Incorrect result for divergent integral

[ppurka]

From google spreadsheet which no one reads X-(

sage: integrate(x^2/exp(-1/(x^2+1))/(x^2+1),(x,-infinity,infinity)).n()
-3.3357920860949157

Wolfram alpha gives the correct answer.

The problem comes from here:

sage: D=1.00000000000000*x^2*e^(1/(x^2 + 1.00000000000000))/(x^2 + 1.00000000000
....: 000)
sage: numerical_integral(D,-oo,+oo)
(-3.3357920860949157, 4.262178610048295e-09)

and in one specific algorithm

sage: numerical_integral(D,-oo,+oo, algorithm="qag")
(-3.3357920860949157, 4.262178610048295e-09)
sage: numerical_integral(D,-oo,+oo, algorithm="qng")
(nan, nan)
sage: numerical_integral(D,-oo,+oo, algorithm="qags")
(nan, nan)

and this case is handled by
gsl_integration_qagi

And the problem can be cut into two parts:

sage: numerical_integral(D,0,+oo)
(-1.6678960430474579, 2.1310893050241475e-09)
sage: numerical_integral(D,-oo,0)
(-1.6678960430474579, 2.1310893050241475e-09)

CC: @kcrisman @sagetrac-jakobkroeker @slel @vinklein @rwst @tscrim

Component: calculus

Author: Frédéric Chapoton

Branch/Commit: u/chapoton/15496 @ 0661926

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/15496

[kcrisman]

comment:1

This is not a symbolic Maxima problem, as the integral is just returned, so it's something about how we evaluate n on these unevaluated integrals.

[fchapoton]

Description changed:

--- 
+++ 
@@ -6,3 +6,12 @@
 ```
 
 Wolfram alpha [gives the correct answer](http://www.wolframalpha.com/input/?i=integrate+x<sup>2/exp(-1/(x</sup>2%2B1))/(x^2%2B1)+from+-infinity+to+infinity).
+
+The problem comes from here:
+
+```
+sage: D=1.00000000000000*x^2*e^(1/(x^2 + 1.00000000000000))/(x^2 + 1.00000000000
+....: 000)
+sage: numerical_integral(D,-oo,+oo)
+(-3.3357920860949157, 4.262178610048295e-09)
+```

[fchapoton]

Description changed:

--- 
+++ 
@@ -15,3 +15,13 @@
 sage: numerical_integral(D,-oo,+oo)
 (-3.3357920860949157, 4.262178610048295e-09)
 ```
+and in one specific algorithm
+
+```
+sage: numerical_integral(D,-oo,+oo, algorithm="qag")
+(-3.3357920860949157, 4.262178610048295e-09)
+sage: numerical_integral(D,-oo,+oo, algorithm="qng")
+(nan, nan)
+sage: numerical_integral(D,-oo,+oo, algorithm="qags")
+(nan, nan)
+```

[fchapoton]

Description changed:

--- 
+++ 
@@ -25,3 +25,5 @@
 sage: numerical_integral(D,-oo,+oo, algorithm="qags")
 (nan, nan)
 ```
+and this case is handled by
+`gsl_integration_qagi`

[fchapoton]

Description changed:

--- 
+++ 
@@ -27,3 +27,12 @@
 ```
 and this case is handled by
 `gsl_integration_qagi`
+
+And the problem can be cut into two parts:
+
+```
+sage: numerical_integral(D,0,+oo)
+(-1.6678960430474579, 2.1310893050241475e-09)
+sage: numerical_integral(D,-oo,0)
+(-1.6678960430474579, 2.1310893050241475e-09)
+```