\[ \int \frac{\cos \alpha+\cos x+1}{\cos \alpha+\cos x} d x=a f(x)+...
0\[
\int \frac{\cos \alpha+\cos x+1}{\cos \alpha+\cos x} d x=a f(x)+b g(x)+c, \alpha \in(0, \pi)
\]
(a) \( a=-\cos \alpha, b=\sin \alpha, f(x)=x+1, g(x)=\ln \left|\frac{\tan \frac{x}{2}-\cot \frac{\alpha}{2}}{\tan \frac{x}{2}+\cot \frac{\alpha}{2}}\right| \)
(b) \( a=\sin \alpha, b=\cos ^{2} \alpha, f(x)=\sin x, g(x)=\tan ^{-1}\left(\tan \frac{x}{2}+\tan \frac{\alpha}{2}\right) \)
(c) \( a=\sin ^{2} \alpha, b=-\cos \alpha, f(x)=\cos x, g(x)=\tan ^{-1}\left(\frac{\tan \frac{x}{2}+\cot \frac{\alpha}{2}}{\tan \frac{x}{2}-\cot \frac{\alpha}{2}}\right) \)
(d) \( a=1, b=\operatorname{cosec} \alpha, f(x)=x, g(x)=\ln \left|\frac{\tan \frac{x}{2}+\cot \frac{\alpha}{2}}{\tan \frac{x}{2}-\cot \frac{\alpha}{2}}\right| \)
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