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

Optimal. Leaf size=16 \[ 2 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+4 x}}\right ) \]

[Out]

2*ArcTanh[x/Sqrt[4*x + x^2]]

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Rubi [A]  time = 0.0038626, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {620, 206} \[ 2 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+4 x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4*x + x^2],x]

[Out]

2*ArcTanh[x/Sqrt[4*x + x^2]]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

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

Rubi steps

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

Mathematica [B]  time = 0.0064175, size = 33, normalized size = 2.06 \[ \frac{2 \sqrt{x} \sqrt{x+4} \sinh ^{-1}\left (\frac{\sqrt{x}}{2}\right )}{\sqrt{x (x+4)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4*x + x^2],x]

[Out]

(2*Sqrt[x]*Sqrt[4 + x]*ArcSinh[Sqrt[x]/2])/Sqrt[x*(4 + x)]

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Maple [A]  time = 0.052, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( x+2+\sqrt{{x}^{2}+4\,x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x+2+(x^2+4*x)^(1/2))

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Maxima [A]  time = 2.49606, size = 23, normalized size = 1.44 \begin{align*} \log \left (2 \, x + 2 \, \sqrt{x^{2} + 4 \, x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x + 2*sqrt(x^2 + 4*x) + 4)

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Fricas [A]  time = 2.28133, size = 43, normalized size = 2.69 \begin{align*} -\log \left (-x + \sqrt{x^{2} + 4 \, x} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 + 4*x) - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x**2 + 4*x), x)

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Giac [A]  time = 1.85389, size = 24, normalized size = 1.5 \begin{align*} -\log \left ({\left | -x + \sqrt{x^{2} + 4 \, x} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-x + sqrt(x^2 + 4*x) - 2))