\[ y''(x)=-\left (n^2-a^2\right ) y(x)-2 n \coth (x) y'(x) \] ✓ Mathematica : cpu = 0.857674 (sec), leaf count = 145
\[\left \{\left \{y(x)\to (-1)^{-n} \left (-\text {sech}^2(x)\right )^{\frac {a+1}{2}} \tanh ^{-n-\frac {1}{2}}(x) \tanh ^2(x)^{-\frac {n}{2}-\frac {1}{4}} \text {sech}^2(x)^{\frac {n-1}{2}} \left (c_1 (-1)^n \tanh ^2(x)^{n+\frac {1}{2}} \, _2F_1\left (\frac {a+n}{2},\frac {1}{2} (a+n+1);n+\frac {1}{2};\tanh ^2(x)\right )+i c_2 \tanh ^2(x) \, _2F_1\left (\frac {1}{2} (a-n+1),\frac {1}{2} (a-n+2);\frac {3}{2}-n;\tanh ^2(x)\right )\right )\right \}\right \}\]
✓ Maple : cpu = 0.145 (sec), leaf count = 36
\[ \left \{ y \left ( x \right ) = \left ( \sinh \left ( x \right ) \right ) ^{-n+{\frac {1}{2}}} \left ( {\it LegendreQ} \left ( a-{\frac {1}{2}},n-{\frac {1}{2}},\cosh \left ( x \right ) \right ) {\it \_C2}+{\it LegendreP} \left ( a-{\frac {1}{2}},n-{\frac {1}{2}},\cosh \left ( x \right ) \right ) {\it \_C1} \right ) \right \} \]