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

Optimal. Leaf size=27 \[ \sqrt{x^2+x+1}-\frac{1}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

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Rubi [A]  time = 0.0280356, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \sqrt{x^2+x+1}-\frac{1}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[1 + x + x^2],x]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

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Rubi in Sympy [A]  time = 1.50343, size = 29, normalized size = 1.07 \[ \sqrt{x^{2} + x + 1} - \frac{\operatorname{atanh}{\left (\frac{2 x + 1}{2 \sqrt{x^{2} + x + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**2+x+1)**(1/2),x)

[Out]

sqrt(x**2 + x + 1) - atanh((2*x + 1)/(2*sqrt(x**2 + x + 1)))/2

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Mathematica [A]  time = 0.013231, size = 27, normalized size = 1. \[ \sqrt{x^2+x+1}-\frac{1}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[1 + x + x^2],x]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

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Maple [A]  time = 0.008, size = 21, normalized size = 0.8 \[ \sqrt{{x}^{2}+x+1}-{\frac{1}{2}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+x+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49646, size = 30, normalized size = 1.11 \[ \sqrt{x^{2} + x + 1} - \frac{1}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^2 + x + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216753, size = 104, normalized size = 3.85 \[ -\frac{8 \, x^{2} - 2 \,{\left (2 \, x - 2 \, \sqrt{x^{2} + x + 1} + 1\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) - 2 \, \sqrt{x^{2} + x + 1}{\left (4 \, x + 1\right )} + 6 \, x + 7}{4 \,{\left (2 \, x - 2 \, \sqrt{x^{2} + x + 1} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^2 + x + 1),x, algorithm="fricas")

[Out]

-1/4*(8*x^2 - 2*(2*x - 2*sqrt(x^2 + x + 1) + 1)*log(-2*x + 2*sqrt(x^2 + x + 1) -
1) - 2*sqrt(x^2 + x + 1)*(4*x + 1) + 6*x + 7)/(2*x - 2*sqrt(x^2 + x + 1) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**2+x+1)**(1/2),x)

[Out]

Integral(x/sqrt(x**2 + x + 1), x)

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GIAC/XCAS [A]  time = 0.222296, size = 36, normalized size = 1.33 \[ \sqrt{x^{2} + x + 1} + \frac{1}{2} \,{\rm ln}\left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^2 + x + 1),x, algorithm="giac")

[Out]

sqrt(x^2 + x + 1) + 1/2*ln(-2*x + 2*sqrt(x^2 + x + 1) - 1)