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

Optimal. Leaf size=42 \[ -\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0853295, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.97279, size = 39, normalized size = 0.93 \[ - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3 \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x + 1)*sqrt(x**2 - x + 1)*atanh(sqrt(x**3 + 1))/(3*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.405498, size = 68, normalized size = 1.62 \[ -\frac{2 \sqrt{x+1} \Pi \left (1+\sqrt [3]{-1};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3} \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]

[Out]

(-2*Sqrt[1 + x]*EllipticPi[1 + (-1)^(1/3), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (
-1)^(1/3))]], (-1)^(1/3)])/(Sqrt[3]*Sqrt[(1 + x)/(1 + (-1)^(1/3))])

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Maple [A]  time = 0.033, size = 33, normalized size = 0.8 \[ -{\frac{2}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{1+x}\sqrt{{x}^{2}-x+1}{\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(1+x)^(1/2)/(x^2-x+1)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^2 - x + 1)*sqrt(x + 1)*x), x)

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Fricas [A]  time = 0.285613, size = 58, normalized size = 1.38 \[ -\frac{1}{3} \, \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(x + 1)*sqrt(x**2 - x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - x + 1} \sqrt{x + 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^2 - x + 1)*sqrt(x + 1)*x), x)