Maxima doesn't do an integral we thought was fixed

[kcrisman]

#11238 is back.

;;; Loading #P"/Users/.../sage/local/lib/ecl/sb-bsd-sockets.fas"
;;; Loading #P"/Users/.../sage/local/lib/ecl/sockets.fas"
;;; Loading #P"/Users/.../sage/local/lib/ecl/defsystem.fas"
;;; Loading #P"/Users/.../sage/local/lib/ecl/cmp.fas"
Maxima 5.34.1 http://maxima.sourceforge.net
using Lisp ECL 13.5.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) display2d:false
;

(%o1) false
(%i2) integrate(exp(-x)*sinh(sqrt(x)),x,0,inf);

(%o2) -%e^(1/4)*(2*gamma_incomplete(1,1)-gamma_incomplete(1/2,1)-sqrt(%pi)-2)/4
 +%e^(1/4)*gamma_incomplete(1,1)/2-%e^(1/4)*gamma_incomplete(1/2,1)/4
 +%e^(1/4)*sqrt(%pi)/4-%e^(1/4)/2
(%i3) domain:complex;

(%o3) complex
(%i4) integrate(exp(-x)*sinh(sqrt(x)),x,0,inf);
^C
Maxima encountered a Lisp error:

 Console interrupt.

Automatically continuing.

[kcrisman]

To be precise,

sage: integrate(exp(-x)*sin(sqrt(x)), x, 0, oo)
integrate(e^(-x)*sin(sqrt(x)), x, 0, +Infinity)

This did work in Sage 5.2.

sage: integrate(exp(-x)*sinh(sqrt(x)), x, 0, oo)
1/2*sqrt(pi)*e^(1/4)

[kcrisman]

​https://sourceforge.net/p/maxima/bugs/2854/

[kcrisman]

Current doctest for this bites.

    Another symbolic integral, from :trac:`11238`, that used to return
    zero incorrectly; with Maxima 5.26.0 one gets
    ``1/2*sqrt(pi)*e^(1/4)``, whereas with 5.29.1, and even more so
    with 5.33.0, the expression is less pleasant, but still has the
    same value.  Unfortunately, the computation takes a very long time
    with the default settings, so we temporarily use the Maxima
    setting ``domain: real``::

        sage: sage.calculus.calculus.maxima('domain: real')
        real
        sage: f = exp(-x) * sinh(sqrt(x))
        sage: t = integrate(f, x, 0, Infinity); t            # long time
        1/4*sqrt(pi)*(erf(1) - 1)*e^(1/4) - 1/4*(sqrt(pi)*(erf(1) - 1) - sqrt(pi) + 2*e^(-1) - 2)*e^(1/4) + 1/4*sqrt(pi)*e^(1/4) - 1/2*e^(1/4) + 1/2*e^(-3/4)
        sage: t.simplify_exp()  # long time
        1/2*sqrt(pi)*e^(1/4)
        sage: sage.calculus.calculus.maxima('domain: complex')
        complex

And the 'long time' is not kidding.

[kcrisman]

Apparently fixed again upstream, though still looooong time...

[charpent]

Sorry, wrong ticket (and I don't know how to delete a comment in Trac...)

[gh-DaveWitteMorris]

Update: Sage v9.2 can evaluate the integral:

sage: integrate(exp(-x)*sin(sqrt(x)), x, 0, oo)                                          
1/2*sqrt(pi)*e^(-1/4)

This is not the same answer as previously reported, because the previous calculation has e^(1/4), not e^(-1/4). However, Maple and Wolfram Alpha agree that there should be a minus sign, so probably this is good.

And the calculation is not ridiculously slow (from %timeit on CoCalc):

3.02 s ± 248 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

I was going to suggest closing this ticket as WorksForMe, but then I noticed that the integration fails if we simply multiply the integrand by a constant:

sage: integrate(-exp(-x)*sin(sqrt(x)), x, 0, oo)
-integrate(e^(-x)*sin(sqrt(x)), x, 0, +Infinity)

[charpent]

FMIW :

sage: var("a")
a
sage: f(x)=exp(-x)*sin(sqrt(x))
sage: integrate(f(x),[x,0,oo])
1/2*sqrt(pi)*e^(-1/4)
sage: integrate(f(x),[x,0,oo]).n()
0.690194223521571
sage: numerical_integral(f(x),0,oo)
(0.6901942235198321, 1.3133617826621702e-07)

The minus sign seems correct...

sage: integrate(a*f(x),[x,0,oo])
a*integrate(e^(-x)*sin(sqrt(x)), x, 0, +Infinity)

Indeed... But :

sage: integrate(a*f(x),[x,0,oo]).unhold()
1/2*sqrt(pi)*a*e^(-1/4)

This might be more general...

Another quirk :

sage: integrate(a*f(x),[x,0,oo], algorithm="mathematica_free")
1/2*sqrt(pi)*a*e^(-1/4)
sage: integrate(f(x),[x,0,oo], algorithm="mathematica_free")
0.690194000000000

Replying to gh-DaveWitteMorris:

Update: Sage v9.2 can evaluate the integral:

sage: integrate(exp(-x)*sin(sqrt(x)), x, 0, oo)                                          
1/2*sqrt(pi)*e^(-1/4)

This is not the same answer as previously reported, because the previous calculation has e^(1/4), not e^(-1/4). However, Maple and Wolfram Alpha agree that there should be a minus sign, so probably this is good.

And the calculation is not ridiculously slow (from %timeit on CoCalc):

3.02 s ± 248 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

I was going to suggest closing this ticket as WorksForMe, but then I noticed that the integration fails if we simply multiply the integrand by a constant:

sage: integrate(-exp(-x)*sin(sqrt(x)), x, 0, oo)
-integrate(e^(-x)*sin(sqrt(x)), x, 0, +Infinity)