\[ y''(x)=-\frac {a y(x)}{\left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.109272 (sec), leaf count = 72
\[\left \{\left \{y(x)\to c_1 \sqrt {x^2+1} e^{i \sqrt {a+1} \tan ^{-1}(x)}+\frac {i c_2 \sqrt {x^2+1} e^{-i \sqrt {a+1} \tan ^{-1}(x)}}{2 \sqrt {a+1}}\right \}\right \}\]
✓ Maple : cpu = 0.061 (sec), leaf count = 65
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sqrt {{x}^{2}+1} \left ( {\frac {x+i}{-x+i}} \right ) ^{{\frac {1}{2}\sqrt {a+1}}}+{\it \_C2}\,\sqrt {{x}^{2}+1} \left ( {\frac {x+i}{-x+i}} \right ) ^{-{\frac {1}{2}\sqrt {a+1}}} \right \} \]