\[ y''(x)=-\frac {\left (a x^2+a-3\right ) y(x)}{4 \left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0140124 (sec), leaf count = 78
\[\left \{\left \{y(x)\to c_1 \sqrt {x^2+1} P_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)+c_2 \sqrt {x^2+1} Q_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)\right \}\right \}\] ✓ Maple : cpu = 0.064 (sec), leaf count = 55
\[\left \{y \left (x \right ) = \left (x^{2}+1\right )^{\frac {1}{4}} \left (c_{1} \left (x +\sqrt {x^{2}+1}\right )^{\frac {\sqrt {-a +1}}{2}}+c_{2} \left (x +\sqrt {x^{2}+1}\right )^{-\frac {\sqrt {-a +1}}{2}}\right )\right \}\]