3.137 \(\int \frac{1}{x \sqrt{3-3 x^2+x^4}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTanh[(Sqrt[3]*(2 - x^2))/(2*Sqrt[3 - 3*x^2 + x^4])]/(2*Sqrt[3])

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Rubi [A]  time = 0.057932, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (2-x^2\right )}{2 \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[3 - 3*x^2 + x^4]),x]

[Out]

-ArcTanh[(Sqrt[3]*(2 - x^2))/(2*Sqrt[3 - 3*x^2 + x^4])]/(2*Sqrt[3])

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Rubi in Sympy [A]  time = 8.57512, size = 36, normalized size = 0.9 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (- 3 x^{2} + 6\right )}{6 \sqrt{x^{4} - 3 x^{2} + 3}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(x**4-3*x**2+3)**(1/2),x)

[Out]

-sqrt(3)*atanh(sqrt(3)*(-3*x**2 + 6)/(6*sqrt(x**4 - 3*x**2 + 3)))/6

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Mathematica [A]  time = 0.0514968, size = 45, normalized size = 1.12 \[ \frac{\log \left (x^2\right )-\log \left (-3 x^2+2 \sqrt{3} \sqrt{x^4-3 x^2+3}+6\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[3 - 3*x^2 + x^4]),x]

[Out]

(Log[x^2] - Log[6 - 3*x^2 + 2*Sqrt[3]*Sqrt[3 - 3*x^2 + x^4]])/(2*Sqrt[3])

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Maple [A]  time = 0.005, size = 31, normalized size = 0.8 \[ -{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{ \left ( -3\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^4-3*x^2+3)^(1/2),x)

[Out]

-1/6*3^(1/2)*arctanh(1/6*(-3*x^2+6)*3^(1/2)/(x^4-3*x^2+3)^(1/2))

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Maxima [A]  time = 0.851726, size = 27, normalized size = 0.68 \[ -\frac{1}{6} \, \sqrt{3} \operatorname{arsinh}\left (-\sqrt{3} + \frac{2 \, \sqrt{3}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 - 3*x^2 + 3)*x),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*arcsinh(-sqrt(3) + 2*sqrt(3)/x^2)

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Fricas [A]  time = 0.287976, size = 111, normalized size = 2.78 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} - 3 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3} x^{2} - 3 \, x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 - 3*x^2 + 3)*x),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log((6*x^2 + sqrt(3)*(2*x^4 - 3*x^2 + 6) - 2*sqrt(x^4 - 3*x^2 + 3)*(
sqrt(3)*x^2 + 3))/(2*x^4 - 2*sqrt(x^4 - 3*x^2 + 3)*x^2 - 3*x^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{x^{4} - 3 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**4-3*x**2+3)**(1/2),x)

[Out]

Integral(1/(x*sqrt(x**4 - 3*x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} - 3 \, x^{2} + 3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 - 3*x^2 + 3)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^4 - 3*x^2 + 3)*x), x)