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#612 defint(1/(sin(x)^2+1),x,0,3*%pi) wrong
INTEGRATE(1/(SIN(x)^2+1),x,0,3*%PI) => 0
Since the integrand is everywhere >= 1/2, the integral
cannot be zero -- in fact
integrate(1/(sin(x)^2+1),x,q,q+k*%pi)
k*pi/sqrt(2) (real q, integral k)
Presumably defint is using the indefinite integral
atan(2*tan(x)/sqrt(2))/sqrt(2) inappropriately.
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Observed in 5.9.3cvs. I find
defint(1/(sin(x)^2+1),x,0,3*%pi); =>
sqrt(2)*atan(sqrt(2)*tan(3*%pi))/2
-sqrt(2)*atan(sqrt(2)*tan(0))/2
but then ratsimp(%) => 0 .
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I think the problem is caused by INFR. I think it's
supposed to compute how many full periods and fractional
periods are in the integral, but it does it incorrectly.
The following replacement:
(defun infr (a)
(let* ((q (igprt (div a (mul 2 '$%pi))))
(r (add a (mul -1 (mul q 2 '$%pi)))))
(cons q r)))
works much better for this example, and we get 3*pi/sqrt(2),
as expected.
In the results shown by Robert, ratsimp produces zero
because atan is using the principal result, but defint is
trying to tell you to be careful by returning
atan(sqrt(2)*tan(3*%pi))...
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Closing this report.
The function INFR has been replaced with a new version that
works for this case.
I've also updated INTSC1 so that it can handle the case
where the limits are q and q + k * 2 * %pi, for k a
numerical integer.
There are still other issues, which we be in a new bug report.