For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$ and set $J _ { \alpha } = \int _ { 0 } ^ { 1 } h _ { \alpha } ( t ) \, \mathrm { d } t$. Deduce that
$$J _ { \alpha } = 2 \ln ( 2 ) - \ln \left( 1 + \alpha ^ { 2 } \right) - 2 \alpha \arctan \left( \frac { 1 } { \alpha } \right).$$