I have the integrals $$ I(\alpha)=\int\limits_\alpha^\infty\sqrt{t^2-\alpha^2}\,\frac{t\:dt}{e^{t-x}+1},\qquad J(\alpha)=\int\limits_\alpha^\infty\frac1{\sqrt{t^2-\alpha^2}}\,\frac{t\:dt}{e^{t-x}+1} $$ and want to expand them analytically up to 4-th order in the small parameter $\alpha$ (here $\alpha,x>0$). At $\alpha=0$ they reduce to Fermi-Dirac integrals, at $\alpha\ll1$ they acquire, at least, quadratic corrections, as seen numerically.
I tried to make the change of variables $u=\sqrt{t^2-\alpha^2}$, $$ I(\alpha)=\int\limits_0^\infty\frac{u^2\:du}{e^{\sqrt{u^2+\alpha^2}-x}+1},\qquad J(\alpha)=\int\limits_0^\infty\frac{du}{e^{\sqrt{u^2+\alpha^2}-x}+1} $$ and expand the resulting integrands $$ \frac1{e^{\sqrt{u^2+\alpha^2}-x}+1}=\frac1{e^{u-x}+1}-\frac{e^{u-x}}{2(e^{u-x}+1)^2}\frac{\alpha^2}{u}+(\ldots)\frac{\alpha^4}{u^3}+\ldots $$ in powers of $\alpha$, but, unfortunately, these expansions contain negative powers of $u$, which make the integrals diverging on the lower limit.
Is it possible to obtain analytically asymptotic expansions of such kind of integrals?