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The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.

You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-2a^2)$, which is quite accurate up to about $a=0.2$, see plot below. Incidentally, the small-$a$ and large-$a$ asymptotics (see Brendan McKay's comment) are the same upon replacement of $a$ by $1/a$.

The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.

You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-2a^2)$, which is quite accurate up to about $a=0.2$, see plot below. Incidentally, the small-$a$ and large-$a$ asymptotics (see Brendan McKay's comment) are the same upon replacement of $a$ by $1/a$.

The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.

You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-2a^2)$, which is quite accurate up to about $a=0.2$, see plot below:

The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.

You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-2a^2)$, which is quite accurate up to about $a=0.2$, see plot below. Incidentally, the small-$a$ and large-$a$ asymptotics (see Brendan McKay's comment) are the same upon replacement of $a$ by $1/a$.

The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.

You could use this for a global approximation. In the small-$a$ region the asymptotics is $ae^{-1/a}$, which is quite accurate up to about $a=0.1$, see plot below:

The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$.

You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-2a^2)$, which is quite accurate up to about $a=0.2$, see plot below: