ODE
\[ \left (x^2+1\right ) y''(x)+y'(x)^2+1=0 \] ODE Classification
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0798301 (sec), leaf count = 33
\[\left \{\left \{y(x)\to -x \cot \left (c_1\right )+\csc ^2\left (c_1\right ) \log \left (-x \sin \left (c_1\right )-\cos \left (c_1\right )\right )+c_2\right \}\right \}\]
Maple ✓
cpu = 0.224 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) ={\frac {x}{{\it \_C1}}}+\ln \left ( x{\it \_C1}-1 \right ) +{\frac {\ln \left ( x{\it \_C1}-1 \right ) }{{{\it \_C1}}^{2}}}+{\it \_C2} \right \} \] Mathematica raw input
DSolve[1 + y'[x]^2 + (1 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2] - x*Cot[C[1]] + Csc[C[1]]^2*Log[-Cos[C[1]] - x*Sin[C[1]]]}}
Maple raw input
dsolve((x^2+1)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1 = 0, y(x),'implicit')
Maple raw output
y(x) = 1/_C1*x+ln(_C1*x-1)+1/_C1^2*ln(_C1*x-1)+_C2