Help with Inverse Tangent Integral
Evaluate
$$\large \int _{ -1 }^{ 1 }{ \left( \cot^{-1}{ \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } \right) } \left( \cot^{-1}{ \frac { x }{ \sqrt { 1-{ \left( { x }^{ 2 } \right) }^{ \left| x \right| } } } } \right) dx\quad$$
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Unfortunately, this time, I am unable to give any of my own inputs since I haven't been able to think of anything at all. I'm truly sorry for this; I would certainly have given all my inputs had I had the fortune to think of something relevant. The only thing which struck me was that the denominator in the first bracket of the integrand was the derivative of the Inverse Sine Function.$$$$ I just cannot get any way of solving this and have been completely stumped with this.$$$$ Would somebody please be so kind as to help me solve this? If possible, please could you refrain from using any special functions other than the Beta, Gamma and Digamma Functions? I would be extremely grateful for your assistance. $$$$Many, many thanks in advance!
EDIT: Please could you answer my comments to Robert Sir's solution? Thanks very much!
$\endgroup$ 21 Answer
$\begingroup$Ah, there's a trick to it. Let your integral be $J = \int_{-1}^1 f(x)\; dx$. By symmetry, this is also $\int_{-1}^1 f(-x)\; dx$. Now note that $$ f(x) + f(-x) = \dfrac{\pi^2}{2} - \pi \arctan\left(\dfrac{1}{\sqrt{1-x^2}}\right)$$ and this can easily be integrated, resulting in $$ J = \dfrac{\pi^2}{2} \left(\sqrt{2}-1\right) $$
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