Separate Complete Elliptic Integral of Third Kind into Real and Imaginary Parts

Question

I'm interested in doing something similar to this question, however I'm interested in the complete elliptic integral of the third kind $$\Pi(n,k)=\int_0^{\pi/2} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}$$ for $k>1$. Playing around with the definition a bit a bit, $$\Pi(n,k)=\int_0^{\arcsin(1/k)} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}-i\int_{\arcsin(1/k)}^{\pi/2} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{k^2\sin^2\theta-1}},$$ I believe that it should be expressible in terms of incomplete elliptic integrals of the third kind as $$\Pi(n,k)=\Pi\left(n,k\mid\arcsin\left(\tfrac{1}{k}\right)\right)+\frac{i}{\sqrt{k^2-1}(n-1)}\Pi\left(\tfrac{n}{n-1},\tfrac{\pi}{2}-\arcsin\left(\tfrac{1}{k}\right)\mid\tfrac{k^2}{k^2-1}\right).$$ (I'm not 100% certain about the second term, however; I obtained it via Mathematica.) However, I don't find this very satisfying since it consists of a bunch of incomplete elliptic integrals. So, my question is, is it possible to express $\Pi(n,k)$ for $k>1$ in terms of complete elliptic integrals, such that it is separated into real and imaginary parts (as in the linked question)?