sqrt(-r_0^2 + r_1^2 + 2*r_0*x - x^2): Maxima gives an antiderivative but *crashes* on definite integral

[EmmanuelCharpentier]

Inspired by this ask.sagemath.org question:

var("x, y, t, r_0, r_1, X", domain="real")
rr=var("rr")
assume(r_0>0,X<r_0,r_1>0,r_1<2*r_0,rr>0)
with assuming(y>0): foo=solve(x^2 + y^2 == r_0^2, y)[0].rhs()
Y_0(x)=foo
with assuming(y>0): foo=solve((r_0-x)^2 + y^2 == r_1^2, y)[0].rhs()
Y_1(x)=foo
X=solve(Y_1(x)^2==Y_0(x)^2,x)[0].rhs()
g(x)=integrate(Y_1(x),x).expand()

So far, so good :

sage: (g(X)-g(r_0-r_1)).expand()
1/4*pi*r_1^2 - 1/2*r_1^2*arcsin(1/2*r_1/r_0) - 1/8*sqrt(4*r_1^2 - r_1^4/r_0^2)*r_1^2/r_0

But integrate(Y_1(x),x,r_0-r_1,X).expand() crashes Maxima (and Sage) :

sage: integrate(Y_1(x),x,r_0-r_1,X)

;;;
;;; Detected access to protected memory, also kwown as 'bus or segmentation fault'.
;;; Jumping to the outermost toplevel prompt
;;;

repeated /ad nauseam/...

This can be reproduced in Maxima. This works as expected:

assume(r_0>0,X<r_0,r_1>0,r_1<2*r_0,rr>0);
assume(y>0);
define(Y_0(x),rhs(solve(x^2+y^2=r_0^2,y)[2]));
define(Y_1(x),rhs(solve((r_0-x)^2+y^2=r_1^2,y)[2]));
X:rhs(solve(Y_0(x)^2=Y_1(x)^2,x)[1]);
Y:Y_0(X);
define(g(x),expand(integrate(Y_1(x),x)));
g(X)-g(r_0-r_1);

But asking for integrate(Y_1(x),x,r_0-r_1,X); crashes Maxima with the same error messages...

This seems different from bugs already reported on definite integration.

FWIW, both giac and fricas give acceptable answers (different in their integration process : one uses arcsin, the other arctan, with similar geometric interpretation...).

Upstream: Not yet reported upstream; Will do shortly.

Component: symbolics

Keywords: definite integral

Issue created by migration from https://trac.sagemath.org/ticket/27816

[EmmanuelCharpentier]

comment:1

This is still true in 8.8.beta7, which has maxima 5.42.2...

[embray]

comment:2

As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).

[oldk1331]

With maxima-5.47.0, it works in sage-10.2 by default (domain:complex;):

sage: integrate(Y_1(x),x,r_0-r_1,X).expand()
-1/2*r_1^2*arctan(sqrt(4*r_0^2 - r_1^2)/r_1) - 1/8*I*sqrt(-4*r_0^2 + r_1^2)*r_1^3/r_0^2

But in maxima, the default domain : real; gives:

(%i10) domain : complex;
(%o10)                              complex
(%i11) integrate(Y_1(x),x,r_0-r_1,X);
                                         2
       (%i (2 log(2) - log(4)) + %pi) r_1
(%o11) -----------------------------------
                        4
                                2      2            2
       2    2         sqrt(4 r_0  - r_1 ) sqrt(4 r_1 )
 - (r_0  r_1  (4 atan(--------------------------------)
                                        2
                                   2 r_1
                                              3         2        2         2
 + %i (4 log(4) - 2 log(16)) + 2 %pi) + %i r_1  sqrt(r_1  - 4 r_0 ))/(8 r_0 )
(%i12) domain : real;
(%o12)                               real
(%i13) integrate(Y_1(x),x,r_0-r_1,X);

Maxima encountered a Lisp error:

 binding-stack overflow at size 10240. Stack can probably be resized.
Proceed with caution.

Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.