\[ y''(x)=-\frac {y(x) \left (-a^2 \left (x^2-1\right )^2-m^2-n (n+1) \left (x^2-1\right )\right )}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.199011 (sec), leaf count = 113
DSolve[Derivative[2][y][x] == -(((-m^2 - n*(1 + n)*(-1 + x^2) - a^2*(-1 + x^2)^2)*y[x])/(-1 + x^2)^2) - (2*x*Derivative[1][y][x])/(-1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2-1\right )^{m/2} \text {HeunC}\left [\frac {1}{4} \left (-a^2-m (m+1)+n^2+n\right ),-\frac {a^2}{4},\frac {1}{2},m+1,0,x^2\right ]+c_2 x \left (x^2-1\right )^{m/2} \text {HeunC}\left [\frac {1}{4} \left (-a^2-(m-n+1) (m+n+2)\right ),-\frac {a^2}{4},\frac {3}{2},m+1,0,x^2\right ]\right \}\right \}\] ✓ Maple : cpu = 0.174 (sec), leaf count = 84
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2*(x^2-1)^2-n*(n+1)*(x^2-1)-m^2)/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = \left (x^{2}-1\right )^{\frac {m}{2}} \left (\HeunC \left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , x^{2}\right ) c_{2} x +\HeunC \left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , x^{2}\right ) c_{1}\right )\]