I am trying to calculate $$\displaystyle\int\limits^{\cssId{upper-bound-mathjax}{\infty}}_{\cssId{lower-bound-mathjax}{-\infty}} \dfrac{x^2}{\left(x^2+a^2\right)\left(x^2+b^2\right)}\,\cssId{int-var-mathjax}{\mathrm{d}x}$$ for $a,b >0$
I recognize that I can split it to multiply of the derivative of $\frac{ln(x^2+a^2)}{2}$ and $\frac{ln(x^2+b^2)}{2}$ but I am stuck on calculate transform Fourier of the $ln$ function
Why not to use $$\dfrac{x^2}{\left(x^2+a^2\right)\left(x^2+b^2\right)}=\frac1 {a^2-b^2}\Bigg[\frac {a^2}{x^2+a^2}-\frac {b^2}{x^2+b^2} \Bigg]$$