Problem: Approximate the expectation of the solution of SDE $$dX_t = \Bigl(-\frac{\sin(t^{-1/2})}{t^2}-\frac{1+\tfrac1t}{2t^{3/2}}\cos(t^{-1/2})-\frac{B_t}{2t^{3/2}}+\frac1{t+1}+B_t^2\bigl(6t^{-3}-\frac1{(t+1)^2}\bigr)-3\frac{B_t^4}{t^4}\Bigr)\,dt + \Bigl(\frac1{\sqrt t}+\frac{2B_t}{t+1}+\frac{4B_t^3}{t^3}\Bigr)\,dB_t.$$ 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.
Solution: $\boxed{\frac{1}{1 + \sqrt{t}}}$