I need to solve this integral:
$$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$$
$$\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) \sqrt{1-p^2 \cos 2 t+2 p \sin t \sqrt{1-p^2 \cos^2 t}}=$$
$$=(\vec{i} \cos t +\vec{j} \sin t) (\sqrt{1-p^2 \cos^2 t}+p \sin t)$$
Here $\vec{V}$ is a constant vector (independent on $t$), $\vec{i}, \vec{j}$ are unit vectors of Cartesian coordinates, $p<1$ is a parameter.
To clarify, I've already done the following two things - solved it numerically using Mathematica and expanded it into Taylor series for small values of $p$, integrating several terms of the series.
I want to know if there is some way to simplify this integral, or obtain some approximation for small values of $p$, not relying completely on Taylor series. Any suggestion would be welcome.
I know, that the first good approximation for small $p$ would be:
$$\sqrt{1-p^2 \cos 2 t+2 p \sin t \sqrt{1-p^2 \cos^2 t}} \approx \sqrt{1-p^2 \cos 2 t+2 p \sin t}$$
Edit
Thanks to Martin's comment, I now edited the title. Some further simplification would be nice.