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Find the number of solutions in $\mathbb{R}$ to the equation $$|x^2-1|=x^\alpha\arctan\left(\frac1{x^2}\right)$$ with $\alpha\in\{-1, 0, 1, 2, 3\}$.

Only pen and paper is allowed.
After observing the evenness of $\arctan\left(\frac1{x^2}\right)$ and that $$\begin{align*} \arctan\left(\frac1{x^2}\right) \sim \frac1{x^2}&\qquad\text{for}\ x \to \pm\infty\\ \arctan\left(\frac1{x^2}\right) \rightarrow \frac\pi2 &\qquad\text{for}\ x \to 0 \end{align*}$$

graphing all the cases is somewhat easy, although time-consuming and error-prone.

Is there a faster (maybe algebraical) way to solve the problem?

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