Optimal. Leaf size=127 \[ -\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right ),\frac{1}{2}\right )}{2 \sqrt{3} x}+\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}} \]
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Rubi [A] time = 0.0481679, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {400, 234, 220, 442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt{6}}-\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 \sqrt{3} x} \]
Antiderivative was successfully verified.
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Rule 400
Rule 234
Rule 220
Rule 442
Rubi steps
\begin{align*} \int \frac{1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=-\left (\frac{1}{2} \int \frac{1}{\left (-1+3 x^2\right )^{3/4}} \, dx\right )+\frac{3}{2} \int \frac{x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, 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}}-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{\sqrt{3} x}\\ &=\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}}-\frac{\sqrt{\frac{x^2}{\left (1+\sqrt{-1+3 x^2}\right )^2}} \left (1+\sqrt{-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac{1}{2}\right )}{2 \sqrt{3} x}\\ \end{align*}
Mathematica [C] time = 0.0153465, size = 68, normalized size = 0.54 \[ \frac{\sqrt [4]{-1} \sqrt{x^2} \left (\Pi \left (-i;\left .\sin ^{-1}\left ((-1)^{3/4} \sqrt [4]{3 x^2-1}\right )\right |-1\right )+\Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} \sqrt [4]{3 x^2-1}\right )\right |-1\right )\right )}{\sqrt{3} x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{9 \, x^{4} - 9 \, x^{2} + 2}, x\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 ) \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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{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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