1.10 Obtain Inverse Laplace transform of a transfer function

Problem: Obtain the inverse Laplace transform for the function

\[ H(s)=\frac {s^{4}+5s^{3}+6s^{2}+9s+30}{s^{4}+6s^{3}+21s^{2}+46s+30}\]

Mathematica

Remove["Global`*"]; 
f = (s^4+5 s^3+6 s^2+9 s+30)/(s^4+6 s^3+21 s^2+46 s+30); 
InverseLaplaceTransform[f,s,t]; 
Expand[FullSimplify[%]]
 

\[ \delta (t)+\left (\frac {1}{234}+\frac {i}{234}\right ) e^{(-1-3 i) t} \left ((73+326 i) e^{6 i t}+(-326-73 i)\right )-\frac {3 e^{-3 t}}{26}+\frac {23 e^{-t}}{18} \]

Matlab

clear all; 
syms s t 
f = (s^4+5*s^3+6*s^2+9*s+30)/(s^4+6*s^3+21*s^2+46*s+30); 
pretty(f)
 

    4      3      2 
   s  + 5 s  + 6 s  + 9 s + 30 
  ----------------------------- 
   4      3       2 
  s  + 6 s  + 21 s  + 46 s + 30
 

pretty(ilaplace(f))
 

                                                    /            399 sin(3 t) \ 
                                        253 exp(-t) | cos(3 t) + ------------ | 
  23 exp(-t)   3 exp(-3 t)                          \                253      / 
  ---------- - ----------- + dirac(t) - --------------------------------------- 
      18           26                                     117
 

Maple

restart; 
interface(showassumed=0): 
p:=(s^4+5*s^3+6*s^2+9*s+30)/(s^4+6*s^3+21*s^2+46*s+30); 
r:=inttrans[invlaplace](p,s,t);
 

\[ Dirac \left ( t \right ) -{\frac {3\,{{\rm e}^{-3\,t}}}{26}}+{ \frac { \left ( -506\,\cos \left ( 3\,t \right ) -798\,\sin \left ( 3\,t \right ) +299 \right ) {{\rm e}^{-t}}}{234}} \]