$Here\ we \ use\\ L^{-1}[\phi(s)]=-\dfrac{1}t.L^{-1}[\phi'(s)] -----(1)$

$L^{-1}\bigg[log\bigg(\dfrac{s+2}{s^2-2s+17}\bigg)\bigg]\\ =L^{-1}[log({s+2})-log({s^2-2s+17})] \\=L^{-1}[log({s+2})]-L^{-1}[log({s^2-2s+17})]( using \ equation (1) ) \\ =-\dfrac1tL^{-1}\bigg[\dfrac d{ds}(log(s+2)\bigg]+\dfrac1tL^{-1}\bigg[\dfrac d{ds}(log(s^2-2s+17)\bigg] \\ =-\dfrac1t L^{-1}\bigg[\dfrac{1}{s+2}\bigg]+\dfrac1t.L^{-1}\bigg[\dfrac{2s-2}{s^2-2s+17}\bigg]$

$=-\dfrac1t e^{-2t}+\dfrac1t.L^{-1}\bigg[\dfrac{2(s-1)}{(s-1)^2+4^2}\bigg] \\ =-\dfrac1t e^{-2t}+2.\dfrac1t.e^tL^{-1}\bigg[\dfrac{s}{s^2+4^2}\bigg]\\-\dfrac1t e^{-2t}+2.\dfrac1t.e^t \dfrac14.cos2t\\ = -\dfrac1t e^{-2t}+2.\dfrac1t.e^t \dfrac14.cos2t\\ =\dfrac1{2t}[2e^t.\ cos2t-e^{-2t}] $