lunes, 24 de noviembre de 2025

Benchmarking USM Transform #3 vs. Mathematica’s Integrate - Part 2

In these tables we benchmark the Unified Substitution Method (USM) change of variables that this arXiv draft calls Transformation 3 (this is the “Transform 1” used in the Mathematica code: see here and here): a half-angle substitution that converts integrals built from tan(½ csc⁻¹((x+b)/a)) and tan(½ sec⁻¹((x+b)/a)) into a rational integrand in a new variable t. The general transformation formula is

$$\int f\!\left(x,\,\tan\left(\tfrac12\csc^{-1}\left(\frac{x+b}{a}\right)\right),\,
\tan\left(\tfrac12\sec^{-1}\left(\frac{x+b}{a}\right)\right)\right)\,dx=
\int f\!\left(a\,\frac{t^{2}+1}{2t} - b,\, t,\, \frac{1-t}{1+t}
\right)\, a\,\frac{t^{2}-1}{2t^{2}}\,dt.$$


For each integrand in the 10 datasets, we compare Transform 1 + back-substitution against Mathematica’s Integrate by timing both methods (total t-USM time, split into the y≥1 and y≤−1 branches, versus Integrate time) and by measuring the structural size of the resulting antiderivatives (LeafCnt and ByteCnt for USM and for Integrate). The integrands are systematically built from tan(½ arccsc(…)) and tan(½ arcsec(…)), with powers, factors of x, x², x³, and rational combinations such as 1/(1+tan²(…)) and products of the two half-angle tangents, so the benchmark probes exactly the niche Transform 1 is designed for, from simple to highly intricate cases. Overall, the data show that on many of the “hard” mixed cases USM is much faster (often by an order of magnitude) while producing antiderivatives of comparable or smaller complexity; on very simple, pattern-friendly integrands Integrate can be faster because USM has a fixed overhead; and across the full test family USM timings are more predictable, with the y≥1 / y≤−1 branch split adding only modest extra cost. In short, Transform 1 is a robust, domain-specific integrator for these arccsc/arcsec tan-half-angle families: it typically yields simpler or similar antiderivatives and large speedups on difficult examples, at the price of some overhead on the easy ones. 

Conclusions from the benchmark

1. On many “hard” mixed cases, Transform 1 is much faster than Integrate.

In datasets like 3, 5, 7, and 8 (the ones with products and rational combinations of both arccsc and arcsec half-angle tangents), Integrate often takes from a few tenths of a second up to several seconds, while t-USM usually stays in the tens to low hundreds of milliseconds.

In some individual examples you get around one order of magnitude speedup: Integrate is in the 1–10 second range while t-USM is still below about 0.2 seconds.

⇒ For structurally complicated expressions in this class, the Transform 1 route is clearly advantageous.

2. On simple or pattern-friendly cases, Integrate can be faster than Transform 1.

In datasets like 1, 4, 9, and 10, several examples have Integrate times of just a few milliseconds, while t-USM has a relatively fixed overhead (often between about 0.02 and 0.06 seconds).

In those cases, Mathematica recognizes a very simple pattern (such as standard tan or rational trig identities) and wins on raw speed.

⇒ Your transform has a nontrivial constant overhead. It shines when the problem is hard for Integrate, but cannot beat Mathematica’s near-instant pattern match on the easy ones.

3. Runtime variability vs. predictability

Integrate is highly variable: sometimes extremely fast, sometimes very slow, even for similar-looking integrands in the same dataset.

t-USM is more stable: most examples sit in a narrow time band, with far fewer extreme slowdowns.

⇒ Transform 1 gives more predictable performance over this whole integrand family, whereas Integrate is opportunistically very fast but with occasional expensive spikes.

4. Result size and complexity stay comparable and reasonable.

The leaf and byte counts show that:

USM antiderivatives are usually similar or somewhat larger in size compared to Integrate’s results, reflecting the mechanical tan-substitution and back-substitution.

There is no systematic explosion in size: the USM expressions stay in the same general range as the ones produced by Integrate.

⇒ From a “how big and messy is the final formula?” standpoint, Transform 1 is competitive and practical, even if it does not always find the most compact form that Integrate sometimes can.

5. The y >= 1 / y <= -1 split is reasonable and not a major cost.

The USM y>=1 and USM y<=-1 times are usually of the same order, with the y >= 1 branch often a bit slower but not dramatically.

Summing them to get t–USM total time roughly doubles the branch time, but that combined cost is still modest compared with the multi-second peaks seen in Integrate.

⇒ The branch-based back-substitution strategy works well in practice and does not dominate the runtime.

Some more specific details
  • USM total time was faster than Integrate in 67 cases.
  • The USM y <= -1 branch alone was faster than Integrate in 74 cases.
  • USM produced a simpler antiderivative (smaller ByteCnt) than Integrate in 40 cases.
  • “Monster” antiderivatives (ByteCnt >= 10,000) occurred 5 times for USM and 24 times for Integrate.
  • The largest ByteCnt observed for a USM antiderivative was 21,616, compared with 150,360 for Integrate.

Benchmark tables

Batch 1 (a = 1, b = 0) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\tan(\frac{1}{2}\csc^{-1}(x))$

0.035177

0.027021

0.008156

0.047215

49

1424

40

1272

2

$\tan^2(\frac{1}{2}\csc^{-1}(x))$

0.003547

0.002061

0.001486

0.022223

46

1328

32

920

3

$x \tan(\frac{1}{2}\csc^{-1}(x))$

0.022767

0.014251

0.008516

0.024667

49

1432

29

840

4

$x \tan^2(\frac{1}{2}\csc^{-1}(x))$

0.018108

0.010469

0.007638

0.040701

50

1480

57

1808

5

$x^2 \tan(\frac{1}{2}\csc^{-1}(x))$

0.039577

0.021322

0.018255

0.022719

78

2304

55

1736

6

$\tan^3(\frac{1}{2}\csc^{-1}(x))$

0.004856

0.002986

0.001871

0.020577

49

1352

34

1032

7

$\frac{\tan^2(\frac{1}{2}\csc^{-1}(x))}{1+\tan^2(\frac{1}{2}\csc^{-1}(x))}$

0.014925

0.009498

0.005427

0.022065

43

1248

26

776

8

$\frac{x \tan(\frac{1}{2}\csc^{-1}(x))}{1+\tan^2(\frac{1}{2}\csc^{-1}(x))}$

0.001671

0.001088

0.000584

0.000562

5

112

5

112

9

$\frac{x^2 \tan^2(\frac{1}{2}\csc^{-1}(x))}{1+\tan^2(\frac{1}{2}\csc^{-1}(x))}$

0.006925

0.004103

0.002822

0.040198

49

1432

29

840

10

$\frac{\tan(\frac{1}{2}\csc^{-1}(x))}{1+\tan^2(\frac{1}{2}\csc^{-1}(x))}$

0.013033

0.007457

0.005577

0.000707

49

1488

6

160

Batch 2 (a = 2, b = 1) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))$

0.028608

0.016822

0.011786

0.060040

73

2136

52

1648

2

$\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))$

0.020553

0.011208

0.009345

0.028131

49

1456

14

432

3

$x \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))$

0.079965

0.043316

0.036649

0.112408

112

3288

61

1856

4

$x \tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))$

0.067947

0.036553

0.031394

0.124146

64

1872

22

648

5

$x^2 \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))$

0.111423

0.059831

0.051593

0.152783

128

3784

66

2008

6

$\tan^3(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))$

0.110670

0.062986

0.047684

0.059734

97

2864

63

1944

7

$\frac{\tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}$

0.029397

0.014407

0.014990

3.089166

73

2136

32

1032

8

$\frac{x \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}$

0.155862

0.092036

0.063826

3.375702

140

4096

89

2768

9

$\frac{\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}{(1+\tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{2})))^2}$

0.055680

0.035450

0.020230

0.239037

104

3040

76

2352

10

$\frac{x^2 \tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{2}))}$

0.117399

0.062708

0.054691

0.193478

72

2160

25

760

Batch 3 (a = 3, b = 0) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))$

0.016477

0.009987

0.006490

0.433624

69

1984

78

2392

2

$\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))$

0.009812

0.005681

0.004131

0.542413

68

2024

75

2328

3

$\tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{3}))$

0.032569

0.017494

0.015075

1.761959

97

2824

2965

90824

4

$\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{3}))$

0.037695

0.019111

0.018584

1.764086

96

2816

212

6488

5

$x \tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))$

0.083580

0.047455

0.036126

1.829866

116

3472

135

4120

6

$x^2 \tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))$

0.192960

0.107316

0.085644

1.250586

172

5056

122

3760

7

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{3}))}$

0.021344

0.011844

0.009500

0.161355

53

1568

56

1696

8

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{3}))}$

0.033939

0.017833

0.016106

0.252749

81

2408

72

2248

9

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{3}))}$

0.030086

0.016910

0.013176

0.120906

70

2064

50

1488

10

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{3}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{3}))}$

0.020298

0.013891

0.006407

0.134716

72

2136

37

1128

Batch 4 (a = 2, b = 0) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\frac{1}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.039622

0.025272

0.014350

0.003285

126

3728

401

12384

2

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.024419

0.012977

0.011442

0.003787

93

2728

365

11256

3

$\frac{\tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.025928

0.012744

0.013184

0.003889

93

2768

365

11296

4

$\frac{\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.035140

0.018781

0.016360

0.003947

124

3632

629

19440

5

$\frac{\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.031646

0.018818

0.012827

0.003794

124

3688

519

16072

6

$\frac{x}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.047375

0.030310

0.017065

0.003241

171

5032

535

16552

7

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.036878

0.021596

0.015282

0.003546

162

4752

221

6728

8

$\frac{x \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.038476

0.022442

0.016035

0.003687

165

4840

220

6664

9

$\frac{x^2}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.057311

0.034182

0.023129

0.003055

221

6520

502

15528

10

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{2})) - \tan(\frac{1}{2}\sec^{-1}(\frac{x}{2}))}$

0.030976

0.017882

0.013095

2.376426

125

3672

319

9928

Batch 5 (a = 3, b = 1) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.043872

0.024213

0.019659

0.008277

132

3888

540

16856

2

$\frac{\tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.036972

0.022722

0.014251

0.008790

134

3992

540

16856

3

$\frac{\tan^2(\frac{1}{2}\csc^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.037858

0.024503

0.013354

0.009292

131

3864

1148

35720

4

$\frac{\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.030831

0.018789

0.012042

6.014513

137

4112

4819

150104

5

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.046732

0.024187

0.022545

1.042959

94

2728

283

8736

6

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.132241

0.079733

0.052507

6.376916

187

5528

932

29240

7

$\frac{x \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.187629

0.118098

0.069531

4.000208

182

5368

934

29312

8

$\frac{x^2 \tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.237800

0.144427

0.093373

2.374396

237

6968

698

21672

9

$\frac{\tan^2(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{1+x}{3}))}$

0.027960

0.015031

0.012929

0.304504

54

1584

86

2720

10

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{1+x}{3})) \tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{1+x}{3}))}$

0.047983

0.020850

0.027133

1.719440

99

2864

1718

52960

Batch 6 (a = 1, b = 2) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$x \tan(\frac{1}{2}\csc^{-1}(2+x))$

0.082802

0.051429

0.031373

0.085258

96

2816

76

2368

2

$x \tan(\frac{1}{2}\sec^{-1}(2+x))$

0.085719

0.055877

0.029842

0.079071

113

3312

59

1840

3

$x^2 \tan(\frac{1}{2}\csc^{-1}(2+x))$

0.094610

0.048842

0.045768

0.059720

142

4208

82

2544

4

$x^2 \tan(\frac{1}{2}\sec^{-1}(2+x))$

0.157842

0.088791

0.069051

0.056525

149

4408

66

2064

5

$x^3 \tan(\frac{1}{2}\csc^{-1}(2+x))$

0.174567

0.095373

0.079194

0.064654

194

5688

86

2672

6

$x^3 \tan(\frac{1}{2}\sec^{-1}(2+x))$

0.204784

0.120930

0.083854

0.057148

176

5224

71

2216

7

$x^2 \tan(\frac{1}{2}\csc^{-1}(2+x)) \tan(\frac{1}{2}\sec^{-1}(2+x))$

0.144057

0.089703

0.054354

1.557722

171

4944

2621

86944

8

$x \tan^2(\frac{1}{2}\csc^{-1}(2+x)) \tan(\frac{1}{2}\sec^{-1}(2+x))$

0.030621

0.015974

0.014647

1.480089

110

3216

147

4784

9

$x \tan(\frac{1}{2}\csc^{-1}(2+x)) \tan^2(\frac{1}{2}\sec^{-1}(2+x))$

0.145077

0.090093

0.054985

2.565992

134

3944

3504

116216

10

$x^2 \tan^2(\frac{1}{2}\csc^{-1}(2+x)) \tan^2(\frac{1}{2}\sec^{-1}(2+x))$

0.119787

0.065501

0.054286

8.066104

185

5440

4526

150360

Batch 7 (a = 2, b = -1) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}$

0.015485

0.008121

0.007364

0.174565

47

1368

56

1704

2

$\frac{\tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}$

0.071097

0.040702

0.030395

0.182227

95

2792

71

2200

3

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}$

0.019971

0.012104

0.007867

0.239892

50

1488

43

1304

4

$\frac{x \tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}$

0.082406

0.040635

0.041771

0.291465

100

2920

88

2728

5

$\frac{\tan^2(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}$

0.002707

0.001527

0.001179

0.197232

44

1272

32

952

6

$\frac{\tan^2(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}$

0.071507

0.036088

0.035419

0.265361

94

2744

84

2560

7

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1))) \tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}$

0.030971

0.016526

0.014445

3.137826

70

2072

393

12344

8

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1))) \tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}$

0.024857

0.012298

0.012559

0.817277

92

2672

1

16

9

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1))) \tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1)))}$

0.081031

0.042717

0.038314

11.304564

118

3472

1061

33304

10

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{1}{2}(x-1))) \tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}{1+\tan(\frac{1}{2}\sec^{-1}(\frac{1}{2}(x-1)))}$

0.058130

0.033846

0.024283

1.569247

97

2832

2842

87608

Batch 8 (a = 4, b = 0) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\tan^4(\frac{1}{2}\csc^{-1}(\frac{x}{4}))$

0.008754

0.005712

0.003041

0.047867

49

1424

44

1264

2

$\tan^4(\frac{1}{2}\sec^{-1}(\frac{x}{4}))$

0.109920

0.060848

0.049072

0.039679

66

1984

19

584

3

$\tan^3(\frac{1}{2}\csc^{-1}(\frac{x}{4})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{4}))$

0.014493

0.008004

0.006489

1.729428

70

2072

142

4248

4

$\tan(\frac{1}{2}\csc^{-1}(\frac{x}{4})) \tan^3(\frac{1}{2}\sec^{-1}(\frac{x}{4}))$

0.097480

0.053342

0.044138

2.323108

122

3600

3761

115416

5

$\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{4})) \tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{4}))$

0.037647

0.020417

0.017231

1.690136

93

2720

212

6488

6

$x \tan^3(\frac{1}{2}\csc^{-1}(\frac{x}{4}))$

0.016296

0.009435

0.006861

0.066551

43

1248

38

1152

7

$x \tan^3(\frac{1}{2}\sec^{-1}(\frac{x}{4}))$

0.126396

0.073583

0.052813

0.106924

139

4072

51

1528

8

$x^2 \tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{4})) \tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{4}))$

0.120098

0.064641

0.055457

2.926456

188

5536

4752

146112

9

$\frac{\tan^3(\frac{1}{2}\csc^{-1}(\frac{x}{4}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{4}))}$

0.019274

0.010572

0.008702

0.063735

52

1472

47

1464

10

$\frac{\tan^3(\frac{1}{2}\sec^{-1}(\frac{x}{4}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{4}))}$

0.061335

0.033892

0.027443

0.080144

96

2800

46

1416

Batch 9 (a = 1, b = -2) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$-\frac{\tan(\frac{1}{2}\csc^{-1}(2-x))}{(1+\tan^2(\frac{1}{2}\csc^{-1}(2-x)))^2}$

1.488667

1.147823

0.340844

0.082347

360

10976

56

1816

2

$\frac{\tan(\frac{1}{2}\sec^{-1}(-2+x))}{(1+\tan^2(\frac{1}{2}\sec^{-1}(-2+x)))^2}$

0.139448

0.082895

0.056553

3.269446

118

3528

58

1816

3

$-\frac{\tan(\frac{1}{2}\csc^{-1}(2-x)) \tan(\frac{1}{2}\sec^{-1}(-2+x))}{(1+\tan^2(\frac{1}{2}\csc^{-1}(2-x))) (1+\tan^2(\frac{1}{2}\sec^{-1}(-2+x)))}$

1.538979

1.354974

0.184005

3.169035

403

12256

36

1128

4

$-\frac{x \tan(\frac{1}{2}\csc^{-1}(2-x))}{(1+\tan^2(\frac{1}{2}\csc^{-1}(2-x)))^2}$

4.217547

3.369883

0.847664

0.117005

617

18304

68

2120

5

$\frac{x \tan(\frac{1}{2}\sec^{-1}(-2+x))}{(1+\tan^2(\frac{1}{2}\sec^{-1}(-2+x)))^2}$

0.354207

0.121863

0.232345

3.556781

170

5024

141

4440

6

$-\frac{x^2 \tan(\frac{1}{2}\csc^{-1}(2-x))}{(1+\tan^2(\frac{1}{2}\csc^{-1}(2-x)))^2}$

4.950009

3.593241

1.356769

0.126037

730

21616

85

2744

7

$\frac{x^2 \tan(\frac{1}{2}\sec^{-1}(-2+x))}{(1+\tan^2(\frac{1}{2}\sec^{-1}(-2+x)))^2}$

0.368285

0.233888

0.134396

3.620825

212

6264

157

4960

8

$\frac{\tan^2(\frac{1}{2}\csc^{-1}(2-x))}{(1+\tan^2(\frac{1}{2}\csc^{-1}(2-x)))^2}$

0.014437

0.009164

0.005273

0.001271

7

216

9

256

9

$\frac{\tan^2(\frac{1}{2}\sec^{-1}(-2+x))}{(1+\tan^2(\frac{1}{2}\sec^{-1}(-2+x)))^2}$

0.033048

0.019206

0.013843

0.001939

17

488

11

312

10

$-\frac{x \tan(\frac{1}{2}\csc^{-1}(2-x)) \tan(\frac{1}{2}\sec^{-1}(-2+x))}{(1+\tan^2(\frac{1}{2}\csc^{-1}(2-x))) (1+\tan^2(\frac{1}{2}\sec^{-1}(-2+x)))}$

3.828501

3.053233

0.775268

3.387147

696

20600

49

1552

Batch 10 (a = 5, b = 0) 

#Integrandt–USM total time (s)USM y>=1 time (s)USM y<=-1 time (s)Integrate time (s)LeafCnt USMByteCnt USMLeafCnt IntegrateByteCnt Integrate

1

$\frac{1}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}$

0.062679

0.050728

0.011951

0.057984

50

1512

27

824

2

$\frac{1}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}$

0.029378

0.016563

0.012815

0.032491

60

1792

16

440

3

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}$

0.013509

0.008641

0.004868

0.000725

50

1464

6

160

4

$\frac{\tan(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}$

0.048435

0.025181

0.023255

0.018859

70

2064

51

1520

5

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{5})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}$

0.018353

0.009823

0.008530

0.047740

53

1568

38

1168

6

$\frac{\tan(\frac{1}{2}\csc^{-1}(\frac{x}{5})) \tan(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}$

0.029087

0.014639

0.014448

0.252519

81

2408

72

2248

7

$\frac{x}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}$

0.013335

0.007846

0.005488

0.051719

52

1520

43

1336

8

$\frac{x}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}$

0.008351

0.004908

0.003442

0.048506

14

408

13

352

9

$\frac{x \tan(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}{1+\tan^2(\frac{1}{2}\csc^{-1}(\frac{x}{5}))}$

0.003713

0.002236

0.001476

0.000690

5

112

5

112

10

$\frac{x \tan(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}{1+\tan^2(\frac{1}{2}\sec^{-1}(\frac{x}{5}))}$

0.049127

0.037491

0.011635

0.057764

73

2168

51

1520


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