Numeric integration returns wrong result for `N[ integrate(1/x,{x,0,1}) ]`

[tranleduy2000]

There is the input, the integral of 1/x from 0 to 1 is a divergent integral:

N[ integrate(1/x,{x,0,1}) ] 

Error result:

𝟣𝟢.𝟥𝟩𝟦𝟩𝟨

Expected result:

Indeterminate

[axkr]

Integrate function switches to NIntegrate function in numeric mode:

• symja_android_library/symja_android_library/matheclipse-core/src/main/java/org/matheclipse/core/reflection/system/Integrate.java

  Line 266 in 87689e1

  IASTAppendable copy = holdallAST.apply(S.NIntegrate);

For the NIntegrate function the LegendreGauss method is the default numeric method for the calculation.

See these JUnit tests:

• symja_android_library/symja_android_library/matheclipse-core/src/test/java/org/matheclipse/core/system/IntegrateTest.java

  Line 606 in 87689e1

  public void testXReciprocalIssue1064() {

Maybe we should use another default method for NIntegrate?
Any suggestions?

[tranleduy2000]

Each numerical integration method has its own strengths and weaknesses. I think the LegendreGauss method is effective in general cases.

[tranleduy2000]

@axkr I suggest to use the Romberg as the default method, since LegendreGauss fails in many common cases:

This algorithm divides the integration interval into equally-sized sub-interval and on each of them performs a Legendre-Gauss quadrature. Because of its non-adaptive nature, this algorithm can converge to a wrong value for the integral (for example, if the function is significantly different from zero toward the ends of the integration interval). In particular, a change of variables aimed at estimating integrals over infinite intervals as proposed here should be avoided when using this class.

https://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/analysis/integration/IterativeLegendreGaussIntegrator.html

[tranleduy2000]

I have some suggestions for determining numerical integral methods:

• If the expression has Abs function, use the LegendreGauss method

  Example input: Integrate[Abs(x^2-2x), {x, -10, 10}] // N
  Result should be 669.3282335875249

• If the expression contains a variable that occurs in the exponent of Power function, use the GaussKronrod method
  Example input: x^x, 3^(2x), E^(-Sin(t))

• Otherwise, use the Romberg method, since it is the optimized version of trapezoid and Simpson methods