How to show that the integral $$\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx$$ equals to $\frac{3 \pi ^2}{160}$? I've already verified this numerically but failed to prove it. The problem came from a Chinese blog (closed), however the blogger didn't show the method to arrive at the result. Does it has some relation with the Ahmed integral? Basic methods, residue cauculus as well as other methods are all welcomed. Thank you!
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