Optimal. Leaf size=102 \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{12 \sqrt [4]{6}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{12 \sqrt [4]{6}}+\frac{x}{3}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{6 \sqrt [4]{6}} \]
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Rubi [A] time = 0.0751092, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {321, 211, 1165, 628, 1162, 617, 204} \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{12 \sqrt [4]{6}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{12 \sqrt [4]{6}}+\frac{x}{3}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{6 \sqrt [4]{6}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^4}{2+3 x^4} \, dx &=\frac{x}{3}-\frac{2}{3} \int \frac{1}{2+3 x^4} \, dx\\ &=\frac{x}{3}-\frac{\int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{3 \sqrt{2}}-\frac{\int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{3 \sqrt{2}}\\ &=\frac{x}{3}-\frac{\int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{6 \sqrt{6}}-\frac{\int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{6 \sqrt{6}}+\frac{\int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{12 \sqrt [4]{6}}+\frac{\int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{12 \sqrt [4]{6}}\\ &=\frac{x}{3}+\frac{\log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{12 \sqrt [4]{6}}-\frac{\log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{12 \sqrt [4]{6}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}\\ &=\frac{x}{3}+\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}-\frac{\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{6 \sqrt [4]{6}}+\frac{\log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{12 \sqrt [4]{6}}-\frac{\log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{12 \sqrt [4]{6}}\\ \end{align*}
Mathematica [A] time = 0.0229548, size = 96, normalized size = 0.94 \[ \frac{1}{72} \left (6^{3/4} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-6^{3/4} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+24 x+2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )-2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 114, normalized size = 1.1 \begin{align*}{\frac{x}{3}}-{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{36}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{72}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{3}\sqrt [4]{6}\sqrt{2}}{36}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48414, size = 167, normalized size = 1.64 \begin{align*} -\frac{1}{36} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{36} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87002, size = 571, normalized size = 5.6 \begin{align*} \frac{1}{72} \cdot 24^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{12} \cdot 24^{\frac{1}{4}} \sqrt{3} \sqrt{2} \sqrt{24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}} - \frac{1}{2} \cdot 24^{\frac{1}{4}} \sqrt{2} x - 1\right ) + \frac{1}{72} \cdot 24^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{12} \cdot 24^{\frac{1}{4}} \sqrt{3} \sqrt{2} \sqrt{-24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}} - \frac{1}{2} \cdot 24^{\frac{1}{4}} \sqrt{2} x + 1\right ) - \frac{1}{288} \cdot 24^{\frac{3}{4}} \sqrt{2} \log \left (24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}\right ) + \frac{1}{288} \cdot 24^{\frac{3}{4}} \sqrt{2} \log \left (-24^{\frac{3}{4}} \sqrt{2} x + 12 \, x^{2} + 4 \, \sqrt{6}\right ) + \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.486485, size = 90, normalized size = 0.88 \begin{align*} \frac{x}{3} + \frac{6^{\frac{3}{4}} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{72} - \frac{6^{\frac{3}{4}} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{72} - \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{36} - \frac{6^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{36} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13857, size = 132, normalized size = 1.29 \begin{align*} -\frac{1}{36} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{36} \cdot 6^{\frac{3}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{72} \cdot 6^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{72} \cdot 6^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{3} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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