$\int{ -\frac{ 3 }{ x }+\frac{ 3x+13 }{ {x}^{2}+2x+5 } } \mathrm{d} x$
Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$$-\int{ \frac{ 3 }{ x } } \mathrm{d} x+\int{ \frac{ 3x+13 }{ {x}^{2}+2x+5 } } \mathrm{d} x$
Use $\int{ \frac{ a }{ x } } \mathrm{d} x=a \times \ln\left({|x|}\right)$ to evaluate the integral$-3\ln\left({|x|}\right)+\int{ \frac{ 3x+13 }{ {x}^{2}+2x+5 } } \mathrm{d} x$
Evaluate the indefinite integral$-3\ln\left({|x|}\right)+\frac{ 3 }{ 2 } \times \ln\left({|{x}^{2}+2x+5|}\right)+5\arctan\left({\frac{ x+1 }{ 2 }}\right)$
Add the constant of integration $C \in ℝ$$\begin{array} { l }-3\ln\left({|x|}\right)+\frac{ 3 }{ 2 } \times \ln\left({|{x}^{2}+2x+5|}\right)+5\arctan\left({\frac{ x+1 }{ 2 }}\right)+C,& C \in ℝ\end{array}$