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

Optimal. Leaf size=45 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0248499, antiderivative size = 45, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} x \left (2-x^2\right )}{2 \sqrt{x^6-3 x^4+3 x^2}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.0183, size = 68, normalized size = 1.51 \[ - \frac{\sqrt{3} x \sqrt{x^{4} - 3 x^{2} + 3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (- 3 x^{2} + 6\right )}{6 \sqrt{x^{4} - 3 x^{2} + 3}} \right )}}{6 \sqrt{x^{6} - 3 x^{4} + 3 x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0420534, size = 78, normalized size = 1.73 \[ \frac{x \sqrt{x^4-3 x^2+3} \left (\log \left (x^2\right )-\log \left (-3 x^2+2 \sqrt{3} \sqrt{x^4-3 x^2+3}+6\right )\right )}{2 \sqrt{3} \sqrt{x^2 \left (x^4-3 x^2+3\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 58, normalized size = 1.3 \[{\frac{x\sqrt{3}}{6}\sqrt{{x}^{4}-3\,{x}^{2}+3}{\it Artanh} \left ({\frac{ \left ({x}^{2}-2 \right ) \sqrt{3}}{2}{\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \right ){\frac{1}{\sqrt{{x}^{2} \left ({x}^{4}-3\,{x}^{2}+3 \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.872848, size = 27, normalized size = 0.6 \[ -\frac{1}{6} \, \sqrt{3} \operatorname{arsinh}\left (-\sqrt{3} + \frac{2 \, \sqrt{3}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.290175, size = 124, normalized size = 2.76 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{6 \, x^{3} + \sqrt{3}{\left (2 \, x^{5} - 3 \, x^{3} + 6 \, x\right )} - 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{5} - 3 \, x^{3} - 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}} x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*log((6*x^3 + sqrt(3)*(2*x^5 - 3*x^3 + 6*x) - 2*sqrt(x^6 - 3*x^4 + 3*
x^2)*(sqrt(3)*x^2 + 3))/(2*x^5 - 3*x^3 - 2*sqrt(x^6 - 3*x^4 + 3*x^2)*x^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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