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**Summary::**Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine.

**[Mentor Note -- thread moved from the technical math forums, so no Homework template is shown]**

Hi,

I'm trying to find the answer to the following integral over the rectangle ##(x^-,x^+)\times(y^-,y^+)##:

[tex]

\int_{y^-}^{y^+}\int_{x^-}^{x^+}v^2\arctan\frac{Lu}{v\sqrt{u^2+v^2+L^2}}\,\mathrm{d}v\,\mathrm{d}u.

[/tex]

For the indefinite double integral, WA gives a complicated answer with logarithms of complex arguments. I know that ##\tan^{-1}z=\frac{i}{2}\log\frac{1-iz}{1+iz}##, but it is not clear how to get this from the WA expression:

[tex]

\log\left\{-\frac{12(L\sqrt{L^2+u^2+v^2}+L^2+u^2-iuv)}{L^2u^3(v+iu)}\right\}

[/tex]

Obviously, this should have a real answer since the integrand is real-valued over the domain of integration.

I don't have access to Mathematica at the moment and the internet integral evaluators I've tried all seem to fail here. I'm not interested in the step-by-step approach (##u##-/trigonometric substitution, integration by parts, partial fractions, etc.). I just want to know the answer to the double integral above.

Can somebody with access to a CAS or Mathematica please compute this double integral over the given rectangular bounds and share the answer? It would be much appreciated.

Thanks,

QM

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