\[ y''(x)=-a^2 n y(x) \sec ^2(a x) \left ((n-1) \sin ^2(a x)+\cos ^2(a x)\right )-a (n-1) \sin (2 a x) \sec ^2(a x) y'(x) \] ✓ Mathematica : cpu = 0.245691 (sec), leaf count = 65
\[\left \{\left \{y(x)\to \frac {2^{-n} \left (2 a c_1-i c_2 e^{2 i a x}\right ) \left (e^{-i a x}+e^{i a x}\right )^n}{a \left (1+e^{2 i a x}\right )}\right \}\right \}\]
✓ Maple : cpu = 0.079 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \cos \left ( ax \right ) \right ) ^{n}+{\it \_C2}\,\sin \left ( ax \right ) \left ( \cos \left ( ax \right ) \right ) ^{n-1} \right \} \]