4.41.9 $(y(x{)}^2+1)y^{″}(x)+(1-2y(x))y^{\prime }(x{)}^2=0$

ODE
$$(y(x{)}^2+1)y^{″}(x)+(1-2y(x))y^{\prime }(x{)}^2=0$$ ODE Classification

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.08824 (sec), leaf count = 14

$$\left\{\{y(x)\to \tan (\log \left(c_1(c_2+x)\right))\}\right\}$$

Maple ✓
cpu = 0.025 (sec), leaf count = 15

$$\{{\mathrm{e}}^{\arctan \left(y\left(x\right)\right)}-\mathit{\_}\mathit{C}\mathit{1}x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[(1 - 2*y[x])*y'[x]^2 + (1 + y[x]^2)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Tan[Log[C[1]*(x + C[2])]]}}

Maple raw input

dsolve((1+y(x)^2)*diff(diff(y(x),x),x)+(1-2*y(x))*diff(y(x),x)^2 = 0, y(x),'implicit')

Maple raw output

exp(arctan(y(x)))-_C1*x-_C2 = 0