Evaluate $$\int \frac {dx}{\sin \frac x2\sqrt {\cos^3 \frac x2}}$$
My try
Write $t=\frac x2$ and hence $dx=2dt$
To change the integral to $$\int \frac {\csc t dt}{\cos^{\frac 32} t}$$
Multiplying both bottom and top by $\csc t$ and then using $\csc^2 t=1+\cot^2 t$ in the numerator the problem simplifies to $$2\int (\sin t)(\cos ^{\frac {-3}{2}} t) dt+2\int \frac {\cot^3 t dt}{\sqrt {\cos t}}$$
Now the first integral is easy to go but I am not getting any idea for the second one. Any help would be very beneficial. New methods are also welcome.