\[ v (v+1) y(x)+y''(x)+\cot (x) y'(x)=0 \] ✓ Mathematica : cpu = 0.112921 (sec), leaf count = 20
DSolve[v*(1 + v)*y[x] + Cot[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
\[\{\{y(x)\to c_1 P_v(\cos (x))+c_2 Q_v(\cos (x))\}\}\] ✓ Maple : cpu = 0.196 (sec), leaf count = 45
dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*cot(x)+v*(v+1)*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} \hypergeom \left (\left [-\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {1}{2}\right ], \cos ^{2}\left (x \right )\right )+c_{2} \cos \left (x \right ) \hypergeom \left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {3}{2}\right ], \cos ^{2}\left (x \right )\right )\]