The closed-form solution for this probability density is in the eye of the beholder as it is expressed with an infinite number of terms.
$$\sum _{k=1}^{\infty} \frac{e^{-\frac{1}{18} \left(2-\sqrt{2 \pi k-\cos ^{-1}(c)}\right)^2}}{6 \sqrt{2 \pi } \sqrt{1-c^2} \sqrt{2 \pi k-\cos ^{-1}(c)}}+\sum _{k=1}^{\infty} \frac{e^{-\frac{1}{18} \left(\sqrt{2 \pi k-\cos ^{-1}(c)}+2\right)^2}}{6 \sqrt{2 \pi } \sqrt{1-c^2} \sqrt{2 \pi k-\cos ^{-1}(c)}}+\sum _{k=0}^{\infty} \frac{e^{-\frac{1}{18} \left(2-\sqrt{\cos ^{-1}(c)+2 \pi k}\right)^2}}{6 \sqrt{2 \pi } \sqrt{1-c^2} \sqrt{\cos ^{-1}(c)+2 \pi k}}+\sum _{k=0}^{\infty} \frac{e^{-\frac{1}{18} \left(\sqrt{\cos ^{-1}(c)+2 \pi k}+2\right)^2}}{6 \sqrt{2 \pi } \sqrt{1-c^2} \sqrt{\cos ^{-1}(c)+2 \pi k}}$$
but a good approximation can be had with 40 to 50 terms per summation.
Here is the Mathematica code:
kmax =.
pdf =
Sum[PDF[NormalDistribution[-2, 3], Sqrt[ArcCos[c] + 2 \[Pi] k]] 1/(2 Sqrt[1 - c^2] Sqrt[2 k \[Pi] + ArcCos[c]]), {k, 0, kmax}] +
Sum[PDF[NormalDistribution[-2, 3], Sqrt[-ArcCos[c] + 2 \[Pi] k]] 1/(2 Sqrt[1 - c^2] Sqrt[2 k \[Pi] - ArcCos[c]]), {k, 1, kmax}] +
Sum[PDF[NormalDistribution[-2, 3], -Sqrt[ArcCos[c] + 2 \[Pi] k]] 1/(2 Sqrt[1 - c^2] Sqrt[2 k \[Pi] + ArcCos[c]]), {k, 0, kmax}] +
Sum[PDF[NormalDistribution[-2, 3], -Sqrt[-ArcCos[c] + 2 \[Pi] k]] 1/(2 Sqrt[1 - c^2] Sqrt[2 k \[Pi] - ArcCos[c]]), {k, 1, kmax}]
I'll put in the details sometime soon.
