ODE
\[ \left (y(x)^2+1\right ) y''(x)+y'(x) \left (y'(x)^2+1\right )=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.192275 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \csc \left (c_1\right ) \sec \left (c_1\right ) W\left (\sin \left (c_1\right ) e^{-\left (c_2+x\right ) \cos ^2\left (c_1\right )-\sin ^2\left (c_1\right )}\right )+\tan \left (c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.469 (sec), leaf count = 56
\[ \left \{ {\frac {-4\,{\it \_C1}\,\ln \left ( \left ( -1-{\it \_C1} \right ) y \left ( x \right ) +i{\it \_C1}-i \right ) - \left ( 1+{\it \_C1} \right ) \left ( \left ( i{\it \_C1}-i \right ) y \left ( x \right ) + \left ( 1+{\it \_C1} \right ) \left ( x+{\it \_C2} \right ) \right ) }{ \left ( 1+{\it \_C1} \right ) ^{2}}}=0,y \left ( x \right ) ={\it \_C1} \right \} \] Mathematica raw input
DSolve[y'[x]*(1 + y'[x]^2) + (1 + y[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Csc[C[1]]*ProductLog[E^(-((x + C[2])*Cos[C[1]]^2) - Sin[C[1]]^2)*Sin[C[1]]]*Sec[C[1]] + Tan[C[1]]}}
Maple raw input
dsolve((1+y(x)^2)*diff(diff(y(x),x),x)+diff(y(x),x)*(1+diff(y(x),x)^2) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1, (-4*_C1*ln((-1-_C1)*y(x)+I*_C1-I)-(1+_C1)*((I*_C1-I)*y(x)+(1+_C1)*(x+_C2)))/(1+_C1)^2 = 0