Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx$

\begin{align} \int\frac{\sqrt{\cos 2x}}{\sin x}\ dx&=\int\frac{\sqrt{\cos^2x-\sin^2x}}{\sin x}\ dx\\ &\stackrel{\color{red}{[1]}}=\int\frac{\sqrt{t^4-6t^2+1}}{t^3+t}\ dt\\ &\stackrel{\color{red}{[2]}}=\frac12\int\frac{\sqrt{u^2-6u+1}}{u^2+u}\ du\\ &\stackrel{\color{red}{[3]}}=\int\frac{(y^2-6y+1)^2}{(y-1)(y-3)(y+1)(y^2+2t-7)}\ dy\\ &\stackrel{\color{red}{[4]}}=\int\left[\frac1{y-1}+\frac1{y-3}-\frac1{y+1}-\frac{16}{y^2+2y-7}\right]\ dt\\ &=\int\left[\frac1{y-1}+\frac1{y-3}-\frac1{y+1}-\frac{16}{(y+1)^2-8}\right]\ dt \end{align} The rest is yours.

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Notes :

$\color{red}{[1]}\;\;\;$Use Weierstrass substitution, $\tan\left(\dfrac{x}{2}\right)=t$.

$\color{red}{[2]}\;\;\;$Use substitution $u=t^2$.

$\color{red}{[3]}\;\;\;$Use Euler substitution, $y-u=\sqrt{u^2-6u+1}\;\color{blue}{\Rightarrow}\;y=\dfrac{u^2-1}{2u-6}$.

$\color{red}{[4]}\;\;\;$Use partial fractions decomposition.

If in the last integral you substitute $t=\frac{1}{\sqrt{2}}\cosh z$, you end with: $$ I = \frac{1}{\sqrt{2}}\int\frac{1}{\frac{\cosh^2 z}{2}-1}dz=-\operatorname{arctanh}(\sqrt{2}\tanh z)=-\operatorname{arctanh}\left(\sqrt{2-\frac{1}{t^2}}\right).$$

$\displaystyle I=\int\frac{\sqrt{2t^2-1}}{t^2-1}dt$, so $\displaystyle t=\frac{1}{\sqrt{2}}\sec\theta, dt=\frac{1}{\sqrt{2}}\sec\theta\tan\theta d\theta$ gives

$\displaystyle I=\frac{1}{\sqrt{2}}\int\frac{\sec\theta\tan^2\theta}{\frac{1}{2}\sec^2\theta-1}d\theta=\sqrt{2}\int\frac{\sec\theta\tan^2\theta}{\sec^2\theta-2}d\theta=\sqrt{2}\int\frac{\sin^2\theta\cos\theta}{\cos^2\theta(1-2\cos^2\theta)}d\theta$.

Next letting $u=\sin\theta, du=\cos\theta d\theta$ gives

$\displaystyle I=\sqrt{2}\int\frac{u^2}{(1-u^2)(2u^2-1)}du=\sqrt{2}\cdot\frac{1}{2}\int\left(\frac{1}{1-u}+\frac{1}{1+u}-\frac{1}{\sqrt{2}u+1}+\frac{1}{\sqrt{2}u-1}\right)du$

$\displaystyle=\frac{1}{\sqrt{2}}\left[\ln\left(\frac{1+u}{1-u}\right)+\frac{1}{\sqrt{2}}\ln\left(\frac{\sqrt{2}u-1}{\sqrt{2}u+1}\right)\right]+C$

$\displaystyle=\frac{1}{\sqrt{2}}\left[\ln\left(\frac{1+\sin\theta}{1-\sin\theta}\right)+\frac{1}{\sqrt{2}}\ln\left(\frac{\sqrt{2}\sin\theta-1}{\sqrt{2}\sin\theta+1}\right)\right]+C=\frac{1}{\sqrt{2}}\left[2\ln\lvert\sec\theta+\tan\theta\rvert+\frac{1}{\sqrt{2}}\ln\left(\frac{\sqrt{2}\sin\theta-1}{\sqrt{2}\sin\theta+1}\right)\right]+C$

$\displaystyle=\sqrt{2}\ln\big|\sqrt{2}t+\sqrt{2t^2-1}\big|+\frac{1}{2}\ln\bigg|\frac{\sqrt{2t^2-1}-t}{\sqrt{2t^2-1}+t}\bigg|+C$

$=\displaystyle\sqrt{2}\ln\bigg|\sqrt{2}\cos x+\sqrt{\cos 2x}\bigg|+\ln\bigg|\frac{\cos x-\sqrt{\cos 2x}}{\sin x}\bigg|+C$