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