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\[\int\left(\frac{1}{u}e^u\right)\,du=\int\left(\frac{1}{u}\sum_{n=1}^\infty u^n\right)\,du\]

\[\int\sum_{n=1}^\infty u^{n-1}\,du\]

Whoops, forgot something!

there is no antiderivative in terms of elementary functions

Expressing the answer in terms of a sum would be fine.

\[\sum_{n=1}^\infty\frac{u^n}{n\cdot n!}=\sum_{n=1}^\infty\frac{\ln(x)^n}{n\cdot n!}\]

Can it be further simplified?

A better question: what does\[\sum_{n=1}^\infty\frac{\ln(x)^n}{n\cdot n!}\]converge to?

By the ratio test it converges.

you should include your constant of integration along with the radious of convergence.