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$\frac{1}{\sqrt{\sin ^{3} x \sin (x+\alpha)}}$


$\frac{1}{\sqrt{\sin ^{3} x \sin (x+\alpha)}}=\frac{1}{\sqrt{\sin ^{3} x(\sin x \cos \alpha+\cos x \sin \alpha)}}$

$=\frac{1}{\sqrt{\sin ^{4} x \cos \alpha+\sin ^{3} x \cos x \sin \alpha}}$

$=\frac{1}{\sin ^{2} x \sqrt{\cos \alpha+\cot x \sin \alpha}}$

$=\frac{\operatorname{cosec}^{2} x}{\sqrt{\cos \alpha+\cot x \sin \alpha}}$

Let $\cos \alpha+\cot x \sin \alpha=t \Rightarrow-\operatorname{cosec}^{2} x \sin \alpha d x=d t$

$\therefore \int \frac{1}{\sin ^{3} x \sin (x+\alpha)} d x=\int \frac{\operatorname{cosec}^{2} x}{\sqrt{\cos \alpha+\cot x \sin \alpha}} d x$

$=\frac{-1}{\sin \alpha} \int \frac{d t}{\sqrt{t}}$

$=\frac{-1}{\sin \alpha}[2 \sqrt{t}]+\mathrm{C}$

$=\frac{-1}{\sin \alpha}[2 \sqrt{\cos \alpha+\cot x \sin \alpha}]+\mathrm{C}$

$=\frac{-2}{\sin \alpha} \sqrt{\cos \alpha+\frac{\cos x \sin \alpha}{\sin x}}+\mathrm{C}$

$=\frac{-2}{\sin \alpha} \sqrt{\frac{\sin x \cos \alpha+\cos x \sin \alpha}{\sin x}}+\mathrm{C}$

$=-\frac{2}{\sin \alpha} \sqrt{\frac{\sin (x+\alpha)}{\sin x}+\mathrm{C}}$

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