Why does Mathematica and Wolfram|Alpha give different results based upon the same code?
I know Wolfram|Alpha's
7.85 is correct.
Artes' guess seems basically right. Here is a way to reach the correct result. First, the antiderivative returned by Mathematica:
If we add to
It turns out that
yields the correct (approximate) result.
Such are the pitfalls of elliptic integrals and their symbolic antiderivatives, I suppose.
The plots below show the discontinuities in the
To get the real-valued antiderivative, piece together different branches:
This does not really answer the question as to why Mathematica's
I observed this behavior while at a training session at Wolfram Headquarters, and there seems to be an issue with using
This is not an answer, but comparing the answer with Maple, and also showing step by step integration using Rubi, which might help point to where Mathematica went wrong.
Rubi 4.2 step by step
In one of those steps, the Integrate went wrong. Need more time to analyze.
Okay, I'm slowly going through the questions involving elliptic integral evaluations. Yet again, none of the software mentioned in this thread have managed to produce a "clean" expression. For the benefit of future readers, here's a tidier closed form for your perusal: