$$f(\phi)=\frac{1}{\pi}\cos^2 \phi\Rightarrow g(x)=\frac{2 [I_1(x)+x I_2(x)]}{x},$$ $$f(\phi)=\frac{4}{\pi}\cos^2 \phi\sin^2\phi\Rightarrow g(x)=\frac{8 (x I_1(x)-3 I_2(x))}{x^2}.$$

Q: Do there exist other known functions, for other probability density functions $f(\phi)$?

Here are three:

$$f(\phi)=\frac{1}{\pi}\cos^2 \phi\Rightarrow g(x)=\frac{2 [I_1(x)+x I_2(x)]}{x},$$ $$f(\phi)=\frac{4}{\pi}\cos^2 \phi\sin^2\phi\Rightarrow g(x)=\frac{8 (x I_1(x)-3 I_2(x))}{x^2},$$ $$f(\phi)=\frac{1}{2\pi I_0(1)}e^{\sin\phi}\Rightarrow g(x)=\frac{I_0\left(\sqrt{x^2+1}\right)}{I_0(1)}.$$