3.295 \(\int \frac{1}{\sqrt{5-4 x-x^2}} \, dx\)

Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

[Out]

-ArcSin[(-2 - x)/3]

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Rubi [A]  time = 0.0066545, antiderivative size = 12, normalized size of antiderivative = 1., 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 = {619, 216} \[ -\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[5 - 4*x - x^2],x]

[Out]

-ArcSin[(-2 - x)/3]

Rule 619

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]

Rule 216

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{5-4 x-x^2}} \, dx &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,-4-2 x\right )\right )\\ &=-\sin ^{-1}\left (\frac{1}{3} (-2-x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0060933, size = 12, normalized size = 1. \[ -\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[5 - 4*x - x^2],x]

[Out]

-ArcSin[(-2 - x)/3]

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Maple [A]  time = 0.003, size = 7, normalized size = 0.6 \begin{align*} \arcsin \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2-4*x+5)^(1/2),x)

[Out]

arcsin(2/3+1/3*x)

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Maxima [A]  time = 1.40417, size = 11, normalized size = 0.92 \begin{align*} -\arcsin \left (-\frac{1}{3} \, x - \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-4*x+5)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-1/3*x - 2/3)

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Fricas [B]  time = 2.10494, size = 74, normalized size = 6.17 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-4*x+5)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt(-x^2 - 4*x + 5)*(x + 2)/(x^2 + 4*x - 5))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x^{2} - 4 x + 5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2-4*x+5)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 - 4*x + 5), x)

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Giac [A]  time = 1.06393, size = 8, normalized size = 0.67 \begin{align*} \arcsin \left (\frac{1}{3} \, x + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-4*x+5)^(1/2),x, algorithm="giac")

[Out]

arcsin(1/3*x + 2/3)