This integral does not converge as $x \to \infty$

Calculus Level 5

For $x > 0$ , define $f(x)$ as follows: $f(x) = \int_{0}^{x} \left( \sqrt{t^2+1} - t \right) \, dt.$ Let $L = \displaystyle\lim_{x\to\infty} \left(f(x) - \dfrac{\ln x}{2}\right).$

If $L = \dfrac{a + \ln{b}}{c}$ for positive integers $a, b,$ and $c$ with $\gcd(a,c) = 1$ , then find the value of $a + b + c$ .