As usual for special functions, there is no closed form until a closed form is defined and referenced.
"The Sophomores Dream Function" on Scribd : JJacquelin's Documents | Scribd
Sphd(1;x) is an antiderivative of x^x .
Before saying "This is a vicious circle", one have to think at the role played by all special functions in maths. On this subject, the paper : "Safari in the country of special functions", pp.18-36, same link on Scribd.