\[ \boxed { {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) -2\,{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -{a}^{2}{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +2\,{a}^{2}y \left ( x \right ) -\sinh \left ( x \right ) =0} \]
Mathematica: cpu = 0.081010 (sec), leaf count = 95 \[ \left \{\left \{y(x)\to \frac {e^{-x} \left (3 a^2 e^{2 x}-a^2-3 e^{2 x}-12 e^x \sinh (x)-6 e^x \cosh (x)+1\right )}{6 (a-2) (a+2) \left (a^2-1\right )}+c_1 e^{-a x}+c_3 e^{a x}+c_2 e^{2 x}\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 467 \[ \left \{ y \left ( x \right ) =-{\frac {6\,\cosh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{ax}}-6\,\cosh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{-ax}}-6\,\sinh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{ax}}-6\,\sinh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{-ax}}-3\,\cosh \left ( \left ( a-1 \right ) x \right ) {{\rm e} ^{-ax}}a+3\,\sinh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{ax}}a +9\,\sinh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{-ax}}a+6\,{ {\rm e}^{2\,x}}\cosh \left ( x \right ) a-3\,\sinh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{-ax}}a-3\,\cosh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{ax}}a+9\,\cosh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{-ax}}a+9\,\cosh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{ax}}a-6\,\sinh \left ( x \right ) {{\rm e}^ {2\,x}}a+2\,\sinh \left ( 3\,x \right ) {{\rm e}^{2\,x}}a-2\,\cosh \left ( 3\,x \right ) {{\rm e}^{2\,x}}a-9\,\sinh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{ax}}a-2\,\sinh \left ( 3\,x \right ) { {\rm e}^{2\,x}}{a}^{3}+2\,\cosh \left ( 3\,x \right ) {{\rm e}^{2\,x}}{a }^{3}-6\,{{\rm e}^{2\,x}}\cosh \left ( x \right ) {a}^{3}-3\,\sinh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{ax}}{a}^{2}+3\,\sinh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{-ax}}{a}^{2}-3\,\cosh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{ax}}{a}^{2}-3\,\cosh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{-ax}}{a}^{2}+3\,\cosh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{ax}}{a}^{2}+3\,\cosh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{-ax}}{a}^{2}+3\,\sinh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{ax}}{a}^{2}-3\,\sinh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{-ax}}{a}^{2}+6\,\sinh \left ( x \right ) {{\rm e}^{2\,x}}{a}^{3}-6\,\sinh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{ax}}-6\,\sinh \left ( \left ( a-1 \right ) x \right ) {{\rm e}^{-ax}}+6\,\cosh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{ax}}-6\,\cosh \left ( \left ( a+1 \right ) x \right ) {{\rm e}^{-ax}}}{12\,a \left ( {a}^{2}-4 \right ) \left ( a-1 \right ) \left ( a+1 \right ) }}+{\it \_C1}\,{{\rm e}^{2\,x}}+{\it \_C2}\,{ {\rm e}^{ax}}+{\it \_C3}\,{{\rm e}^{-ax}} \right \} \]