Estimating $\int_{\mathbb{R}}|\frac{\log|x|}{1+x^4}|\ dx$

Question

I'd like to show that $\frac{\log|x|}{1+x^4} \in L^1(\mathbb{R})$, but I have trouble with estimating $$\int_{\mathbb{R}}\left|\frac{\log|x|}{1+x^4}\right|\ dx.$$ Writing $(1+x^4) = (x^2 - i)(x^2 + i)$ or splitting the integral into $$\int_{\mathbb{R}}\left|\frac{\log|x|}{1+x^4}\right|\ dx = \int_{0 < |x| < 1}\frac{|\log|x||}{1+x^4}\ dx + \int_{|x| > 1}\frac{|\log|x||}{1+x^4}\ dx$$ doesn't seem to help me either. Is it true that $$\frac{|\log|x||}{1+x^4} < \frac{1}{x^2}$$ for any $|x| > 1$? If that's true, how so? What can I do with $$\int_{0 < |x| < 1}\frac{|\log|x||}{1+x^4}\ dx?$$

Answer 1

for $|x|>1$ it is clear that $\ln|x|<x^2$ since $\ln|x|<|x-1|$ so you can use this. In your case you can use this to show that: $$\int\limits_{|x|>1}\left|\frac{\ln|x|}{x^4+1}\right|dx<\int\limits_{|x|>1}\frac1{x^2}dx$$ now you need to try and do the other proportion of the integral where $|x|<1$