This integral was found on a dead forum so I cannot obtain any further information on it.
$$\int_{0}^\infty \frac{\tan^{-1}{x} \tan^{-1}{\frac{x}{2}} \tan^{-1}{\frac{x}{3}}}{x^2+1} \, \text{d}x$$
I could not find a closed form using ISC++
The equivalent form
$$\int_0^{\frac{\pi}{2}} \theta \tan^{-1}{\left(\frac{\tan{\theta}}{2}\right)} \tan^{-1}{\left(\frac{\tan{\theta}}{3}\right)} \, \text{d}x$$
does not seem any more promising.
Perhaps a carefully considered contour could solve this problem but I do not have enough familiarity with complex analysis to come up with an argument.