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 = 0.0212163, antiderivative size = 33, normalized size of antiderivative = 1., 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 444
Rule 63
Rule 212
Rule 206
Rule 203
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*}
Mathematica [A] time = 0.0057361, size = 33, normalized size = 1. \[ -\frac{1}{3} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{1}{3} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57283, size = 55, normalized size = 1.67 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57698, size = 132, normalized size = 4. \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.23762, size = 42, normalized size = 1.27 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18501, size = 57, normalized size = 1.73 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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