in my probability research I encounter the following integral equation for continuous non-negative $f: (0,\pi/4] \to \mathbb R$:
$$ f(\varphi) = \int_0^{\pi/4} \frac {4} {\pi} \sin \varphi_0 \cos \varphi_0 \sin \varphi \cos \varphi \, (\cos \varphi_0 \sqrt{1+\cos^2 \varphi_0} ( \frac {1} {4\cos^4\varphi_0 \sin^2 \varphi + \cos^4 \varphi} + \frac {1} {4\cos^4\varphi_0 \cos^2 \varphi + \sin^4 \varphi} + \frac {1} {1-4\cos^4\varphi_0 \sin^2 \varphi \cos^2 \varphi}) + \sin \varphi_0 \sqrt{1+\sin^2 \varphi_0} ( \frac {1} {4\sin^4\varphi_0 \cos^2 \varphi + \sin^4 \varphi} + \frac {1} {4\sin^4\varphi_0 \sin^2 \varphi + \cos^4 \varphi} + \frac {1} {1-4\sin^4\varphi_0 \sin^2 \varphi \cos^2 \varphi}) \, f(\varphi_0) \, d\varphi_0.$$
The integral kernel is a density for fixed $\varphi_0$ and $f$ integrates to $1$. I have shown that such a density function $f$ exists uniquely and a numerically computed picture but I'm really stuck on finding a suitable analytic approach to solve for $f$.
I highly appreciate any suggestions.
EDIT: Plot of solution $f$ (which is the Lebesgue density function of some stationary distribution):
To me, it looks like $f$ is point-symmetric at $(\pi/8,4/\pi)$, but I can not prove it. Perhaps it is to hard to find any closed form expression, series representation or anything exactly matching $f$. Are there chances to have an approximation $g \in C((0,\pi/4])$ with $\lVert f-g\rVert_{\infty}$ small measured in the supremum norm?