##### 4.12.45 $$\sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3$$

ODE
$\sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.38074 (sec), leaf count = 63

$\left \{\left \{y(x)\to -\frac {1+\sqrt {-2 \sinh ^{-1}(x)+1-2 c_1}}{2 \left (\sinh ^{-1}(x)+c_1\right )}\right \},\left \{y(x)\to \frac {-1+\sqrt {-2 \sinh ^{-1}(x)+1-2 c_1}}{2 \left (\sinh ^{-1}(x)+c_1\right )}\right \}\right \}$

Maple
cpu = 0.028 (sec), leaf count = 49

$\left [y \left (x \right ) = \frac {-1+\sqrt {1-2 \textit {\_C1} -2 \arcsinh \left (x \right )}}{2 \arcsinh \left (x \right )+2 \textit {\_C1}}, y \left (x \right ) = -\frac {1+\sqrt {1-2 \textit {\_C1} -2 \arcsinh \left (x \right )}}{2 \left (\arcsinh \left (x \right )+\textit {\_C1} \right )}\right ]$ Mathematica raw input

DSolve[Sqrt[1 + x^2]*(1 + y[x])*y'[x] == y[x]^3,y[x],x]

Mathematica raw output

{{y[x] -> -1/2*(1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(ArcSinh[x] + C[1])}, {y[x]
 -> (-1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(2*(ArcSinh[x] + C[1]))}}

Maple raw input

dsolve((1+y(x))*diff(y(x),x)*(x^2+1)^(1/2) = y(x)^3, y(x))

Maple raw output

[y(x) = 1/2*(-1+(1-2*_C1-2*arcsinh(x))^(1/2))/(arcsinh(x)+_C1), y(x) = -1/2*(1+(
1-2*_C1-2*arcsinh(x))^(1/2))/(arcsinh(x)+_C1)]