Do I need to know special functions?

You don't need to know special functions to solve this problem.

Take $\displaystyle J = \int _{ 0 }^{ 1 }{ \frac { \tan ^{ -1 }{ x } }{ x } dx }$

Put $x=tan(\frac{\theta}{2})$ , to get :

$\displaystyle J = \frac { 1 }{ 2 } \int _{ 0 }^{ \pi /2 }{ \frac { \theta }{ \sin { \theta } } d\theta } = \frac{I}{2}$

$\large \boxed{\Rightarrow \frac{I}{J} = 2}$

By the way the value of $J = G$ where G is the catalan's constant.

Another(I think!) one solution.

First change the variable $\displaystyle \tan^{-1}x=y$

then $\displaystyle J$ becomes $\displaystyle \int_{0}^{\pi/4} \frac{y}{\sin y \cos y} dy=\int_{2y=0}^{\pi/2} \frac{1}{2}\frac{2y}{\sin 2y} d 2y=\frac{1}{2} I$

So $\displaystyle \frac{I}{J}=2$