I need to solve the following integral to calculate the effect of retarded electromagnetic fields on a test charge:
$\int\limits_0^\zeta\frac{(\psi-(1+x)\sin(\psi+\alpha))(\frac{\psi^2}{2\beta^2(1+x)}+x)}{(\psi-\beta^2(1+x)\sin(\psi +\alpha))^3}*e^{-\frac{(z-R\alpha)^2}{2}}*\frac{d\alpha}{d\psi} d\psi$
You can also make this challenge lighter by removing the exponential term. Here $\alpha, \psi$ and $x$ depend on each other through this system of equations:
$x = \delta\bigg(z(1-\cos(\zeta))-(z-R\alpha)(1-\cos(\zeta-\psi-\alpha))\bigg)$
$1+(1+x)^2-2(1+x)\cos(\psi+\alpha)=\frac{\psi^2}{\beta^2}$
where $\zeta,z,\delta$ and $R$ are just parameters that can be freely chosen (even infinity if you would be pleased by that). In theory this is possible, plug the expression for $x$ into the last equation and find the function $\alpha(\psi)$ and use that to solve the integral. But if this analytically possible I leave to you guys! Even solving the system of equations that form the relation between $x,\alpha$ and $\psi$ would be extremely helpful.