#4522 in definite integrals each log(expr) is erroneously evaluated as log(abs(expr))

[Max]

Please compare the results of

expr: -1/17*sqrt(-52/17*I + 47/17)/(2*x - sqrt(4*I + 1) - 1);
q : integrate(expr, x);
subst(x=1,q) - subst(x=0,q);

and

integrate(expr, x, 0, 1);

They are

(sqrt(47/17-(52*I)/17)*log((-sqrt(4*I+1))-1))/34 -(sqrt(47/17-(52*I)/17)*log(1-sqrt(4*I+1)))/34

-(sqrt(47/17-(52*I)/17)*(log(abs(sqrt(4*I+1)-1))/2 -log(sqrt(4*I+1)+1)/2))/17

Apparently, the first is correct, while the second is not. Please see comparison with how other CAS handle this integral at https://ask.sagemath.org/question/78955/

[Barton Willis]

Here is a putative fix:

(defun antideriv (a ivar)
"Either return an explicit antiderivative (not an integral 
nounform) of 'a' with respect to 'ivar' or return nil"
  (let ((limitp nil)
        (ans nil)
        ($logabs nil)
        (generate-atan2 nil))
    (setq ans (sinint a ivar))
    (if (among '%integrate ans) nil (simplify ans))))

The only real change is to set the option variable logabs to nil, but I also inserted a docstring and reformatted the code.

This change causes five testsuite failures--I'm working through these changes now.

[Barton Willis]

With this proposed fix, one of the five regressions is rtest15, test 210. This test is

ratsimp(ev(logcontract((integrate(1/(x^8-1),x,0,1/2))),algebraic) + 
(sqrt(2)*log((2^(3/2)+5)/(2^(3/2)-5)/-1)
      +2*sqrt(2)*atan((sqrt(2)+2)/2)
      +2*sqrt(2)*atan((sqrt(2)-2)/2)+log(9)
      +4*atan(1/2))
 /16);
0;

The correct numerical value of integrate(1/(x^8-1),x,0,1/2) is about -0.500217463872434252. With my fix, I get

(%i9) ratsimp(ev(logcontract((integrate(1/(x^8-1),x,0,1/2))),algebraic));

(%o9) -((sqrt(2)*log((2^(3/2)+5)/(2^(3/2)-5))
 +2^(3/2)*atan((sqrt(2)+2)/2)+2^(3/2)*atan((sqrt(2)-2)/2)+log(9)+4*atan(1/2)
 -sqrt(2)*%i*%pi)
 /16)
(%i10) expand(float(%));

(%o10) -0.5002174638724343

So I might claim success. But sans the call to logcontract, I get

(%i11) integrate(1/(x^8-1),x,0,1/2);

(%o11) (sqrt(2)*(-log(2^(3/2)+5)+log(2^(3/2)-5)-2*atan((sqrt(2)+2)/2)
                                -2*atan((sqrt(2)-2)/2)+%i*%pi)
 -2*log(3)-4*atan(1/2)+2*%i*%pi)
 /16
 -(%i*%pi)/8
(%i12) expand(float(%));

(%o12) 0.5553603672697958*%i-0.5002174638724343

And this is wrong. With Maxima 5.47, with or without the "extra" call to logcontract, the answer is correct.

Generally, I think it's best when tests have a no or minimal amount of calls to simplification functions on result.

[Barton Willis]

Another testsuite failure is rtestint test 86. This test is

ratsimp(integrate(1/(x^2-3),x,0,1) - sqrt(3)*log(2-sqrt(3))/6);

With my proposed fix, the result is

(%i34) integrate(1/(x^2-3),x,0,1);

(%o34) log(sqrt(3)-2)/(2*sqrt(3))-(%i*%pi)/(2*sqrt(3))
(%i35) expand(float(%));

(%o35) -0.3801729981504731

So the result is correct, but not manifestly real.

Another failure is rtest_sqrt test 299. This test is

integrate(sqrt(t)*(t+1)^(1/2),t,0,1);
 (-log(sqrt(2)+1)+log(1-sqrt(2))+3*2^(3/2))/8-(%i*%pi)/8$

With the proposed fix, the result is

(%i5) integrate(sqrt(t)*(t+1)^(1/2),t,0,1);

(%o5) -(log(sqrt(2)+1)/8)+log(sqrt(2)-1)/8+3/2^(3/2)
(%i6) float(%);

(%o6) 0.8403167750249352

This is correct, and better than the result from Maxima 5.47.

[Barton Willis]

Fixed by Commit [6f1fe7]. Thanks for this bug report. Closing this ticket.