Optimal. Leaf size=61 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}} \]
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Rubi [A] time = 0.0089125, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin{align*} \int \frac{1}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{2 \sqrt{6}}\\ \end{align*}
Mathematica [C] time = 0.0260957, size = 127, normalized size = 2.08 \[ \frac{2 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \sqrt [4]{3 x^2-1} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \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}{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 24.0225, size = 281, normalized size = 4.61 \begin{align*} \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{\sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \, x}\right ) + \frac{1}{24} \, \sqrt{6} \log \left (-\frac{9 \, x^{4} - 6 \, \sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} x^{3} + 12 \, \sqrt{3 \, x^{2} - 1} x^{2} - 4 \, \sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} x + 12 \, x^{2} - 4}{9 \, x^{4} - 12 \, x^{2} + 4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \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}{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}{\left (3 \, x^{2} - 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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