An integral including the complete elliptic integral

Question

Let $\lambda>1$, $0<\alpha<\pi$, how to evaluate: $$I=\int_0^\pi K\left(\sqrt{\frac{2\sin\theta\sin\alpha}{\lambda-cos(\theta+\alpha)}}\right)\frac{\sin\theta\cos\theta d\theta}{\sqrt{\lambda-cos(\theta+\alpha)}}$$ Where $K$ is the complete elliptic integral. Let $$u=\sqrt{\frac{2\sin\theta\sin\alpha}{\lambda-\cos(\theta+\alpha)}}$$ then: $$du=\frac{2\sin\alpha}{u}\frac{cos\theta(\lambda-cos(\theta+\alpha))-sin\theta\sin(\theta+\alpha)}{(\lambda-cos(\theta+\alpha))^2}d\theta=\frac{2\sin\alpha}{u}\frac{\lambda cos\theta-cos\alpha}{(\lambda-cos(\theta+\alpha))^2}d\theta$$ Therefore: $$du=\frac{\lambda cos\theta-cos\alpha}{u}\frac{u^4}{2\sin\alpha\sin^2\theta}d\theta$$ or: $$d\theta=\frac{2\sin\alpha}{u^3}\frac{\sin^2\theta}{\lambda cos\theta-cos\alpha}du$$ ....