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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]