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

Optimal. Leaf size=12 \[ \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

ArcSinh[(1 + 2*x)/Sqrt[3]]

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Rubi [A]  time = 0.0075392, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {619, 215} \[ \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x + x^2],x]

[Out]

ArcSinh[(1 + 2*x)/Sqrt[3]]

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{1+x+x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{3}}} \, dx,x,1+2 x\right )}{\sqrt{3}}\\ &=\sinh ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0055488, size = 12, normalized size = 1. \[ \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x + x^2],x]

[Out]

ArcSinh[(1 + 2*x)/Sqrt[3]]

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \begin{align*}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(2/3*3^(1/2)*(x+1/2))

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Maxima [A]  time = 1.45002, size = 15, normalized size = 1.25 \begin{align*} \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(1/3*sqrt(3)*(2*x + 1))

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Fricas [A]  time = 2.02633, size = 51, normalized size = 4.25 \begin{align*} -\log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-2*x + 2*sqrt(x^2 + x + 1) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x**2 + x + 1), x)

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Giac [A]  time = 1.07154, size = 24, normalized size = 2. \begin{align*} -\log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-2*x + 2*sqrt(x^2 + x + 1) - 1)