Is there a neat closed-form solution to the integral of $$\int \frac{\mathrm{d}x}{\sqrt{a^2\csc^2(x)-b^2}}$$ for some constant $a$ and $b$?
2026-04-11 13:15:56.1775913356
Integral of reciprocal square root of $\operatorname{cosec}(x)$
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The integral has a closed form: $$\int_{0}^{x}\frac{1}{\sqrt{a^{2}\csc\left(t\right)^{2}-b^{2}}}\mathrm{d}t=\frac{1}{b}\ln\left(\frac{b\cos\left(x\right)-\sin\left(x\right)\sqrt{\csc\left(x\right)^{2}a^{2}-b^{2}}}{b-a}\right)$$
Incidentally, Wolfram also solves it but in a different way.