# Symbolic Integral

I am in trouble with the following integral:

$$rtartaruga(r)=\int \frac{\Sigma(r') }{\Delta(r') \sqrt{\Delta th}} d r'$$ Where: $$\Delta(r) =\left(a^2+r^2\right) \left(\frac{r^2}{l^2}+1\right)-2 m r$$ $$\Sigma(r) =\sqrt{\text{\Delta th} \left(a^2+r^2\right)^2-a^2 \Delta \sin ^2(\theta )}$$ And $$\Delta th$$ is a constant in $$r$$: $$\text{\Delta th}=1-\frac{a^2 \cos ^2(\theta )}{l^2}$$.

In the following region of parameters: $$|a| and $$m>m_c$$ (where $$m_c$$ is a certain positive constant $$\Delta (r)$$ has two positive roots and the integrand is divergent in these points.

I would really like to have an analytical solution for the above integral so i try to evaluate it symbolically:

\[CapitalSigma] =
Sqrt[(r^2 + a^2)^2 \[CapitalDelta]th -
a^2 \[CapitalDelta] Sin[\[Theta]]^2];

\[CapitalDelta] = (r^2 + a^2) (1 + r^2/l^2) - 2 m r;

\[CapitalDelta]th = 1 - a^2/l^2 Cos[\[Theta]]^2;

mc = l/(3 Sqrt[6]) (Sqrt[(1 + a^2/l^2) + 12/l^2 a^2] + 2 a^2/l^2 + 2)*
Sqrt[(Sqrt[(1 + a^2/l^2) + 12/l^2 a^2] - a^2/l^2 - 1)];

Clear[rtartaruga]

rtartaruga[r_] =
Integrate[\[CapitalSigma]/Sqrt[\[CapitalDelta]th]*1/\[CapitalDelta],
r, Assumptions -> {{r, m, \[Theta], a, l} \[Element] Reals,
Abs[a] < l, m > mc, r > 0, l > 0, \[Theta] >= 0, \[Theta] <= Pi},
PrincipalValue -> True]


The output is a very long one and includes special functions. When i try to evaluate it at some point:

a = 1/2;
l = 2;
m = 1;
\[Theta] = Pi/4;
N[rtartaruga[0.4]]
Clear[a, l, m, \[Theta]]


I get:

-0.274199 - 0.47465 I


But the result has to be real as the integrand!

I know that there are other similar questions on this issue, and i understand that the problem is about branch cuts on the complex plane, but practically i can't find a way to get out my problem (if it's possible).