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 = 0.353484 (sec), leaf count = 33
\[\left \{\left \{y(x)\to -x \cot (c_1)+\csc ^2(c_1) \log (-x \sin (c_1)-\cos (c_1))+c_2\right \}\right \}\]
Maple ✓
cpu = 0.911 (sec), leaf count = 27
\[\left [y \left (x \right ) = \frac {x}{\textit {\_C1}}+\ln \left (\textit {\_C1} x -1\right )+\frac {\ln \left (\textit {\_C1} x -1\right )}{\textit {\_C1}^{2}}+\textit {\_C2}\right ]\] 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))
Maple raw output
[y(x) = x/_C1+ln(_C1*x-1)+1/_C1^2*ln(_C1*x-1)+_C2]