\[ y''(x)=-\frac {a y(x)}{\left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0773902 (sec), leaf count = 104
\[\left \{\left \{y(x)\to \frac {i c_2 \sqrt {x^2+1} (1-i x)^{\sqrt {a+1}} (1+i x)^{-\sqrt {a+1}} e^{i \sqrt {a+1} \tan ^{-1}(x)}}{2 \sqrt {a+1}}+c_1 \sqrt {x^2+1} e^{i \sqrt {a+1} \tan ^{-1}(x)}\right \}\right \}\] ✓ Maple : cpu = 0.137 (sec), leaf count = 59
\[ \left \{ y \left ( x \right ) =\sqrt {{x}^{2}+1} \left ( \left ( {\frac {x+i}{-x+i}} \right ) ^{-{\frac {1}{2}\sqrt {a+1}}}{\it \_C2}+ \left ( {\frac {x+i}{-x+i}} \right ) ^{{\frac {1}{2}\sqrt {a+1}}}{\it \_C1} \right ) \right \} \]