0 123

N.S.

1 0.93 0.96 1.18 1.11 1.91 0.00 0.00 1.62

time (sec)

N/A 0.257 0.646 23.145 0.253 0.274 0.000 0.000 13.361

Problem 44

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A A F F B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

51 49 45 60 56 87 0 0 84

N.S.

1 0.96 0.88 1.18 1.10 1.71 0.00 0.00 1.65

time (sec)

N/A 0.244 0.438 3.501 0.254 0.290 0.000 0.000 12.662

Problem 45

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A A A A F A B

verified

N/A Yes Yes Yes TBD TBD TBD TBD TBD

size

25 25 26 26 25 41 0 47 73

N.S.

1 1.00 1.04 1.04 1.00 1.64 0.00 1.88 2.92

time (sec)

N/A 0.205 0.025 0.494 0.231 0.258 0.000 0.459 12.411

Problem 46

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

51 51 59 0 0 0 0 0 0

N.S.

1 1.00 1.16 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.217 0.482 0.000 0.000 0.000 0.000 0.000 0.000

Problem 47

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

51 51 59 0 0 0 0 0 0

N.S.

1 1.00 1.16 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.229 0.542 0.000 0.000 0.000 0.000 0.000 0.000

Problem 48

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C F F F F F F(-1)

verified

N/A Yes No N/A TBD TBD TBD TBD TBD

size

79 79 784 0 0 0 0 0 0

N.S.

1 1.00 9.92 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.237 12.599 0.000 0.000 0.000 0.000 0.000 0.000

Problem 49

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A A F F F F F F(-1)

verified

N/A Yes Yes N/A TBD TBD TBD TBD TBD

size

77 77 69 0 0 0 0 0 0

N.S.

1 1.00 0.90 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.216 0.400 0.000 0.000 0.000 0.000 0.000 0.000

Problem 50

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C F F F F F F(-1)

verified

N/A Yes No N/A TBD TBD TBD TBD TBD

size

73 73 264 0 0 0 0 0 0

N.S.

1 1.00 3.62 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.223 1.431 0.000 0.000 0.000 0.000 0.000 0.000

Problem 51

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C F F F F(-1) F F(-1)

verified

N/A Yes No N/A TBD TBD TBD TBD TBD

size

79 79 477 0 0 0 0 0 0

N.S.

1 1.00 6.04 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.226 3.096 0.000 0.000 0.000 0.000 0.000 0.000

Problem 52

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad

grade

N/A A C F F F F F F(-1)

verified

N/A Yes No N/A TBD TBD TBD TBD TBD

size

83 83 306 0 0 0 0 0 0

N.S.

1 1.00 3.69 0.00 0.00 0.00 0.00 0.00 0.00

time (sec)

N/A 0.236 3.030 0.000 0.000 0.000 0.000 0.000 0.000

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