I have the following integral: $$\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$$ where $P_3(t)$ is a third-degree polynomial with all coefficients different from zero and $k$ a generic constant. This integral, once known coefficients and the constant $k$, can be integrated numerically, but I wonder if, in some research work, this integral was approximated analytically (under some assumptions on coefficients and the constant $k$).

Thanks in advance.

**Edit (suggested by IgorKhavkine's comment).**

If $P(t)=a+bt+ct^2+dt^3$, $a$ is very large respect other constants. The coefficient $d$ is $<0$ and very small respect to $c,b$. Coefficients $c,b$ are comparable.