I've been trying to compute the following form, however, I'm not getting any results. Any suggestions?
$$\int \frac{e^{-x}}{\sqrt{x^2+a^2}} \,dx$$
2026-03-26 01:34:45.1774488885
Integration of an exponential: $\int \frac{e^{-x}}{\sqrt{x^2+a^2}} \,dx$
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As Parcly Taxel commented, there is no hope for a closed form, except for $a=0$.
In this last case, assuming $\sqrt {x^2}=x$, this would reduce to $$\int \frac{e^{-x}}{x}\,dx=\text{Ei}(-x)$$ where appears the exponential integral function.
For your problem, we could write $$\frac 1 {\sqrt{x^2+a^2}}=\frac 1x \left(1+\frac {a^2}{x^2} \right)^{-\frac 12}=\sum_{n=0}^\infty \binom{-\frac{1}{2}}{n}\frac{a^{2 n}} { x^{2 n+1}}$$ and face the problem of computing $$I_n=\int \frac{e^{-x}}{ x^{2 n+1}}\,dx=-\Gamma (-2 n,x)$$ where appears the incomplete gamma function.