3.845 \(\int \frac{1}{\sqrt{-15-8 x-x^2}} \, dx\)

Optimal. Leaf size=4 \[ \sin ^{-1}(x+4) \]

[Out]

ArcSin[4 + x]

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Rubi [A]  time = 0.0051104, antiderivative size = 4, 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} \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-15 - 8*x - x^2],x]

[Out]

ArcSin[4 + x]

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{-15-8 x-x^2}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4}}} \, dx,x,-8-2 x\right )\right )\\ &=\sin ^{-1}(4+x)\\ \end{align*}

Mathematica [A]  time = 0.0055258, size = 4, normalized size = 1. \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-15 - 8*x - x^2],x]

[Out]

ArcSin[4 + x]

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Maple [A]  time = 0.003, size = 5, normalized size = 1.3 \begin{align*} \arcsin \left ( 4+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2-8*x-15)^(1/2),x)

[Out]

arcsin(4+x)

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Maxima [B]  time = 1.89915, size = 11, normalized size = 2.75 \begin{align*} -\arcsin \left (-x - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-8*x-15)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-x - 4)

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Fricas [B]  time = 1.50147, size = 77, normalized size = 19.25 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 8 \, x - 15}{\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-8*x-15)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt(-x^2 - 8*x - 15)*(x + 4)/(x^2 + 8*x + 15))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x^{2} - 8 x - 15}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2-8*x-15)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 - 8*x - 15), x)

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Giac [A]  time = 1.13153, size = 5, normalized size = 1.25 \begin{align*} \arcsin \left (x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-8*x-15)^(1/2),x, algorithm="giac")

[Out]

arcsin(x + 4)