TITLE: Hypergeometric integral $\int_0^1 dx ~e^{-(\frac{1}{x}+\frac{1}{1-x})}(x)^{b-a-1}(1-x)^{a-1}$.
QUESTION [7 upvotes]: I have an this integral 

$$\int_0^1 dx ~e^{-(\frac{1}{x}+\frac{1}{1-x})}(x)^{b-a-1}(1-x)^{a-1}$$

how can I calculate the integral in form of Bessel function or Hypergeometric function.
As mentioned in Arfken (special function-chapter)
$$M(a,c;x)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1 e^{xt}t^{a-1}(1-t)^{c-a-1}dt$$
and 
$M(a,c;x)= e^x M(c-a,c;-x) $
But I couldn't find  useful variable to simplify the integral

REPLY [3 votes]: At least there is this particular closed form
$$
\int_0^1e^{-\left(\frac{1}{x}+\frac{1}{1-x}\right)} \frac{dx}{x^{3/2}(1-x)^{3/2}}=2\frac{\sqrt{\pi}}{e^4}.
$$
The general case also should have a closed form and I remember seeing it somewhere...