Harald Schilly <[email protected]> writes:
> this report came in from the "report a problem" link for 4.1.1.
>
> -------------------------
>
> Sage gives an incorrect value when calculating a definite integral
> analytically:
>
> sage: integrate(cos(x)^2 * (1 + sin(x)^2)^-3,x,0,pi/2)
> 21/64*pi*sqrt(2)
> sage: _.n()
> 1.45782096408321
>
> The correct answer is 7/64*pi*sqrt(2) = 0.48594. Sage gets this when
> doing the integral numerically, or when going not quite to pi/2:
>
> sage: numerical_integral(cos(x)^2 * (1 + sin(x)^2)^-3,0,pi/2)
> (0.48594032136107129, 5.3950213336880916e-15)
> sage: integrate(cos(x)^2 * (1 + sin(x)^2)^-3,x,0,pi/2-0.0001).n()
> 0.485940321361
>
> The integrand is perfectly well-behaved at pi/2. The problem may be
> related to the fact that the indefinite integral contains a term like
> arctan(sqrt(2)*tan(x)), which is ill-defined at x=pi/2.
For what it's worth, fricas seems to give a correct answer, although not
as simple as it could be (please excuse the ascii art, view with fixed
width font...):
(1) -> r := integrate(cos(x)^2 * (1 + sin(x)^2)^-3,x=0..%pi/2, "noPole")
+-+
\|2 4
7atan(----) + 7atan(----)
4 +-+
\|2
(1) -------------------------
+-+
16\|2
Type: Union(f1: OrderedCompletion(Expression(Integer)),...)
(2) -> numeric %
(2) 0.4859403213 6107130827
Type: Float
What I find interesting is that fricas does not even detect equality
with the correctly simplified answer:
(3) -> normalize(r - 7/64*%pi*sqrt(2))
+-+
\|2 4
14atan(----) + 14atan(----) - 7%pi
4 +-+
\|2
(3) ----------------------------------
+-+
32\|2
Type: Expression(Integer)
(4) -> numeric %
(4) 0.0
Type: Float
Martin
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