This relates a bit to the power semi-circle distribution from Does the distribution $f(x) \propto (1-x^2)^{n/2}$ have a name?

That distribution is for the marginal distribution of a coordinate uniformly distributed on an sphere.

The distribution that you have is more general and is the marginal distribution of a coordinate distributed in a sphere as a von Mises-Fisher distribution. More specifically the coordinate aligned along the direction of the parameter $\boldsymbol{\mu}$.

Short outline for a derivation: If that parameter is aligned along a single axis $x_1$, ie $\boldsymbol{\mu} = \{\mu,0,0,\dots,0\}$, then you the exponential factor from the von Mises Fisher density $e^{\boldsymbol{\mu} \cdot \textbf{x}} = e^{\mu x_1}$ at cordinate $x_1$. And you get a factor $(1-x_1)^{n/2}$ for the surface area of the sphere at the coordinate $x_1$.

This relates a bit to the power semi-circle distribution from Does the distribution $f(x) \propto (1-x^2)^{n/2}$ have a name?

That distribution is for the marginal distribution of a coordinate from a point uniformly distributed on an sphere.

The distribution that you have is more general and is the marginal distribution of a coordinate from a point distributed on a sphere as a von Mises-Fisher distribution. More specifically the coordinate aligned along the direction of the parameter $\boldsymbol{\mu}$.

Short outline for a derivation: If that parameter is aligned along a single axis $x_1$, ie $\boldsymbol{\mu} = \{\mu,0,0,\dots,0\}$, then you the exponential factor from the von Mises Fisher density $e^{\boldsymbol{\mu} \cdot \textbf{x}} = e^{\mu x_1}$ at cordinate $x_1$. And you get a factor $(1-x_1)^{n/2}$ for the surface area of the sphere at the coordinate $x_1$.