\[ y''(x)=y(x) \left (-\text {csch}^2(x)\right ) \left (-a^2 \sinh ^2(x)-(n-1) n\right ) \] ✓ Mathematica : cpu = 1.17538 (sec), leaf count = 127
\[\left \{\left \{y(x)\to \frac {(-1)^{-n} \left (-\text {sech}^2(x)\right )^{a/2} \tanh ^2(x)^{-\frac {n}{2}-\frac {1}{4}} \left (c_1 (-1)^n \tanh ^2(x)^{n+\frac {1}{2}} \, _2F_1\left (\frac {a+n}{2},\frac {1}{2} (a+n+1);n+\frac {1}{2};\tanh ^2(x)\right )+i c_2 \tanh ^2(x) \, _2F_1\left (\frac {1}{2} (a-n+1),\frac {1}{2} (a-n+2);\frac {3}{2}-n;\tanh ^2(x)\right )\right )}{\sqrt {\tanh (x)}}\right \}\right \}\]
✓ Maple : cpu = 0.267 (sec), leaf count = 97
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \sinh \left ( x \right ) \right ) ^{n}{\mbox {$_2$F$_1$}(-{\frac {a}{2}}+{\frac {n}{2}},{\frac {a}{2}}+{\frac {n}{2}};\,{\frac {1}{2}};\,{\frac {\cosh \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}+{{\it \_C2}\, \left ( \sinh \left ( x \right ) \right ) ^{n} \left ( 2\,\cosh \left ( 2\,x \right ) +2 \right ) ^{{\frac {3}{4}}}{\mbox {$_2$F$_1$}({\frac {1}{2}}+{\frac {a}{2}}+{\frac {n}{2}},{\frac {1}{2}}-{\frac {a}{2}}+{\frac {n}{2}};\,{\frac {3}{2}};\,{\frac {\cosh \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}\sqrt [4]{2\,\cosh \left ( 2\,x \right ) -2}{\frac {1}{\sqrt {\sinh \left ( 2\,x \right ) }}}} \right \} \]