\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-2\,{\frac {n\cosh \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{\sinh \left ( x \right ) }}- \left ( -{a}^{2}+{n}^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.841107 (sec), leaf count = 273 \[ \left \{\left \{y(x)\to \frac {c_2 (-1)^{\frac {1}{2} (-2 n-1)+1} \tanh ^2(x)^{\frac {1}{4} (-2 n-1)+1} \left (\tanh ^2(x)-1\right )^{\frac {1}{2} \left (\frac {a+n}{2}+\frac {1}{2} (a+n+1)+\frac {1}{2} (-2 n-1)+1\right )} \, _2F_1\left (\frac {1}{2} (-2 n-1)+\frac {a+n}{2}+1,\frac {1}{2} (-2 n-1)+\frac {1}{2} (a+n+1)+1;\frac {1}{2} (-2 n-1)+2;\tanh ^2(x)\right ) \exp \left (\frac {1}{2} (n-1) \log \left (1-\tanh ^2(x)\right )-n \log (\tanh (x))\right )}{\sqrt {\tanh (x)}}+\frac {c_1 \tanh ^2(x)^{\frac {1}{4} (2 n+1)} \left (\tanh ^2(x)-1\right )^{\frac {1}{2} \left (\frac {a+n}{2}+\frac {1}{2} (a+n+1)+\frac {1}{2} (-2 n-1)+1\right )} \, _2F_1\left (\frac {a+n}{2},\frac {1}{2} (a+n+1);\frac {1}{2} (2 n+1);\tanh ^2(x)\right ) \exp \left (\frac {1}{2} (n-1) \log \left (1-\tanh ^2(x)\right )-n \log (\tanh (x))\right )}{\sqrt {\tanh (x)}}\right \}\right \} \]
Maple: cpu = 0.109 (sec), leaf count = 43 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \sinh \left ( x \right ) \right ) ^{-n+{\frac {1}{2}}}{\it LegendreP} \left ( a-{\frac {1}{2}},n-{\frac {1}{2}},\cosh \left ( x \right ) \right ) +{\it \_C2}\, \left ( \sinh \left ( x \right ) \right ) ^{-n+{\frac {1}{2}}}{\it LegendreQ} \left ( a-{\frac {1}{2}},n-{\frac {1}{2}},\cosh \left ( x \right ) \right ) \right \} \]