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

Optimal. Leaf size=10 \[ \sin ^{-1}\left (\frac {1+x}{\sqrt {2}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222} \begin {gather*} \sin ^{-1}\left (\frac {x+1}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\sqrt {1-2 x-x^2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{8}}} \, dx,x,-2-2 x\right )}{2 \sqrt {2}}\\ &=\sin ^{-1}\left (\frac {1+x}{\sqrt {2}}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(10)=20\).
time = 0.05, size = 23, normalized size = 2.30 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-2 x-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - 2*x - x^2],x]

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/Sqrt[1 - 2*x - x^2],x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.12, size = 10, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.36, size = 11, normalized size = 1.10 \begin {gather*} -\arcsin \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(-1/2*sqrt(2)*(x + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).
time = 0.34, size = 21, normalized size = 2.10 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} - 2 x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-x**2 - 2*x + 1), x)

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Giac [A]
time = 0.00, size = 11, normalized size = 1.10 \begin {gather*} \arcsin \left (\frac {x+1}{\sqrt {2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/2*sqrt(2)*(x + 1))

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Mupad [B]
time = 0.09, size = 11, normalized size = 1.10 \begin {gather*} \mathrm {asin}\left (\frac {\sqrt {8}\,\left (2\,x+2\right )}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - x^2 - 2*x)^(1/2),x)

[Out]

asin((8^(1/2)*(2*x + 2))/8)

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