Optimal. Leaf size=40 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (2-x^2\right )}{2 \sqrt {3-3 x^2+x^4}}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, 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 = {1128, 738, 212}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (2-x^2\right )}{2 \sqrt {x^4-3 x^2+3}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 1128
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {3-3 x^2+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {3-3 x+x^2}} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {3 \left (2-x^2\right )}{\sqrt {3-3 x^2+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (2-x^2\right )}{2 \sqrt {3-3 x^2+x^4}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 33, normalized size = 0.82 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {x^2-\sqrt {3-3 x^2+x^4}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 31, normalized size = 0.78
method | result | size |
default | \(-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (-3 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}-3 x^{2}+3}}\right )}{6}\) | \(31\) |
elliptic | \(-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (-3 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}-3 x^{2}+3}}\right )}{6}\) | \(31\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+2 \sqrt {x^{4}-3 x^{2}+3}+2 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{6}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 20, normalized size = 0.50 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \operatorname {arsinh}\left (-\sqrt {3} + \frac {2 \, \sqrt {3}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 47, normalized size = 1.18 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-\frac {3 \, x^{2} + 2 \, \sqrt {3} {\left (x^{2} - 2\right )} + 2 \, \sqrt {x^{4} - 3 \, x^{2} + 3} {\left (\sqrt {3} + 2\right )} - 6}{x^{2}}\right ) \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}{x \sqrt {x^{4} - 3 x^{2} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.90, size = 55, normalized size = 1.38 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} - \sqrt {x^{4} - 3 \, x^{2} + 3}\right ) - \frac {1}{6} \, \sqrt {3} \log \left (-x^{2} + \sqrt {3} + \sqrt {x^{4} - 3 \, x^{2} + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 33, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {3}\,\left (\ln \left (x^2-\frac {2\,\sqrt {3}\,\sqrt {x^4-3\,x^2+3}}{3}-2\right )+\ln \left (\frac {1}{x^2}\right )\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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