Optimal. Leaf size=45 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0122878, 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, Rules used = {1996, 1904, 206} \[ -\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{x^6-3 x^4+3 x^2}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1996
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x^2 \left (3-3 x^2+x^4\right )}} \, dx &=\int \frac{1}{\sqrt{3 x^2-3 x^4+x^6}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{x \left (6-3 x^2\right )}{\sqrt{3 x^2-3 x^4+x^6}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{x \left (6-3 x^2\right )}{2 \sqrt{3} \sqrt{3 x^2-3 x^4+x^6}}\right )}{2 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0033544, size = 73, normalized size = 1.62 \[ -\frac{x \sqrt{x^4-3 x^2+3} \tanh ^{-1}\left (\frac{6-3 x^2}{2 \sqrt{3} \sqrt{x^4-3 x^2+3}}\right )}{2 \sqrt{3} \sqrt{x^2 \left (x^4-3 x^2+3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 58, normalized size = 1.3 \begin{align*}{\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 ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (x^{4} - 3 \, x^{2} + 3\right )} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3697, size = 142, normalized size = 3.16 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{3 \, x^{3} + 2 \, \sqrt{3}{\left (x^{3} - 2 \, x\right )} + 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}{\left (\sqrt{3} + 2\right )} - 6 \, x}{x^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (x^{4} - 3 \, x^{2} + 3\right )} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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