\[ 2 a^2 y''(x)+a^4 y(x)-\cosh (a x)+y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.159916 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \frac {\cos ^2(a x) \cosh (a x)+\sin ^2(a x) \cosh (a x)}{4 a^4}+c_1 \cos (a x)+c_2 x \cos (a x)+c_3 \sin (a x)+c_4 x \sin (a x)\right \}\right \}\] ✓ Maple : cpu = 0.371 (sec), leaf count = 49
\[ \left \{ y \left ( x \right ) ={\frac { \left ( 1+{{\rm e}^{2\,ax}} \right ) {{\rm e}^{-ax}}+8\,{a}^{4} \left ( \left ( {\it \_C3}\,x+{\it \_C1} \right ) \cos \left ( ax \right ) +\sin \left ( ax \right ) \left ( {\it \_C4}\,x+{\it \_C2} \right ) \right ) }{8\,{a}^{4}}} \right \} \]