Optimal. Leaf size=165 \[ -\frac{11 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right ),\frac{1}{2}\right )}{8 x}-\frac{2 \sqrt [4]{3 x^2-1}}{x}-\frac{\sqrt [4]{3 x^2-1}}{6 x^3}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right ) \]
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Rubi [A] time = 0.166966, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {443, 325, 234, 220, 400, 442} \[ -\frac{2 \sqrt [4]{3 x^2-1}}{x}-\frac{\sqrt [4]{3 x^2-1}}{6 x^3}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{11 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{8 x} \]
Antiderivative was successfully verified.
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Rule 443
Rule 325
Rule 234
Rule 220
Rule 400
Rule 442
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\int \left (-\frac{1}{2 x^4 \left (-1+3 x^2\right )^{3/4}}-\frac{3}{4 x^2 \left (-1+3 x^2\right )^{3/4}}+\frac{9}{4 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}}\right ) \, dx\\ &=-\left (\frac{1}{2} \int \frac{1}{x^4 \left (-1+3 x^2\right )^{3/4}} \, dx\right )-\frac{3}{4} \int \frac{1}{x^2 \left (-1+3 x^2\right )^{3/4}} \, dx+\frac{9}{4} \int \frac{1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac{3 \sqrt [4]{-1+3 x^2}}{4 x}-2 \left (\frac{9}{8} \int \frac{1}{\left (-1+3 x^2\right )^{3/4}} \, dx\right )-\frac{5}{4} \int \frac{1}{x^2 \left (-1+3 x^2\right )^{3/4}} \, dx+\frac{27}{8} \int \frac{x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac{2 \sqrt [4]{-1+3 x^2}}{x}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{15}{8} \int \frac{1}{\left (-1+3 x^2\right )^{3/4}} \, dx-2 \frac{\left (3 \sqrt{3} \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{4 x}\\ &=-\frac{\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac{2 \sqrt [4]{-1+3 x^2}}{x}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{3 \sqrt{3} \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 )}{4 x}-\frac{\left (5 \sqrt{3} \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{4 x}\\ &=-\frac{\sqrt [4]{-1+3 x^2}}{6 x^3}-\frac{2 \sqrt [4]{-1+3 x^2}}{x}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{11 \sqrt{3} \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 )}{8 x}\\ \end{align*}
Mathematica [C] time = 0.045282, size = 52, normalized size = 0.32 \[ \frac{\left (1-3 x^2\right )^{3/4} F_1\left (-\frac{3}{2};\frac{3}{4},1;-\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )}{6 x^3 \left (3 x^2-1\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \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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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 )} x^{4}}\,{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^{8} - 9 \, x^{6} + 2 \, x^{4}}, 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}{x^{4} \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 )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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