I am working on a project modeling the motion of a particle given certain constants, a,c,n, and $\theta$. I am not sure how to approach it and Wolfram Alpha was unable to solve it
The integral is as follows
$$ \int\tan\left(\,\arcsin\left(\,\exp\left(\,-ac\,\mathrm{e}^{nx} - \frac{c^{2}}{2}\,\mathrm{e}^{2nx}\,\right)\,\right) + \theta\,\right)\,\mathrm{d}x $$
The constants have the following possible ranges of values: $$a > 0$$ $$c < 0$$ $$ 0 < n < 1$$ $$ 0 < \theta < \pi$$
The integral should have some solution, as I am fairly certain that it is physically meaningful
To add some context: This integral is to solve the differential equation $\frac{dy}{dx}$ given initial state $y = 0$
The ultimate goal is to find an equation for $\theta$ in terms of y, x, and the other constants
Edit: Due to the complexity of the integral, I have been for looking for a way to express the integrand as $f(\theta) * f(u)$ as opposed to $f(u + \theta)$ (where u is the arcsin function). If I could find a way to do that, the integral would simplify to $f(\theta) * \int f(u) dx$.
Then $y = f(\theta) * \int_{x_0}^{x_f} f(u) dx$ and I could find theta via $\theta = f^{-1}(\frac{y}{\int_{x_0}^{x_f} f(u) dx})$ since a computer could estimate the definite integral
Unfortunately, I have played quite a bit with the trig Identities and I don't know if there is a way to simplify $tan(u + \theta)$ to any two factors $f(\theta) * f(u)$