s/(s4-a4)=s/[(s-a)(s+a)(s2+a2)] which can be expressed as partial fractions:
s/(s4-a4)=A/(s-a)+B/(s+a)+(Cs+D)/(s2+a2),
s/(s4-a4)=[A(s+a)(s2+a2)+B(s-a)(s2+a2)+(Cs+D)(s2-a2)]/(s4-a4).
Equating numerators:
s=As3+Aa2s+Aas2+Aa3+Bs3+Ba2s-Bas2-Ba3+Cs3-Ca2s+Ds2-Da2.
Equating like powers of s:
s3: A+B+C=0;
s2: Aa-Ba+D=0, D=a(B-A);
s: Aa2+Ba2-Ca2=1, A+B-C=1/a2;
constant: Aa3-Ba3-Da2=0, Aa-Ba-D=0, D=a(A-B).
D=a(B-A)=a(A-B), B-A=A-B, 2A=2B, A=B, D=0.
C=-(A+B)=-2A; A+B-C=1/a2⇒2A+2A=4A=1/a2, A=B=1/(4a2), C=-1/(2a2), D=0.
s/(s4-a4)=1/[4a2(s-a)]+1/[4a2(s+a)]-s/[2a2(s2+a2)].
This looks like a Laplace transform: (1/(4a2))[(1/(s-a))+1/(s+a)]-(1/(2a2))(s/(s2+a2)):
(1/(4a2))(eat+e-at)+(1/(2a2))cos(at).