#2732 wrong answer for similar to gaussian integral

[dan hayes]

wxMaxima version: 13.4.0
Maxima version: 5.31.1
Maxima build date: 2013-09-24 09:49:12
Host type: i686-pc-mingw32
Lisp implementation type: GNU Common Lisp (GCL)
Lisp implementation version: GCL 2.6.8

(assume(a>0,b>0), t1:integrate(exp(-a^2*x^2-b^2/x^2),x,0,inf), t2:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,0)
, t3:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,inf),ldisp([t1,t2,t3]));

By symmetry of x and -x the first two results, t1 and t2 , should be same which they correctly are. But then obviously the 3rd result, t3, should just be twice t1 (or t2) but instead it adds a term with the the positive exponent in the exponential to t1 which is incorrect. Also maxima should be able to do closed form integrals of the form integrate(x^(2*i)*exp(-a*x^2-b/x^2),x,0,inf) for any integer i by differentiating i times the closed form answer of

integrate(exp(-a*x^2-b/x^2),x,0,inf)

with respect to a for i>0 or b for i<0 but it does not do so.

[Rupert Swarbrick]

For reference, here's the missing transcript (with current git master):

(%i1) assume(a>0, b>0);
(%o1)                           [a > 0, b > 0]
(%i2) t1:integrate(exp(-a^2*x^2-b^2/x^2),x,0,inf);
                                          - 2 a b
                              sqrt(%pi) %e
(%o2)                         -------------------
                                      2 a
(%i3) t2:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,0);
                                          - 2 a b
                              sqrt(%pi) %e
(%o3)                         -------------------
                                      2 a
(%i4) t3:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,inf);
                                2 a b               - 2 a b
                    sqrt(%pi) %e        sqrt(%pi) %e
(%o4)               ----------------- + -------------------
                           2 a                  2 a
(%i5) print([t1,t2,t3]);
             - 2 a b              - 2 a b
 sqrt(%pi) %e         sqrt(%pi) %e
[-------------------, -------------------, 
         2 a                  2 a
                                                  2 a b               - 2 a b
                                      sqrt(%pi) %e        sqrt(%pi) %e
                                      ----------------- + -------------------] 
                                             2 a                  2 a
                   - 2 a b              - 2 a b
       sqrt(%pi) %e         sqrt(%pi) %e
(%o5) [-------------------, -------------------, 
               2 a                  2 a
                                                   2 a b               - 2 a b
                                       sqrt(%pi) %e        sqrt(%pi) %e
                                       ----------------- + -------------------]
                                              2 a                  2 a

[dan hayes]

Note the commercial version Macsyma gives the correct answer of
sqrt(%pi)exp(-2ab)/a for integrate(exp(-a^2x^2-b^2/x^2),x,-inf,inf);

I see that the source code for that now defunct project is made publicly available so perhaps those knowledgeable could use or find out how Macsyma did it.

[Dan Gildea]

Fixed by [96309ddcd0dd398aade5f01eb1006512c23c2cf4]

src/defint.lisp:
o discontinuities-denom:
  recursively traverse exp in order to find discontinuities such as
  erfc(1/x-x) at x=0

tests/rtestint.mac:
o add integrate(exp(-x^2-1/x^2),x,-inf,inf);