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 = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 1904
Rule 1996
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*}
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Mathematica [A] time = 0.01, size = 73, normalized size = 1.62 \begin {gather*} -\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 )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 40, normalized size = 0.89 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{x^3-\sqrt {x^6-3 x^4+3 x^2}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 55, normalized size = 1.22 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 60, normalized size = 1.33 \begin {gather*} \frac {\sqrt {3} \log \left (x^{2} + \sqrt {3} - \sqrt {x^{4} - 3 \, x^{2} + 3}\right ) - \sqrt {3} \log \left (-x^{2} + \sqrt {3} + \sqrt {x^{4} - 3 \, x^{2} + 3}\right )}{6 \, \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 58, normalized size = 1.29 \begin {gather*} \frac {\sqrt {x^{4}-3 x^{2}+3}\, \sqrt {3}\, x \arctanh \left (\frac {\left (x^{2}-2\right ) \sqrt {3}}{2 \sqrt {x^{4}-3 x^{2}+3}}\right )}{6 \sqrt {\left (x^{4}-3 x^{2}+3\right ) x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {{\left (x^{4} - 3 \, x^{2} + 3\right )} x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2\,\left (x^4-3\,x^2+3\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x^{2} \left (x^{4} - 3 x^{2} + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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