Problem: Approximate the expectation function of the solution of SDE $$dX_t=\exp\bigl(\cos(t)B_t\bigr)\Bigl(-\frac{2t}{(1+t^2)^2}-\frac{B_t\sin(t)}{1+t^2}+\frac{\cos^2(t)}{2(1+t^2)}\Bigr)\,dt+\frac{\cos(t)}{1+t^2}\exp\bigl(\cos(t)B_t\bigr)\,dB_t.$$ in the limit $t \to \infty$ Please provide your answer in LaTeX \boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \exp, but just explicitly write e^.
Solution: $\boxed{\frac{1}{1 + t^2} e^{\frac{t}{2}\cos^2(t)}}$