Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f(x)=\ln x+\int_0^x \sqrt{1+\sin t} \, dt$. Then which of the following statement$(s)$ is (are) true?
$(A)$ $f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$
$(B)$ $f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$,but not differentiable on $(0, \infty)$
$(C)$ there exists $\alpha>1$ such that $|f^{\prime}(x)|<|f(x)|$ for all $x \in(\alpha, \infty)$
$(D)$ there exists $\beta>0$ such that $|f(x)|+|f^{\prime}(x)| \leq \beta$ for all $x \in(0, \infty)$

• A
  $(B, C)$
• B
  $(B, D)$
• C
  $(A, D)$
• D
  $(A, B)$