3.1.28 \(\int \frac {1}{\sqrt {4 x+x^2}} \, dx\) [28]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {634, 212} \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+4 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4*x + x^2],x]

[Out]

2*ArcTanh[x/Sqrt[4*x + x^2]]

Rule 212

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

Rule 634

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {4 x+x^2}} \, dx &=2 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(16)=32\).
time = 0.03, size = 37, normalized size = 2.31 \begin {gather*} \frac {2 \sqrt {x} \sqrt {4+x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {4+x}}\right )}{\sqrt {x (4+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4*x + x^2],x]

[Out]

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

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Maple [A]
time = 0.37, size = 14, normalized size = 0.88

method	result	size
meijerg	\(2 \arcsinh \left (\frac {\sqrt {x}}{2}\right )\)	\(9\)
default	\(\ln \left (2+x +\sqrt {x^{2}+4 x}\right )\)	\(14\)
trager	\(\ln \left (2+x +\sqrt {x^{2}+4 x}\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 17, normalized size = 1.06 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {x^{2} + 4 \, x} + 4\right ) \end {gather*}

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 = 1.45, size = 17, normalized size = 1.06 \begin {gather*} -\log \left (-x + \sqrt {x^{2} + 4 \, x} - 2\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{2} + 4 x}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
time = 1.97, size = 33, normalized size = 2.06 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 4 \, x} {\left (x + 2\right )} + 2 \, \log \left ({\left | -x + \sqrt {x^{2} + 4 \, x} - 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^2 + 4*x)*(x + 2) + 2*log(abs(-x + sqrt(x^2 + 4*x) - 2))

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Mupad [B]
time = 0.51, size = 11, normalized size = 0.69 \begin {gather*} \ln \left (x+\sqrt {x\,\left (x+4\right )}+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x + x^2)^(1/2),x)

[Out]

log(x + (x*(x + 4))^(1/2) + 2)

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