\[ a^2 y(x)^2 \left (\log ^2(y(x))-1\right )+y'(x)^2=0 \] ✓ Mathematica : cpu = 0.291367 (sec), leaf count = 185
DSolve[a^2*(-1 + Log[y[x]]^2)*y[x]^2 + Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\left \{y(x)\to \exp \left (-\frac {1}{2} \sqrt {-e^{2 i a x-2 c_1}-e^{2 c_1-2 i a x}+2}\right )\right \},\left \{y(x)\to \exp \left (\frac {1}{2} \sqrt {-e^{2 i a x-2 c_1}-e^{2 c_1-2 i a x}+2}\right )\right \},\left \{y(x)\to \exp \left (-\frac {1}{2} \sqrt {-e^{-2 i a x-2 c_1}-e^{2 i a x+2 c_1}+2}\right )\right \},\left \{y(x)\to \exp \left (\frac {1}{2} \sqrt {-e^{-2 i a x-2 c_1}-e^{2 i a x+2 c_1}+2}\right )\right \}\right \}\] ✓ Maple : cpu = 0.226 (sec), leaf count = 49
dsolve(diff(y(x),x)^2+a^2*y(x)^2*(ln(y(x))^2-1) = 0,y(x))
\[y \left (x \right ) = {\mathrm e}^{\RootOf \left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )}\]