Problem: Approximate the expectation function of the solution of SDE $$ dX_t=\exp\Bigl(\tfrac{t}{\sqrt3}\sin\bigl(\tfrac{t}{3}\bigr)B_t\Bigr)\Bigl(-\tfrac{2B_t^2}{t^3(1+t^2)}-\tfrac{2t(1+\tfrac{B_t^2}{t^2})}{(1+t^2)^2}+\tfrac{(1+\tfrac{B_t^2}{t^2})B_t}{1+t^2}(\tfrac{\sin\tfrac{t}{3}}{\sqrt3}+\tfrac{t\cos\tfrac{t}{3}}{3\sqrt3})+\tfrac12\tfrac{1+\tfrac{B_t^2}{t^2}}{1+t^2}\tfrac{t^2}{3}\sin^2\bigl(\tfrac{t}{3}\bigr)\Bigr)\,dt+\tfrac{1+\tfrac{B_t^2}{t^2}}{1+t^2}\tfrac{t}{\sqrt3}\sin\bigl(\tfrac{t}{3}\bigr)\exp\Bigl(\tfrac{t}{\sqrt3}\sin\bigl(\tfrac{t}{3}\bigr)B_t\Bigr)\,dB_t.$$ 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^. Do not use X_0. Do not use enlarge any parenthesis. Keep paranthesis () without modifying anything
Solution: $\boxed{\frac{1}{1 + t^2} e^{\frac{t^3}{6} \sin^2(t/3)}}$