I need help to integrate $$\int{\sin^2(nx)\csc^2(x)dx}$$ Integral of each part is easy $$\int{\sin^2(nx)}dx=\frac{x}{2}+\frac{\sin(2nx)}{4n}+C$$ $$\int{\csc^2(x)}dx=-\cot(x)+C$$ I then tried to use the integral by parts but failed to make it work. Also tried Walfram calculator but it returned a very complicated expression that's not what I wanted.

Can you help to resolve this indefinite integration? Definite integral $$\int_{2\pi}^{N\pi}{\sin^2(nx)\csc^2(x)}dx$$ where $N$ is big natural number, is also helpful.


This answer concerns the indefinite integral $\displaystyle\int\sin^2(nx)\csc^2 x\ dx$.

Hint 1:- Observe that, $$\displaystyle\int\cos^2(nx)\csc^2 x\ dx+\displaystyle\int\sin^2(nx)\csc^2 x\ dx=\displaystyle\int\csc^2 x\ dx$$

$$\displaystyle\int\cos^2(nx)\csc^2 x\ dx-\displaystyle\int\sin^2(nx)\csc^2 x\ dx=\displaystyle\int\cos(2nx)\csc^2 x\ dx$$

For, $\displaystyle\int\cos(2nx)\csc^2 x\ dx$ observe that, $$\displaystyle\int\cos(2nx)\csc^2 x\ dx=\Im \left(\int e^{2inx}\csc^2 x\ dx\right)$$

Hint 2:- For evaluating $\displaystyle\int\cos(2nx)\csc^2 x\ dx$ you may also use the following formula, $$\boxed{\cos (2nx)=\displaystyle\sum_{k=0}^{2n}\binom{2n}{k}\cos^kx\sin^{2n-k} x\cos\left(\dfrac{1}{2}(2n-k)\right)}$$


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