\[ a^4 y(x)+2 a^2 y''(x)-\cosh (a x)+y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.267774 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \frac {\cos ^2(a x) \cosh (a x)+\sin ^2(a x) \cosh (a x)}{4 a^4}+c_3 \sin (a x)+c_4 x \sin (a x)+c_1 \cos (a x)+c_2 x \cos (a x)\right \}\right \}\] ✓ Maple : cpu = 0.51 (sec), leaf count = 51
\[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{-ax}}+ \left ( 8\,{\it \_C3}\,x+8\,{\it \_C1} \right ) {a}^{4}\cos \left ( ax \right ) + \left ( 8\,{\it \_C4}\,x+8\,{\it \_C2} \right ) {a}^{4}\sin \left ( ax \right ) +{{\rm e}^{ax}}}{8\,{a}^{4}}} \right \} \]