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How do i evaluate $\displaystyle \int \frac{dx}{\cos(x+a) \cos(x+b)}\text{ ?}$

My thought process

This integral is not a standard integral in the form it is in. Integration by parts does not reduce it to any standard form (it seems). Integration by the method of partial fractions is not applicable (it seems). The only two elementary methods left are algebraic manipulation and integration by substitution. I can neither think of any suitable algebraic manipulation nor can i think of any suitable substitution to simplify the integrand. Please help. Please explain your thought process also (why you did what you did). In many cases i have seen people bring in new stuff into the integrand out of thin air.

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Hint

Assuming $a \neq b$. Multiply and divide by the constant $\sin(a-b)$ and use the fact that $$\sin(a-b)=\sin((x+a)-(x+b))$$ to split the integral into $\tan$.

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