I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll 1$, $n<2$ and $$\cos\theta=\frac{z}{\sqrt{R^2+z^2}}.$$ I was thinking of Taylor Expanding the integrand for $z<R$, and integrating the result from $-R$ to $R$, but the answer seems not good when compared numerically. Any good approximation if not the exact answer would work. Any ideas? Thanks
Note that for $b=0$, an exact expression for the above integral exists and is given by $$\mathcal{I}=-\frac{\sqrt{\pi}\Gamma[n/4-1/2]}{\sqrt{a}\Gamma[n/4]}R^{1-n/2}.$$