I recently saw an interesting integral at this link.
It seems to be another case where the integrator (Maple 15 (X86 64 LINUX)) somehow looses his/her bearings.
When I try the following command: int(1/(sin(theta)+cos(theta))/sqrt(sin(theta)*cos(theta)), theta=0..Pi/2); I get an answer that is fairly complicated given that the value of the integral is simply Pi.
But what is worse, even that answer, when evaluated numerically, turns out to be wrong. I get 1.641651398.
On the other hand, this command does produce the right answer:evalf(Int(1/(sin(theta)+cos(theta))/sqrt(sin(theta)*cos(theta)), theta=0..Pi/2));
Can anyone confirm this? Axel Vogt explained to me never to use the combination evalf/int on integrals. But I would think that is precisely what new users are likely to do! Furthermore, I can think of many cases where there is a reasonable expectation that a closed-form non-numeric answer exists for some integrals, and a user would expect Maple to produce that answer. This particular integral can be reduced to a Beta function integral in several ways and the actual computagtion is not that difficult.
Best regards, Marko Riedel