analytic result of an integral

Question

I am working on a physics project and I encounter an integral that I need to get analytic results about. Otherwise I will have to numerically compute the second integral, which significantly increases the amount of computer work.

Here is the integral: $\int_0^\pi e^{ia\cos\phi}(\sin\phi)^2 d\phi$.

I am not sure if it can be analytically computed. It does look like definition of the spherical Bessel function of first kind if it's $\sin\phi\cos\phi$ rather than $(\sin\phi)^2$. Any help would be appreciated.

Answer 1

Mathematica says: $$ \frac{\pi J_1(\left| a\right| )}{\left| a\right| }. $$