3.145 \(\int \frac{1}{\sqrt{8+4 x+x^2}} \, dx\)

Optimal. Leaf size=8 \[ \sinh ^{-1}\left (\frac{x+2}{2}\right ) \]

[Out]

ArcSinh[(2 + x)/2]

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Rubi [A]  time = 0.0062037, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {619, 215} \[ \sinh ^{-1}\left (\frac{x+2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[8 + 4*x + x^2],x]

[Out]

ArcSinh[(2 + x)/2]

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 215

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{8+4 x+x^2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{16}}} \, dx,x,4+2 x\right )\\ &=\sinh ^{-1}\left (\frac{2+x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0049086, size = 8, normalized size = 1. \[ \sinh ^{-1}\left (\frac{x+2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[8 + 4*x + x^2],x]

[Out]

ArcSinh[(2 + x)/2]

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Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \begin{align*}{\it Arcsinh} \left ( 1+{\frac{x}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+4*x+8)^(1/2),x)

[Out]

arcsinh(1+1/2*x)

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Maxima [A]  time = 1.40643, size = 8, normalized size = 1. \begin{align*} \operatorname{arsinh}\left (\frac{1}{2} \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+4*x+8)^(1/2),x, algorithm="maxima")

[Out]

arcsinh(1/2*x + 1)

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Fricas [B]  time = 2.10724, size = 49, normalized size = 6.12 \begin{align*} -\log \left (-x + \sqrt{x^{2} + 4 \, x + 8} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+4*x+8)^(1/2),x, algorithm="fricas")

[Out]

-log(-x + sqrt(x^2 + 4*x + 8) - 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 4 x + 8}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+4*x+8)**(1/2),x)

[Out]

Integral(1/sqrt(x**2 + 4*x + 8), x)

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Giac [B]  time = 1.06002, size = 24, normalized size = 3. \begin{align*} -\log \left (-x + \sqrt{x^{2} + 4 \, x + 8} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+4*x+8)^(1/2),x, algorithm="giac")

[Out]

-log(-x + sqrt(x^2 + 4*x + 8) - 2)