Paneer Definite Masala 3

Calculus Level 5

$\large \int_0^\infty \dfrac{ \cos xt}{1+t^2} \, dt$

If the integral above is equal to $\dfrac ab$ , where $a$ and $b$ are coprime positive integers, find $a+b+ab$ .

Take $x=\ln \frac{3\pi}5$