Evaluating the integral $\int_0^{\pi/2}\frac{\cos^{3/2}x}{\sin x\sqrt{\lambda-2\cos x-2\ln(1-\cos x)+2\ln(\sin x)}}\,\mathrm dx$

Question

I'm trying to find this indefinite integral

$$\int\frac{\cos^{3/2} x}{\sin x\sqrt{(\lambda-2\cos x-2\ln(1-\cos x)+2\ln(\sin x))}} \,\mathrm dx$$

Or $$\int_0^{\pi/2}\frac{\cos^{3/2}x}{\sin x\sqrt{\lambda-2\cos x-2\ln(1-\cos x)+2\ln(\sin x)}}\,\mathrm dx$$

Where $\lambda>2$

Which sub I take !?

I don't have any ideas for this type integration

Please give me ideas