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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 N.S. in the table below, which stands for normalized size is defined as \(\frac {\text {antiderivative leaf size}}{\text {optimal antiderivative leaf size}}\). To help make the table fit, Mathematica was abbreviated to MMA and IntegrateAlgebraic to I.A.

Problem 1

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B F

verified

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

size

112 112 112 109 108 142 148 150 115 0

N.S.

1 1.00 1.00 0.97 0.96 1.27 1.32 1.34 1.03 0.00

time (sec)

N/A 0.158 0.052 0.001 0.445 0.998 0.094 1.121 2.383 0.000

Problem 2

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B A A A A B F

verified

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

size

126 126 123 246 162 163 146 208 174 0

N.S.

1 1.00 0.98 1.95 1.29 1.29 1.16 1.65 1.38 0.00

time (sec)

N/A 0.211 0.088 0.005 0.443 0.695 0.580 1.189 2.537 0.001

Problem 3

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B A A B A B F

verified

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

size

84 84 85 196 104 117 507 112 105 0

N.S.

1 1.00 1.01 2.33 1.24 1.39 6.04 1.33 1.25 0.00

time (sec)

N/A 0.086 0.062 0.009 0.439 1.316 20.494 1.251 2.970 0.001

Problem 4

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A F(-1) F(-1) A B F

verified

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

size

108 108 102 179 134 0 0 162 127 0

N.S.

1 1.00 0.94 1.66 1.24 0.00 0.00 1.50 1.18 0.00

time (sec)

N/A 0.110 0.082 0.010 0.443 0.000 0.000 1.137 4.168 0.001

Problem 5

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A F(-1) F(-1) B B F

verified

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

size

163 163 164 164 310 0 0 363 317 0

N.S.

1 1.00 1.01 1.01 1.90 0.00 0.00 2.23 1.94 0.00

time (sec)

N/A 0.212 0.201 0.010 0.489 0.000 0.000 1.189 6.622 0.002

Problem 6

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B F

verified

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

size

23 23 23 20 19 19 20 22 19 0

N.S.

1 1.00 1.00 0.87 0.83 0.83 0.87 0.96 0.83 0.00

time (sec)

N/A 0.011 0.006 0.007 0.430 1.709 0.153 1.251 0.076 0.000

Problem 7

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B F

verified

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

size

43 43 33 34 34 53 32 31 29 0

N.S.

1 1.00 0.77 0.79 0.79 1.23 0.74 0.72 0.67 0.00

time (sec)

N/A 0.037 0.024 0.009 0.433 1.440 0.161 1.176 0.123 0.000

Problem 8

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

227 227 205 301 239 649 274 338 413 297

N.S.

1 1.00 0.90 1.33 1.05 2.86 1.21 1.49 1.82 1.31

time (sec)

N/A 0.257 0.275 0.010 0.991 1.294 37.787 1.405 0.161 0.201

Problem 9

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

146 146 145 176 152 405 167 201 263 202

N.S.

1 1.00 0.99 1.21 1.04 2.77 1.14 1.38 1.80 1.38

time (sec)

N/A 0.098 0.183 0.010 0.963 1.591 27.596 1.362 2.621 0.126

Problem 10

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

77 77 81 89 91 219 92 105 136 105

N.S.

1 1.00 1.05 1.16 1.18 2.84 1.19 1.36 1.77 1.36

time (sec)

N/A 0.024 0.159 0.008 0.979 0.901 25.991 1.328 0.091 0.060

Problem 11

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

54 54 55 46 60 111 54 57 45 57

N.S.

1 1.00 1.02 0.85 1.11 2.06 1.00 1.06 0.83 1.06

time (sec)

N/A 0.017 0.048 0.007 0.979 0.866 5.984 1.244 0.072 0.032

Problem 12

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B F(-2) A A A B A

verified

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

size

101 101 101 196 0 449 97 112 2368 111

N.S.

1 1.00 1.00 1.94 0.00 4.45 0.96 1.11 23.45 1.10

time (sec)

N/A 0.114 0.122 0.016 0.000 1.217 27.334 1.352 2.871 0.173

Problem 13

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A F(-2) B B A B A

verified

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

size

127 127 124 192 0 1018 1204 142 1827 151

N.S.

1 1.00 0.98 1.51 0.00 8.02 9.48 1.12 14.39 1.19

time (sec)

N/A 0.109 0.439 0.019 0.000 1.345 133.870 1.423 0.599 0.497

Problem 14

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

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

verified

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

size

208 208 260 424 0 2216 0 300 4852 268

N.S.

1 1.00 1.25 2.04 0.00 10.65 0.00 1.44 23.33 1.29

time (sec)

N/A 0.274 0.602 0.020 0.000 3.164 0.000 1.394 4.543 1.081

Problem 15

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

226 226 204 301 238 641 274 338 413 343

N.S.

1 1.00 0.90 1.33 1.05 2.84 1.21 1.50 1.83 1.52

time (sec)

N/A 0.252 0.281 0.011 0.974 1.403 37.946 1.394 2.528 0.178

Problem 16

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

145 145 146 176 152 403 167 201 263 202

N.S.

1 1.00 1.01 1.21 1.05 2.78 1.15 1.39 1.81 1.39

time (sec)

N/A 0.094 0.179 0.010 0.980 1.071 26.173 1.368 0.089 0.128

Problem 17

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

77 77 87 89 90 217 92 105 136 91

N.S.

1 1.00 1.13 1.16 1.17 2.82 1.19 1.36 1.77 1.18

time (sec)

N/A 0.024 0.132 0.009 0.979 0.925 25.966 1.262 2.487 0.076

Problem 18

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A A A B A

verified

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

size

54 54 55 46 60 111 54 57 45 57

N.S.

1 1.00 1.02 0.85 1.11 2.06 1.00 1.06 0.83 1.06

time (sec)

N/A 0.016 0.050 0.007 0.974 0.782 6.083 1.201 0.069 0.035

Problem 19

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A F(-2) A A A B A

verified

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

size

101 101 100 103 0 450 100 112 2355 101

N.S.

1 1.00 0.99 1.02 0.00 4.46 0.99 1.11 23.32 1.00

time (sec)

N/A 0.119 0.227 0.016 0.000 1.473 24.251 1.219 2.819 0.170

Problem 20

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A F(-2) B B A B A

verified

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

size

128 128 122 137 0 1008 1149 142 1814 139

N.S.

1 1.00 0.95 1.07 0.00 7.88 8.98 1.11 14.17 1.09

time (sec)

N/A 0.116 0.196 0.017 0.000 1.355 142.242 1.323 2.952 0.482

Problem 21

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

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

verified

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

size

205 205 259 221 0 2211 0 301 4839 248

N.S.

1 1.00 1.26 1.08 0.00 10.79 0.00 1.47 23.60 1.21

time (sec)

N/A 0.279 0.532 0.019 0.000 3.638 0.000 1.401 4.632 1.256

Problem 22

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B A

verified

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

size

111 111 89 132 105 65 484 63 345 116

N.S.

1 1.00 0.80 1.19 0.95 0.59 4.36 0.57 3.11 1.05

time (sec)

N/A 0.045 0.089 0.036 0.974 1.318 35.797 1.245 7.778 0.096

Problem 23

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B A

verified

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

size

87 87 81 111 83 57 393 53 269 100

N.S.

1 1.00 0.93 1.28 0.95 0.66 4.52 0.61 3.09 1.15

time (sec)

N/A 0.034 0.031 0.015 0.993 1.289 25.601 1.341 5.922 0.089

Problem 24

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B A

verified

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

size

63 63 73 90 61 49 269 40 191 84

N.S.

1 1.00 1.16 1.43 0.97 0.78 4.27 0.63 3.03 1.33

time (sec)

N/A 0.023 0.030 0.016 0.962 1.312 20.771 1.275 4.527 0.078

Problem 25

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B A

verified

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

size

37 37 61 70 41 43 133 28 118 51

N.S.

1 1.00 1.65 1.89 1.11 1.16 3.59 0.76 3.19 1.38

time (sec)

N/A 0.013 0.026 0.019 0.955 1.307 11.713 1.239 3.447 0.107

Problem 26

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A B C A B A

verified

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

size

29 29 53 69 41 47 71 44 47 45

N.S.

1 1.00 1.83 2.38 1.41 1.62 2.45 1.52 1.62 1.55

time (sec)

N/A 0.016 0.025 0.019 0.953 0.808 25.620 1.233 2.985 0.099

Problem 27

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A C B B A

verified

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

size

45 45 29 25 42 27 107 88 24 37

N.S.

1 1.00 0.64 0.56 0.93 0.60 2.38 1.96 0.53 0.82

time (sec)

N/A 0.010 0.013 0.004 0.958 0.803 15.238 1.271 2.747 0.044

Problem 28

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A C B B A

verified

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

size

73 73 37 33 62 35 189 130 32 55

N.S.

1 1.00 0.51 0.45 0.85 0.48 2.59 1.78 0.44 0.75

time (sec)

N/A 0.019 0.016 0.005 0.962 1.326 17.839 1.242 2.731 0.045

Problem 29

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A C B B A

verified

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

size

97 97 45 41 84 43 274 175 40 72

N.S.

1 1.00 0.46 0.42 0.87 0.44 2.82 1.80 0.41 0.74

time (sec)

N/A 0.027 0.018 0.007 0.973 0.833 23.154 1.406 2.766 0.046

Problem 30

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A C B B A

verified

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

size

121 121 53 49 106 51 359 217 48 88

N.S.

1 1.00 0.44 0.40 0.88 0.42 2.97 1.79 0.40 0.73

time (sec)

N/A 0.038 0.019 0.006 0.966 0.906 33.865 1.315 2.829 0.049

Problem 31

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A C A B A

verified

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

size

39 39 48 44 21 40 117 43 65 49

N.S.

1 1.00 1.23 1.13 0.54 1.03 3.00 1.10 1.67 1.26

time (sec)

N/A 0.007 0.016 0.018 0.962 0.810 35.801 1.269 4.080 0.055

Problem 32

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A A A A C A B A

verified

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

size

39 39 48 44 21 40 117 43 444 49

N.S.

1 1.00 1.23 1.13 0.54 1.03 3.00 1.10 11.38 1.26

time (sec)

N/A 0.011 0.006 0.005 0.976 1.076 72.778 1.291 5.273 0.064

Problem 33

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B B B A B B F

verified

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

size

167 167 149 726 474 877 8221 1665 819 0

N.S.

1 1.00 0.89 4.35 2.84 5.25 49.23 9.97 4.90 0.00

time (sec)

N/A 0.131 0.231 0.010 0.542 0.865 8.124 1.028 2.945 0.113

Problem 34

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B F B F(-1) F B F

verified

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

size

362 362 220 894 0 1659 0 0 1895 0

N.S.

1 1.00 0.61 2.47 0.00 4.58 0.00 0.00 5.23 0.00

time (sec)

N/A 0.362 0.478 0.010 0.000 0.927 0.000 0.000 4.487 0.122

Problem 35

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B F B F(-1) F B F

verified

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

size

507 507 279 2343 0 3441 0 0 3720 0

N.S.

1 1.00 0.55 4.62 0.00 6.79 0.00 0.00 7.34 0.00

time (sec)

N/A 0.586 0.702 0.017 0.000 1.180 0.000 0.000 6.752 0.123

Problem 36

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B F B F(-1) F B F

verified

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

size

363 360 227 906 0 1608 0 0 1890 0

N.S.

1 0.99 0.63 2.50 0.00 4.43 0.00 0.00 5.21 0.00

time (sec)

N/A 0.397 0.548 0.010 0.000 0.705 0.000 0.000 4.277 0.120

Problem 37

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A B F B F(-1) F B F

verified

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

size

188 186 181 509 0 905 0 0 869 0

N.S.

1 0.99 0.96 2.71 0.00 4.81 0.00 0.00 4.62 0.00

time (sec)

N/A 0.098 0.128 0.010 0.000 0.842 0.000 0.000 3.407 0.079

Problem 38

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B B

verified

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

size

79 79 57 139 87 78 313 101 244 179

N.S.

1 1.00 0.72 1.76 1.10 0.99 3.96 1.28 3.09 2.27

time (sec)

N/A 0.142 0.066 0.043 0.967 0.884 82.403 1.339 7.441 0.172

Problem 39

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B A

verified

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

size

63 63 45 117 57 67 282 76 232 117

N.S.

1 1.00 0.71 1.86 0.90 1.06 4.48 1.21 3.68 1.86

time (sec)

N/A 0.061 0.035 0.020 0.970 0.940 49.786 1.315 6.988 0.126

Problem 40

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C F(-2) B A

verified

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

size

48 48 48 96 57 81 245 0 122 95

N.S.

1 1.00 1.00 2.00 1.19 1.69 5.10 0.00 2.54 1.98

time (sec)

N/A 0.184 0.060 0.028 0.969 0.936 55.199 0.000 3.924 0.126

Problem 41

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C F(-2) B A

verified

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

size

48 48 48 97 57 84 221 0 114 93

N.S.

1 1.00 1.00 2.02 1.19 1.75 4.60 0.00 2.38 1.94

time (sec)

N/A 0.183 0.060 0.023 0.971 0.872 49.853 0.000 3.741 0.166

Problem 42

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C F(-2) B A

verified

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

size

71 71 56 108 98 65 218 0 312 112

N.S.

1 1.00 0.79 1.52 1.38 0.92 3.07 0.00 4.39 1.58

time (sec)

N/A 0.188 0.050 0.022 0.967 0.657 80.293 0.000 5.854 0.257

Problem 43

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C A B B

verified

N/A Yes NO TBD TBD TBD TBD TBD TBD NO

size

87 151 149 137 100 73 308 105 318 230

N.S.

1 1.74 1.71 1.57 1.15 0.84 3.54 1.21 3.66 2.64

time (sec)

N/A 0.146 0.349 0.026 0.434 0.907 78.825 1.303 12.354 0.171

Problem 44

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A B B C B A C A B B

verified

N/A Yes NO TBD TBD TBD TBD TBD TBD NO

size

52 135 126 120 90 61 277 80 312 112

N.S.

1 2.60 2.42 2.31 1.73 1.17 5.33 1.54 6.00 2.15

time (sec)

N/A 0.071 0.225 0.018 0.426 0.923 48.292 1.253 12.400 0.143

Problem 45

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A B B C A A C A B A

verified

N/A Yes NO TBD TBD TBD TBD TBD TBD NO

size

55 135 128 95 56 73 240 71 118 91

N.S.

1 2.45 2.33 1.73 1.02 1.33 4.36 1.29 2.15 1.65

time (sec)

N/A 0.184 0.418 0.023 0.962 0.744 47.403 1.295 3.975 0.148

Problem 46

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A B A C A A C A B A

verified

N/A Yes NO TBD TBD TBD TBD TBD TBD NO

size

55 135 89 96 56 82 216 83 118 89

N.S.

1 2.45 1.62 1.75 1.02 1.49 3.93 1.51 2.15 1.62

time (sec)

N/A 0.182 0.185 0.023 0.965 1.025 45.925 1.347 3.859 0.116

Problem 47

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C B B A

verified

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

size

83 129 82 103 61 69 212 145 316 107

N.S.

1 1.55 0.99 1.24 0.73 0.83 2.55 1.75 3.81 1.29

time (sec)

N/A 0.189 0.144 0.023 0.972 0.881 74.795 1.350 9.891 0.115

Problem 48

Optimal Rubi MMA Maple Maxima Fricas Sympy Giac Mupad I.A.

grade

A A A C A A C B B A

verified

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

size

116 171 94 123 86 90 219 197 304 168

N.S.

1 1.47 0.81 1.06 0.74 0.78 1.89 1.70 2.62 1.45

time (sec)

N/A 0.221 0.133 0.023 0.983 0.945 129.782 1.348 9.436 0.157

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