\[ y''(x)=-\frac {a y(x)}{\left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0659549 (sec), leaf count = 104
DSolve[Derivative[2][y][x] == -((a*y[x])/(1 + x^2)^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {i c_2 \sqrt {x^2+1} (1-i x)^{\sqrt {a+1}} (1+i x)^{-\sqrt {a+1}} e^{i \sqrt {a+1} \tan ^{-1}(x)}}{2 \sqrt {a+1}}+c_1 \sqrt {x^2+1} e^{i \sqrt {a+1} \tan ^{-1}(x)}\right \}\right \}\] ✓ Maple : cpu = 0.062 (sec), leaf count = 59
dsolve(diff(diff(y(x),x),x) = -a/(x^2+1)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {x^{2}+1}\, \left (\left (\frac {x +i}{-x +i}\right )^{-\frac {\sqrt {a +1}}{2}} c_{2}+\left (\frac {x +i}{-x +i}\right )^{\frac {\sqrt {a +1}}{2}} c_{1}\right )\]