\begin{array}{rcl}
I_a&=&\displaystyle
\int_0^a\frac{dt}{t+1}-\int_0^a\frac{t-2}{t^2-t+1}\,dt
\vspace{0.5em}\\
&=&\displaystyle
\ln(a+1)-\ln(\sqrt{a^2-a+1})-\sqrt 3(\arctan((2a-1)/\sqrt 3)+\pi/6)
\vspace{0.5em}\\
&=&\displaystyle
\frac 12\ln\left(1+\frac{3a}{a^2-a+1}\right)-\sqrt 3(\arctan((2a-1)/\sqrt 3)+\pi/6)