\(\int \frac {1}{\sqrt {1-2 x-x^2}} \, dx\) [88]

Optimal result

Integrand size = 14, antiderivative size = 10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\arcsin \left (\frac {1+x}{\sqrt {2}}\right ) \]

[Out]

arcsin(1/2*(1+x)*2^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222} \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\arcsin \left (\frac {x+1}{\sqrt {2}}\right ) \]

[In]

Int[1/Sqrt[1 - 2*x - x^2],x]

[Out]

ArcSin[(1 + x)/Sqrt[2]]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{8}}} \, dx,x,-2-2 x\right )}{2 \sqrt {2}} \\ & = \arcsin \left (\frac {1+x}{\sqrt {2}}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(10)=20\).

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=2 \arctan \left (\frac {x}{-1+\sqrt {1-2 x-x^2}}\right ) \]

[In]

Integrate[1/Sqrt[1 - 2*x - x^2],x]

[Out]

2*ArcTan[x/(-1 + Sqrt[1 - 2*x - x^2])]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

method	result	size
default	\(\arcsin \left (\frac {\left (1+x \right ) \sqrt {2}}{2}\right )\)	\(10\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-2 x +1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\)	\(39\)

[In]

int(1/(-x^2-2*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsin(1/2*(1+x)*2^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=-2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 1} - 1}{x}\right ) \]

[In]

integrate(1/(-x^2-2*x+1)^(1/2),x, algorithm="fricas")

[Out]

-2*arctan((sqrt(-x^2 - 2*x + 1) - 1)/x)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\operatorname {asin}{\left (\frac {\sqrt {2} \left (x + 1\right )}{2} \right )} \]

[In]

integrate(1/(-x**2-2*x+1)**(1/2),x)

[Out]

asin(sqrt(2)*(x + 1)/2)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=-\arcsin \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \]

[In]

integrate(1/(-x^2-2*x+1)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-1/2*sqrt(2)*(x + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (9) = 18\).

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.70 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 1} {\left (x + 1\right )} + \arcsin \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \]

[In]

integrate(1/(-x^2-2*x+1)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 - 2*x + 1)*(x + 1) + arcsin(1/2*sqrt(2)*(x + 1))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx=\mathrm {asin}\left (\frac {\sqrt {8}\,\left (2\,x+2\right )}{8}\right ) \]

[In]

int(1/(1 - x^2 - 2*x)^(1/2),x)

[Out]

asin((8^(1/2)*(2*x + 2))/8)