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?
