Problem: Approximate the expectation function of the solution of SDE $$dX_t =\exp\!\Bigl(\tfrac{\sqrt2\,\sin(t)}{t^{1/4}}\,B_t\Bigr)\Bigl(-\frac{2t}{(1+t^2)^2}+\frac{B_t}{1+t^2}\Bigl(\frac{\sqrt2\,\cos(t)}{t^{1/4}}-\frac{\sqrt2\,\sin(t)}{4\,t^{5/4}}\Bigr)+\frac{\sin^2(t)}{t^{1/2}(1+t^2)}\Bigr)\,dt+\frac{\sqrt2\,\sin(t)}{t^{1/4}(1+t^2)}\exp\!\Bigl(\tfrac{\sqrt2\,\sin(t)}{t^{1/4}}\,B_t\Bigr)\,dB_t.$$ 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: $\boxed{\frac{1}{1 + t^2} e^{\sqrt{t} \sin^2(t)}}$