two different results with same numeric integral

[kcrisman]

Reported by Nicolas Privault via Rado Kirov:

sage: integrate(x^(0.5)*e^(-x-x^2),x,0,infinity).n()
0.886226925452758

sage: integrate(x^(1/2)*e^(-x-x^2),x,0,infinity).n()
0.32015709036013135

which give two different results, and the second one is correct.

[kcrisman]

Here is what is going on.

(%i16) integrate(x^(1/2)*%e^(-x-x^2),x,0,inf);
                           inf
                          /                 2
                          [              - x  - x
(%o16)                    I    sqrt(x) %e         dx
                          ]
                          /
                           0

(%i18) integrate(x^(.5)*%e^(-x-x^2),x,0,inf);
<snip lots of these>
rat: replaced 0.5 by 1/2 = 0.5
                                   sqrt(%pi)
(%o18)                             ---------
                                       2

[kcrisman]

The other thing was correct, I believe, so I'm not sure what the poster thought was wrong with it.

sage: numerical_integral(x^(0.5)*e^(-x),0,100000)
(0.88622692749267307, 4.3485198683228606e-07)

[kcrisman]

This is now reported at ​this Maxima ticket.

[kcrisman]

#13718 may be related.

[chapoton]

Still not working in 6.2.beta3. Moreover one also gets

sage: integrate(e^(-x-x^2+log(x)/2),x,0,infinity)
integrate(e^(-x^2 - x + 1/2*log(x)), x, 0, +Infinity)
sage: integrate(e^(-x-x^2+log(x)/RDF(2)),x,0,infinity)
  ____
\/ pi 
------
  2   

[rws]

Not working in 6.5rc0.