Wrong integral of exp(x^3) from 1 to 2.

[sagetrac-tmonteil]

The following was reported on this ask question:

sage: f(x) = exp(x^3)
sage: f.integrate(x, 1, 2)
-1/3*gamma(1/3, -1) + 1/3*gamma(1/3, -8)
sage: f.integrate(x, 1, 2).n()
-138.557717219510 - 238.442320120796*I

numerical_integral and sympy seem to agree on the result:

sage: numerical_integral(f,1,2)
(275.5109837633117, 3.0587863771115628e-12)

sage: import sympy
sage: f = f._sympy_()
sage: g = f.integrate() ; g
exp(-I*pi/3)*gamma(1/3)*lowergamma(1/3, x**3*exp_polar(I*pi))/(9*gamma(4/3))
sage: h = g.subs(x,2) - g.subs(x,1)
sage: h.n()
275.510983763312 - 1.68744844960818e-21*I

CC: @kcrisman @mforets

Component: symbolics

Author: Marcelo Forets

Branch/Commit: 82161f2

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/17968

[kcrisman]

comment:1

Usually such problems are due to our conversions with such special functions, but apparently this time it might be Maxima's integration that is wrong:

(%i5) display2d: false;

(%o5) false
(%i6) integrate(exp(x^3),x,1,2);

(%o6) (gamma_incomplete(1/3,-8)-gamma_incomplete(1/3,-1))/3
(%i7) %, numer; 

(%o7) 0.3333333333333333*(-715.7985328820091*%i-413.2664756449672)

which seems to be the same issue as we have. Possibly related is #17328 or #11164. W|A gives the same antiderivative but for the definite integral uses the exponential integral E sub 2/3 of -1 and -8, which implies to me some kind of branch cut thing as usual.

[mforets]

comment:2

This seems to be fixed by now:

sage: version()
SageMath version 7.5, Release Date: 2017-01-11
sage: integrate(exp(x^3), (x, 1, 2))
-1/3*(-1)^(2/3)*gamma(1/3, -1) + 1/3*(-1)^(2/3)*gamma(1/3, -8)
sage: integrate(exp(x^3), (x, 1, 2)).n()
275.510983763312 - 7.21644966006352e-14*I

should this ticket be closed?

[sagetrac-tmonteil]

comment:4

Replying to @mforets:

This seems to be fixed by now:

sage: version()
SageMath version 7.5, Release Date: 2017-01-11
sage: integrate(exp(x^3), (x, 1, 2))
-1/3*(-1)^(2/3)*gamma(1/3, -1) + 1/3*(-1)^(2/3)*gamma(1/3, -8)
sage: integrate(exp(x^3), (x, 1, 2)).n()
275.510983763312 - 7.21644966006352e-14*I

should this ticket be closed?

Certainly not. We should add a doctest (with a pointer to this ticket) to ensure that the issue will not reappear.

[mforets]

Branch: u/mforets/wrong_integral_of_exp_x_3__from_1_to_2_

[mforets]

comment:6

Replying to @sagetrac-tmonteil:

Replying to @mforets:

This seems to be fixed by now:

sage: version()
SageMath version 7.5, Release Date: 2017-01-11
sage: integrate(exp(x^3), (x, 1, 2))
-1/3*(-1)^(2/3)*gamma(1/3, -1) + 1/3*(-1)^(2/3)*gamma(1/3, -8)
sage: integrate(exp(x^3), (x, 1, 2)).n()
275.510983763312 - 7.21644966006352e-14*I

should this ticket be closed?

Certainly not. We should add a doctest (with a pointer to this ticket) to ensure that the issue will not reappear.

got it!

i copied the doctest which is just a couple of lines above ("Check that 11737 is fixed::..."). the difference is that the numerical noise in the imag part of the integral of exp(x^3) did not "magically vanish" by setting prec=54, as it is the case for the integral in the 11737 test. hence, i took the real_part().

---

New commits:

69a3e10

add doctest for ticket 17968

[mforets]

Commit: 69a3e10

[kcrisman]

comment:8

Apparently doesn't apply for some reason.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

bbd9432	merge w/integral.py
00fc6d3	trac:17968 doctest

[sagetrac-git]

Changed commit from 69a3e10 to 00fc6d3

[mforets]

comment:10

OK, i've merged with the most recent beta and solved a conflict in integral.py, maybe that was the reason, we'll see..

[tscrim]

comment:11

This is not really a bug:

        sage: N(integrate(exp(x^3), (x, 1, 2)))    # known bug (non-zero imag part)
        275.510983763312

Instead, it is more of an issue with the numerics from the indefinite integral:

sage: integral(exp(x^3), x)
-1/3*x*gamma(1/3, -x^3)/(-x^3)^(1/3)

Also author name.

[sagetrac-git]

Changed commit from 00fc6d3 to 82161f2

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

82161f2

check real AND imag part

[mforets]

comment:13

@sagetrac-travis: yes, you're right, i've changed it.

[mforets]

Author: Marcelo Forets

[tscrim]

comment:14

LGTM.

[tscrim]

Reviewer: Travis Scrimshaw

[vbraun]

Changed branch from u/mforets/wrong_integral_of_exp_x_3__from_1_to_2_ to 82161f2