Definite integral $\int_{\frac{1}{a}}^a \frac{\arctan(x)}{x}$ I have to fund the value of the above integral $$\int_{\frac{1}{a}}^a \frac{\arctan(x)}{x}$$ for $a=2014$. 
I just saw that integral of $\frac{\arctan(x)}{x}$ does not have a closed form. So I used the substitution $x=tan(\theta)$ and then used integration by parts but did not get a result. 

Since $\;a,\,1/a>0\;$ , we can assume $\;x>0\;$ , so
$$\arctan x+\arctan\frac1x=\frac\pi2\implies\text{substituting, as commented}\;\;u:=\frac1x\implies dx=-\frac{du}{u^2}$$
we get that
$$I:=\int_{1/a}^a\frac{\arctan x}xdx=\int_a^{1/a}\frac{\arctan\frac1u}{\frac1u}\left(-\frac{du}{u^2}\right)=\int_{1/a}^a\frac{\frac\pi2-\arctan u}udu$$
and now try to finish from here.
