Answer
$${\tan ^{ - 1}}s + \frac{1}{{s - 1}} - \frac{1}{{{{\left( {s - 1} \right)}^2}}} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{2s + 2}}{{\left( {{s^2} + 1} \right){{\left( {s - 1} \right)}^3}}}} ds \cr
& {\text{Decompose the integrand }}\frac{{2s + 2}}{{\left( {{s^2} + 1} \right){{\left( {s - 1} \right)}^3}}}{\text{ into partial fractions}} \cr
& \frac{{2s + 2}}{{\left( {{s^2} + 1} \right){{\left( {s - 1} \right)}^3}}} = \frac{{As + B}}{{{s^2} + 1}} + \frac{C}{{s - 1}} + \frac{D}{{{{\left( {s - 1} \right)}^2}}} + \frac{E}{{{{\left( {s - 1} \right)}^3}}} \cr
& {\text{multiply by }}\left( {{s^2} + 1} \right){\left( {s - 1} \right)^3}{\text{ and simplify}} \cr
& 2s + 2 = \left( {As + B} \right){\left( {s - 1} \right)^3} + C{\left( {s - 1} \right)^2}\left( {{s^2} + 1} \right) + D\left( {s - 1} \right)\left( {{s^2} + 1} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + E\left( {{s^2} + 1} \right) \cr
& {\text{Expanding the equation }} \cr
& 2s + 2 = \left( {As + B} \right)\left( {{s^3} - 3{s^2} + 3s - 1} \right) + C\left( {{s^4} + 2{s^2} - 2{s^3} - 2s + 1} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + D\left( {{s^3} - {s^2} + s - 1} \right) + E{s^2} + E \cr
& \cr
& 2s + 2 = A{s^4} - 3A{s^3} + 3A{s^2} - As + B{s^3} - 3B{s^2} + 3Bs - B \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + C{s^4} + 2C{s^2} - 2C{s^3} - 2Cs + C + D{s^3} - D{s^2} + Ds - D + E{s^2} + E \cr
& \cr
& {\text{Group terms}} \cr
& 2s + 2 = \left( {A{s^4} + C{s^4}} \right) + \left( { - 3A{s^3} + B{s^3} - 2C{s^3} + D{s^3}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {3A{s^2} - 3B{s^2} + 2C{s^2} - D{s^2} + E{s^2}} \right) + \left( { - As + 3Bs - 2Cs + Ds} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - B\, + C - D + E \cr
& {\text{Equating coefficients}} \cr
& \,\,\,\,\,\,\,\,A + C = 0 \cr
& \,\,\,\,\,\,\, - 3A + B - 2C + D = 0 \cr
& \,\,\,\,\,\,\,\,\,\,3A - 3B + 2C - D + E = 0 \cr
& \,\,\,\,\,\,\,\,\, - A + 3B - 2C + D = 2 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - B\, + C - D + E = 2 \cr
& {\text{Solving the system of equations by using a calculator}}{\text{, we obtain}} \cr
& A = 0,\,\,\,B = 1,\,\,C = 0,\,\,\,D = - 1,\,\,\,E = 2 \cr
& \cr
& \frac{{As + B}}{{{s^2} + 1}} + \frac{C}{{s - 1}} + \frac{D}{{{{\left( {s - 1} \right)}^2}}} + \frac{E}{{{{\left( {s - 1} \right)}^3}}} = \frac{1}{{{s^2} + 1}} + \frac{0}{{s - 1}} - \frac{1}{{{{\left( {s - 1} \right)}^2}}} + \frac{2}{{{{\left( {s - 1} \right)}^3}}} \cr
& \frac{{2s + 2}}{{\left( {{s^2} + 1} \right){{\left( {s - 1} \right)}^3}}} = \frac{1}{{{s^2} + 1}} - \frac{1}{{{{\left( {s - 1} \right)}^2}}} + \frac{2}{{{{\left( {s - 1} \right)}^3}}} \cr
& \cr
& {\text{Therefore}}{\text{,}} \cr
& \int {\frac{{2s + 2}}{{\left( {{s^2} + 1} \right){{\left( {s - 1} \right)}^3}}}} ds = \int {\left( {\frac{1}{{{s^2} + 1}} - \frac{1}{{{{\left( {s - 1} \right)}^2}}} + \frac{2}{{{{\left( {s - 1} \right)}^3}}}} \right)} ds \cr
& {\text{integrating}} \cr
& = {\tan ^{ - 1}}s + \frac{1}{{s - 1}} - \frac{1}{{{{\left( {s - 1} \right)}^2}}} + C \cr} $$
