\[ y''(x)=y(x) \left (-\text {csch}^2(x)\right ) \left ((1-n) n-a^2 \sinh ^2(x)\right ) \] ✓ Mathematica : cpu = 0.531367 (sec), leaf count = 231
DSolve[Derivative[2][y][x] == -(Csch[x]^2*((1 - n)*n - a^2*Sinh[x]^2)*y[x]),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 (-1)^{\frac {1}{2} (-2 n-1)+1} \tanh ^2(x)^{\frac {1}{4} (-2 n-1)+1} \left (\tanh ^2(x)-1\right )^{\frac {1}{2} \left (\frac {a+n}{2}+\frac {1}{2} (a+n+1)+\frac {1}{2} (-2 n-1)+1\right )-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 n-1)+\frac {a+n}{2}+1,\frac {1}{2} (-2 n-1)+\frac {1}{2} (a+n+1)+1,\frac {1}{2} (-2 n-1)+2,\tanh ^2(x)\right )}{\sqrt {\tanh (x)}}+\frac {c_1 \tanh ^2(x)^{\frac {1}{4} (2 n+1)} \left (\tanh ^2(x)-1\right )^{\frac {1}{2} \left (\frac {a+n}{2}+\frac {1}{2} (a+n+1)+\frac {1}{2} (-2 n-1)+1\right )-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {a+n}{2},\frac {1}{2} (a+n+1),\frac {1}{2} (2 n+1),\tanh ^2(x)\right )}{\sqrt {\tanh (x)}}\right \}\right \}\] ✓ Maple : cpu = 0.399 (sec), leaf count = 82
dsolve(diff(diff(y(x),x),x) = -(-a^2*sinh(x)^2-n*(n-1))/sinh(x)^2*y(x),y(x))
\[y \left (x \right ) = \frac {\sinh \left (x \right )^{n +\frac {1}{2}} \sqrt {\cosh \left (x \right )}\, \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {a}{2}+\frac {n}{2}, \frac {1}{2}+\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) \cosh \left (x \right ) c_{2}+\operatorname {hypergeom}\left (\left [\frac {a}{2}+\frac {n}{2}, -\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}\right )}{\sqrt {\sinh \left (2 x \right )}}\]