\[ \boxed { {\it d4y} \left ( x \right ) +2\,{a}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +{a}^{4}y \left ( x \right ) -\cosh \left ( ax \right ) =0} \]
Mathematica: cpu = 0.247531 (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 = 0.250 (sec), leaf count = 46 \[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{ax}}+{{\rm e}^{-ax}}}{8 \,{a}^{4}}}+{\it \_C1}\,\cos \left ( ax \right ) +{\it \_C2}\,\sin \left ( ax \right ) +{\it \_C3}\,\cos \left ( ax \right ) x+{\it \_C4}\, \sin \left ( ax \right ) x \right \} \]