I have to integrate
$$ \int {\sqrt {{{\left[ 1 - {r k \cos \left( {(w - t )s + p} \right) } \right]}^2} + {{\left( {r w } \right)}^2}}} \mathrm{d}s. $$
I have tried solving it with Mathematica, but the answer was in a complex(real and imaginary) form, which was not what I wanted, because I need this integral to determine the arc length of some curve, which should be real only.
I thought I didn't put enough assumptions, so I put every assumption that I could think of, like,
Integrate[Surd[(1-r*k*Cos[(w - t)*s + p])^2 + (r*w)^2, 2], s, Assumptions -> {r, k, t, w, p, s} \[Element] Reals && r >= 0 && 0 <= k <= 1/r],
but Mathematica still kept giving me the same complex answer.
One day, I put $-\left( {r w } \right)^2$ instead of $+ \left( {r w } \right)^2$ by accident, like,
$$ \int {\sqrt {{{\left[ 1 - {r k \cos \left( {(w - t )s + p} \right) } \right]}^2} - {{\left( {r w } \right)}^2}}} \mathrm{d}s. $$
then all the $i$s disappeared and I had a real output. I was happy for a short while until I realized that I used a wrong sign.
Since that time, I have been getting more and more convinced that what I have is actually the correct solution, and there was no real solution to this integral in the first place. But is it really so? Is this really the only solution to this integral? Is there not any real solution to this integral? If there is one, how could I find it? Please help me.