\[ a^4 y(x)+2 a^2 y''(x)-\cosh (a x)+y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.181197 (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 = 0.725 (sec), leaf count = 49
\[\left \{y \left (x \right ) = \frac {8 \left (\left (c_{3} x +c_{1}\right ) \cos \left (a x \right )+\left (c_{4} x +c_{2}\right ) \sin \left (a x \right )\right ) a^{4}+\left ({\mathrm e}^{2 a x}+1\right ) {\mathrm e}^{-a x}}{8 a^{4}}\right \}\]