$\displaystyle \int_{-\infty}^{\infty} \exp\left\{\frac{1}{2}\sin^{2}\left(tx\right)\right\} \phi\left(x\right)\, dx$ where $\phi$ is the density function of a standard normal distribution.
And also $\displaystyle \int_{-\infty}^{\infty} \exp\left\{\frac{1}{2}\cos^{2}\left(tx\right)\right\} \phi\left(x\right)\, dx$.
I've encountered this as the expectation of $\displaystyle \exp\left\{\frac{1}{2}\cos^{2}\left(tX\right)\right\}$ where $X \sim \mathcal{N}(0,1)$. I'm sure this integral is finite but not sure if the result could be represented in a combination of elementary functions. How would I be able to know this and if there is closed form, how can I compute it?