\[ y''(x)=-\frac {a y(x)}{\left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.101317 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \frac {1}{2} \sqrt {x^2+1} e^{-i \sqrt {a+1} \tan ^{-1}(x)} \left (2 c_1 e^{2 i \sqrt {a+1} \tan ^{-1}(x)}+\frac {i c_2}{\sqrt {a+1}}\right )\right \}\right \}\]
✓ Maple : cpu = 0.077 (sec), leaf count = 59
\[ \left \{ y \left ( x \right ) =\sqrt {{x}^{2}+1} \left ( \left ( {\frac {x+i}{-x+i}} \right ) ^{-{\frac {1}{2}\sqrt {a+1}}}{\it \_C2}+ \left ( {\frac {x+i}{-x+i}} \right ) ^{{\frac {1}{2}\sqrt {a+1}}}{\it \_C1} \right ) \right \} \]