I am interested in evaluating

$$\int\frac{dx}{(1-\sin x)^2}$$

without using tedious techniques like the Weierstrass Substitution or trivial observations like conjugating the denominator. Looking for more clever and varied approaches. I already know how to evaluate the integral using the former two approaches, but would like to see some more unique ones as well.


Rewrite the integral,

$$I=\int\frac{dx}{(1-\sin x)^2}=\int\frac{dx}{(\cos\frac x2-\sin\frac x2)^4}=\frac14 \int\frac{dx}{\cos^4(\frac x2+\frac\pi4)}$$

Let $t=\frac x2+\frac\pi4$, $$I=\frac12\int \sec^4t\,dt=\frac12\int (1+\tan^2 t)\,d(\tan t) = \frac12\tan t+ \frac16 \tan^3 t + C$$

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