\[ a^4 y(x)+2 a^2 y''(x)-\cosh (a x)+y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.1603 (sec), leaf count = 66
DSolve[-Cosh[a*x] + a^4*y[x] + 2*a^2*Derivative[2][y][x] + Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {\cos ^2(a x) \cosh (a x)+\sin ^2(a x) \cosh (a x)}{4 a^4}+c_1 \cos (a x)+c_2 x \cos (a x)+c_3 \sin (a x)+c_4 x \sin (a x)\right \}\right \}\] ✓ Maple : cpu = 0.649 (sec), leaf count = 49
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+2*a^2*diff(diff(y(x),x),x)+a^4*y(x)-cosh(a*x)=0,y(x))
\[y \left (x \right ) = \frac {\left (1+{\mathrm e}^{2 a x}\right ) {\mathrm e}^{-a x}+8 \left (\left (c_{3} x +c_{1}\right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (x c_{4}+c_{2}\right )\right ) a^{4}}{8 a^{4}}\]