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

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

[Out]

2*ArcTanh[x/Sqrt[x + x^2]]

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Rubi [A]  time = 0.0031101, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {620, 206} \[ 2 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x + x^2],x]

[Out]

2*ArcTanh[x/Sqrt[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{x+x^2}} \, dx &=2 \operatorname{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*}

Mathematica [B]  time = 0.0046658, size = 29, normalized size = 2.07 \[ \frac{2 \sqrt{x} \sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x (x+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[x + x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \begin{align*} \ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.942062, size = 20, normalized size = 1.43 \begin{align*} \log \left (2 \, x + 2 \, \sqrt{x^{2} + x} + 1\right ) \end{align*}

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 = 1.23835, size = 46, normalized size = 3.29 \begin{align*} -\log \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \end{align*}

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

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 [A]  time = 1.11361, size = 24, normalized size = 1.71 \begin{align*} -\log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-2*x + 2*sqrt(x^2 + x) - 1))