3.2.54 \(\int \frac {1}{\sqrt {x+x^2}} \, dx\) [154]

Optimal. Leaf size=14 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {634, 212} \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x + x^2],x]

[Out]

2*ArcTanh[x/Sqrt[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 {x+x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right )\\ &=2 \tanh ^{-1}\left (\frac {x}{\sqrt {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. \(2(14)=28\).
time = 0.02, size = 37, normalized size = 2.64 \begin {gather*} \frac {2 \sqrt {x} \sqrt {1+x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x}}\right )}{\sqrt {x (1+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[x + x^2],x]

[Out]

(2*Sqrt[x]*Sqrt[1 + x]*ArcTanh[Sqrt[x]/Sqrt[1 + x]])/Sqrt[x*(1 + x)]

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Maple [A]
time = 0.06, size = 12, normalized size = 0.86

method	result	size
meijerg	\(2 \arcsinh \left (\sqrt {x}\right )\)	\(7\)
default	\(\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )\)	\(12\)
trager	\(-\ln \left (2 \sqrt {x^{2}+x}-1-2 x \right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 6.86, size = 15, normalized size = 1.07 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {x^{2} + x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x + 2*sqrt(x^2 + x) + 1)

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Fricas [A]
time = 0.52, size = 17, normalized size = 1.21 \begin {gather*} -\log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-2*x + 2*sqrt(x^2 + x) - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{2} + x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x**2 + x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).
time = 0.45, size = 33, normalized size = 2.36 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} + x} {\left (2 \, x + 1\right )} + \frac {1}{8} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(x^2 + x)*(2*x + 1) + 1/8*log(abs(-2*x + 2*sqrt(x^2 + x) - 1))

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Mupad [B]
time = 0.17, size = 11, normalized size = 0.79 \begin {gather*} \ln \left (x+\sqrt {x\,\left (x+1\right )}+\frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + x^2)^(1/2),x)

[Out]

log(x + (x*(x + 1))^(1/2) + 1/2)

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