### 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);

`

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