Optimal. Leaf size=33 \[ -\frac {1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {444, 63, 212, 206, 203} \[ -\frac {1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 212
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right )\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\frac {1}{3} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ -\frac {1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 41, normalized size = 1.24 \[ -\frac {1}{3} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 42, normalized size = 1.27 \[ -\frac {1}{3} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.53, size = 125, normalized size = 3.79 \[ -\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \sqrt {3 x^{2}-1}}{3 x^{2}-2}\right )}{6}+\frac {\ln \left (\frac {-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.85, size = 41, normalized size = 1.24 \[ -\frac {1}{3} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 25, normalized size = 0.76 \[ -\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{3}-\frac {\mathrm {atanh}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.73, size = 42, normalized size = 1.27 \[ \frac {\log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{6} - \frac {\log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{6} - \frac {\operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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