\[ y''(x)=-\frac {a y(x)}{\left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0991388 (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 = 0.07 (sec), leaf count = 59
\[ \left \{ y \left ( x \right ) =\sqrt {{x}^{2}+1} \left ( \left ( {\frac {x+i}{-x+i}} \right ) ^{-{\frac {1}{2}\sqrt {a+1}}}{\it \_C2}+ \left ( {\frac {x+i}{-x+i}} \right ) ^{{\frac {1}{2}\sqrt {a+1}}}{\it \_C1} \right ) \right \} \]