<s>[INST] Evaluating $\mathcal{P}\int_0^\infty \frac{dx}{(1 + a^2x^2)(x^2 - b^2)}$ I am trying to evaluate this integral 
$$\mathcal{P}\int_0^\infty \frac{dx}{(1 + a^2x^2)(x^2 - b^2)}$$
with $\mathcal{P}$ the principal value and $a,b>0$.
I already know the answer to be $$ - \frac{a\pi}{2(1 + a^2b^2)}$$
after fiddling with Mathematica putting numbers for $a$ and $b$.
The poles of this function are at $x=\pm b, \pm \frac{i}{a^2}$ so I cannot find a proper contour.
Any help in getting the answer will be appreciated!
 [/INST] Use that $$\frac{1}{(1+a^2x^2)(x^2-b^2)}=1/2\,{\frac {1}{ \left( 1+{a}^{2}{b}^{2} \right) b \left( -b+x
 \right) }}-{\frac {{a}^{2}}{ \left( 1+{a}^{2}{b}^{2} \right)  \left( 
1+{a}^{2}{x}^{2} \right) }}-1/2\,{\frac {1}{ \left( 1+{a}^{2}{b}^{2}
 \right) b \left( b+x \right) }}
$$
</s>