Integrate $\frac{\sin^3 \frac{x}{2}}{\cos \frac{x}{2} \sqrt{\cos x+\cos^2 x+\cos^3 x}}$ Evaluate $$\int \frac{\sin^3 \frac{x}{2}}{\cos \frac{x}{2} \sqrt{\cos x+\cos^2 x+\cos^3 x}}dx$$ I saw terms like $1+t+t^2$ in the denominator , so I thought of $t^3-1$ and then converting back into half angle but it doesn't help me. I am unable to simplify it further. Please tell me a better way to start. Thanks.

I might be wrong but if you combine these two identities :
$$ \sin^2(x) = 1 - \cos^2(x) $$
$$ \cos(x) = \cos(\frac{x}{2} + \frac{x}{2}) = \cos^2(\frac{x}{2}) - \sin^2(\frac{x}{2}) = 2\cos^2(\frac{x}{2}) - 1 $$
You can expand the expressions until only the sine function in the numerator remains and everything else is powers of cosine.
$$ \sin^3(\frac{x}{2}) = \sin(\frac{x}{2})(1 - \cos^2(\frac{x}{2}))$$
Then if you make the substitution :
$$ t = \cos(\frac{x}{2}) $$
 $$ dt = -\frac{1}{2}\sin(\frac{x}{2})dx $$
It should remove all the trigonometric functions.
I don't know if this will work to solve the integral, but I hope it helps!
