$\int_0^e\frac{dx}{x(\ln^2{x}+2\ln x+5)}=|\text{subs}:\; t=\ln x; x= e^t; dx = e^t dt|$ $ =\int_\infty^1 \frac{dt}{t^2+2t+5} = \int_\infty^1\frac{dt}{(t+1)^2+4} $ $= \left[{\frac12\arctan\frac{t+1}2}\right]_\infty^1\; =\;\; \frac{\pi}{8}-\frac12\arctan\left(\lim_{t\to0} \frac{t+1}2\right)\; = \;\;\frac{\pi}8 - \frac{\pi}4 \;= \;-\frac{\pi}8$
The result isn't right, but I can't see my mistakes. Do you see where did I do wrong?