How can I evaluate $\int e^u \; u \; \sqrt{\ln \frac{1}{u}\,} \; \mathrm{d}u$?

Question

How do I solve the following integration problem:

$$\int e^u \; u \; \sqrt{\ln \frac{1}{u}\,} \; du \;\;\;\;\; ?$$

Answer 1

If the integration range is $(0,1)$, from $$ \int_{0}^{1} u^k \sqrt{-\log u}\,du = \frac{\sqrt{\pi}}{2(1+k)^{3/2}} \tag{1} $$ it follows that: $$ \int_{0}^{1} u e^u\sqrt{-\log u}\,du = \frac{\sqrt{\pi}}{2}\sum_{k\geq 0}\frac{1}{k!(k+2)^{3/2}}\tag{2} $$ where the RHS of $(2)$ is a fast-converging series. If the integration range is not really specified, you may use differentiation under the integral sign to write your integral as the semiderivative of an incomplete $\Gamma$ function.