It may be useful that
\(
\int \log(x)^k t^k dt=\log (x)^{k+1} (-(k+1) \log (x))^{-k-1} (-\Gamma (k+1,-(k+1) \log (x))) \)
In other words,
\(
\int t^tdt=\sum _{k=0}^{\infty } \frac{t^{-k-1} x^{k+1} \log ^k(t) ((-k-1) \log (t))^{-k} \Gamma (k+1,(-k-1) \log (t))}{(k+1) !}+C \)
\(
\int \log(x)^k t^k dt=\log (x)^{k+1} (-(k+1) \log (x))^{-k-1} (-\Gamma (k+1,-(k+1) \log (x))) \)
In other words,
\(
\int t^tdt=\sum _{k=0}^{\infty } \frac{t^{-k-1} x^{k+1} \log ^k(t) ((-k-1) \log (t))^{-k} \Gamma (k+1,(-k-1) \log (t))}{(k+1) !}+C \)
