<s>[INST] How do we show that $\int_{0}^{1}{\arctan(x\sqrt{x^2+2})\over \sqrt{x^2+2}}\cdot{dx\over x^2+1}=\left({\pi\over 6}\right)^2$ I was going through the the top votes questions and I saw this quite of interesting post posted by @Sangchul Lee. I was just messing around with it and found a slightest Variation of Ahmed's integral.
$$\int_{0}^{1}{\arctan(x\sqrt{x^2+2})\over \sqrt{x^2+2}}\cdot{dx\over x^2+1}=\left({\pi\over 6}\right)^2\tag1$$
May be substitution might work
$u=x(x^2+2)^{1/2}$ the $du=x(x^2+2)^{-1/2}+(x^2+2)^{1/2}dx$
$$\int_{\sqrt{2}}^{4\sqrt{3}\over3}{x^2\arctan{u}\over x^3+u^2}\cdot{x^2\over u^2-x^2}du$$
More difficult then before. I can't remove the x variable from the integral.
Any suggestion on how to prove $(1)$?
 [/INST] Substitute : $u=\arctan\left(x\sqrt{x^2+2}\right)$
You will get : $$\dfrac{\mathrm{d}u}{\mathrm{d}x}=\dfrac{\sqrt{x^2+2}+\frac{x^2}{\sqrt{x^2+2}}}{x^2\left(x^2+2\right)+1}$$
Thus :
$$\int_{0}^{1}{\arctan(x\sqrt{x^2+2})\over \sqrt{x^2+2}}\cdot{dx\over x^2+1}=\int_{0}^{\arctan(\sqrt{3})}{u\over 2}\cdot{du}=\left[\frac{u^2}{4}\right]^{\frac{\pi}{3}}_0=\left({\pi\over 6}\right)^2$$
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