\[ y''(x)=-\frac {\left (a x^2+a-3\right ) y(x)}{4 \left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0211583 (sec), leaf count = 70
\[\left \{\left \{y(x)\to \sqrt {x^2+1} \left (c_1 P_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)+c_2 Q_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)\right )\right \}\right \}\]
✓ Maple : cpu = 0.074 (sec), leaf count = 55
\[ \left \{ y \left ( x \right ) =\sqrt [4]{{x}^{2}+1} \left ( \left ( x+\sqrt {{x}^{2}+1} \right ) ^{-{\frac {1}{2}\sqrt {1-a}}}{\it \_C2}+ \left ( x+\sqrt {{x}^{2}+1} \right ) ^{{\frac {1}{2}\sqrt {1-a}}}{\it \_C1} \right ) \right \} \]