I spent more than an hour trying to fix the formatting of the question below such that the Mathematica and Wolfram Language forum would accept it. I had no success, even though I am aware of threads such as this one.
I even tried to post the entire question inside <pre>...</pre> as below, so that someone with high reputation could modify it afterwards, but the error persisted.
I'd greatly appreciate it if someone helped me post the question.
Today I encountered a phenomenon in the free online version of WolframAlpha which looks to me like a bug. It would be great if someone could check whether the phenomenon also occurs when using Mathematica directly. I am not sure whether it might actually be normal behavior and I am just using the software incorrectly.
I am considering the integral
$$
I(a):=\int_{\mathbb R}\int_{\mathbb R}\frac{|1-xy|^{a-1}}{((1+x^2)(1+y^2))^{(a+1)/2}}\,\mathrm{d}x\, \mathrm{d} y
$$
for $a\geq 1$.
The free WolframAlpha is able to compute $I(a)$ for some $a$, for example, for $a=16$ one gets [this result][1], which claims that
$$
I(16)=1.9141.
$$
If one replaces `abs(1-xy)^(a-1)` in the input by an artificial `((1-xy)^2)((a-1)/2)`, then one obtains a [different result][2], namely
$$
I(16)=1.99968.
$$
Now, one might think that the difference is simply explained by the fact that WolframAlpha cancels the powers and in reality computes the integral $I(a)$ with `abs(1-xy)^(a-1)` replaced by `(1-xy)^(a-1)` but this is not the case: For the latter integral one gets a third, [different value][3].
What strikes me is that **the correct result for $I(16)$ seems to be the one where
`((1-xy)^2)((a-1)/2)` is used**, i.e., $I(16)=1.99968$.
Indeed, using basic Lie theoretic tools one can prove that
$$
I(a)=\frac{1}{4}\int_{0}^{2\pi}\int_{0}^{2\pi}|\cos(x-y)|^{a-1}\,\mathrm{d}x\, \mathrm{d} y,
$$
and WolframAlpha computes the value $I(16)=1.99968$ for the latter integral regardless of whether one writes `abs(cos(x-y))^(a-1)` or `(cos(x-y)^2)^((a-1)/2)` in the input. See [here][4] and [here][5].
**Is this a bug?**
The value $a=16$ is not important here, I just picked a value where WolframAlpha actually computes a result both with `abs(1-xy)^(a-1)`
and `((1-xy)^2)((a-1)/2)`.
Whenever that is the case for an even value of $a$, the phenomenon occurs.
When $a$ is odd, so that the absolute value does not matter, WolframAlpha always computes the same value of $I(a)$ in all variants of the input.
[1]: https://www.wolframalpha.com/input/?i=integrate%20%20%28%281%2Bx%5E2%29%281%2By%5E2%29%29%5E%28-%2816%2B1%29%2F2%29abs%281-xy%29%5E%2816-1%29%20dx%20dy%20from%20x%3D-infty%20to%20infty%2C%20y%3D-infty%20to%20infty
[2]: https://www.wolframalpha.com/input/?i=integrate%20%20%28%281%2Bx%5E2%29%281%2By%5E2%29%29%5E%28-%2816%2B1%29%2F2%29%28%281-xy%29%5E2%29%5E%28%2816-1%29%2F2%29%20dx%20dy%20from%20x%3D-infty%20to%20infty%2C%20y%3D-infty%20to%20infty
[3]: https://www.wolframalpha.com/input/?i=integrate%20%20%28%281%2Bx%5E2%29%281%2By%5E2%29%29%5E%28-%2816%2B1%29%2F2%29%281-xy%29%5E%2816-1%29%20dx%20dy%20from%20x%3D-infty%20to%20infty%2C%20y%3D-infty%20to%20infty
[4]: https://www.wolframalpha.com/input/?i=integrate%201%2F4%20%7Ccos%28x-y%29%7C%5E%2816-1%29%20dx%20dy%20from%20x%3D0%20to%202pi%2C%20y%3D0%20to%202pi
[5]: https://www.wolframalpha.com/input/?i=integrate%201%2F4%20%28cos%28x-y%29%5E2%29%5E%28%2816-1%29%2F2%29%20dx%20dy%20from%20x%3D0%20to%202pi%2C%20y%3D0%20to%202pi
[6]: https://www.wolframalpha.com/input/?i=integrate%201%2F%28pi%29%5E2%20%28%281%2Bx%5E2%29%281%2By%5E2%29%29%5E%28%28-28%29%2F2-1%2F2%29abs%281-xy%29%5E%28-%28-28%29-1%29%20dx%20dy%20from%20x%3D-infty%20to%20infty%2C%20y%3D-infty%20to%20infty
[7]: https://www.wolframalpha.com/input/?i=integrate%20%20%28%281%2Bx%5E2%29%281%2By%5E2%29%29%5E%28-%2828%2B1%29%2F2%29%28%281-xy%29%5E2%29%5E%28%2828-1%29%2F2%29%20dx%20dy%20from%20x%3D-infty%20to%20infty%2C%20y%3D-infty%20to%20infty
[8]: https://www.wolframalpha.com/input/?i=integrate%201%2F4%20abs%28cos%28x-y%29%29%5E%2828-1%29%20dx%20dy%20from%20x%3D0%20to%202pi%2C%20y%3D0%20to%202pi
[9]: https://www.wolframalpha.com/input/?i=integrate%201%2F4%20%28cos%28x-y%29%5E2%29%5E%28%2828-1%29%2F2%29%20dx%20dy%20from%20x%3D0%20to%202pi%2C%20y%3D0%20to%202pi
