$I=\int\frac{\sin\alpha dx}{\sin(x+\frac{\pi}{2})}-\int\frac{\cos\alpha d\cos x}{\cos^2 x}$
$=\int\frac{\sin\alpha dx}{2\tan(\frac{x}{2}+\frac{\pi}{4})\cos^2(\frac{x}{2}+\frac{\pi}{4})}+\frac{\cos\alpha}{\cos x}$
$=\int\frac{\sin\alpha d\tan(\frac{x}{2}+\frac{\pi}{4})}{\tan(\frac{x}{2}+\frac{\pi}{4})}+\frac{\cos\alpha}{\cos x}$
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