\[ a^4 y(x)+2 a^2 y''(x)-\cosh (a x)+y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.27041 (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 = 0.526 (sec), leaf count = 46
\[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{ax}}+{{\rm e}^{-ax}}}{8\,{a}^{4}}}+{\it \_C1}\,\cos \left ( ax \right ) +{\it \_C2}\,\sin \left ( ax \right ) +{\it \_C3}\,x\cos \left ( ax \right ) +{\it \_C4}\,x\sin \left ( ax \right ) \right \} \]