Optimal. Leaf size=113 \[ \frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{32\ 6^{3/4}}-\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{32\ 6^{3/4}}+\frac {x^3}{8 \left (3 x^4+2\right )}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac {\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{16\ 6^{3/4}} \]
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Rubi [A] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, 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 = {290, 297, 1162, 617, 204, 1165, 628} \[ \frac {x^3}{8 \left (3 x^4+2\right )}+\frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{32\ 6^{3/4}}-\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{32\ 6^{3/4}}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac {\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{16\ 6^{3/4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^2}{\left (2+3 x^4\right )^2} \, dx &=\frac {x^3}{8 \left (2+3 x^4\right )}+\frac {1}{8} \int \frac {x^2}{2+3 x^4} \, dx\\ &=\frac {x^3}{8 \left (2+3 x^4\right )}-\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{16 \sqrt {3}}+\frac {\int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{16 \sqrt {3}}\\ &=\frac {x^3}{8 \left (2+3 x^4\right )}+\frac {1}{96} \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{96} \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\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}{32\ 6^{3/4}}+\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}{32\ 6^{3/4}}\\ &=\frac {x^3}{8 \left (2+3 x^4\right )}+\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}-\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{16\ 6^{3/4}}\\ &=\frac {x^3}{8 \left (2+3 x^4\right )}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{16\ 6^{3/4}}+\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}-\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{32\ 6^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 107, normalized size = 0.95 \[ \frac {1}{192} \left (\sqrt [4]{6} \log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} \log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )+\frac {24 x^3}{3 x^4+2}-2 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 202, normalized size = 1.79 \[ -\frac {4 \cdot 54^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac {1}{18} \cdot 54^{\frac {3}{4}} \sqrt {2} x + \frac {1}{54} \cdot 54^{\frac {3}{4}} \sqrt {2} \sqrt {9 \, x^{2} + 3 \cdot 54^{\frac {1}{4}} \sqrt {2} x + 3 \, \sqrt {6}} - 1\right ) + 4 \cdot 54^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac {1}{18} \cdot 54^{\frac {3}{4}} \sqrt {2} x + \frac {1}{54} \cdot 54^{\frac {3}{4}} \sqrt {2} \sqrt {9 \, x^{2} - 3 \cdot 54^{\frac {1}{4}} \sqrt {2} x + 3 \, \sqrt {6}} + 1\right ) + 54^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (9 \, x^{2} + 3 \cdot 54^{\frac {1}{4}} \sqrt {2} x + 3 \, \sqrt {6}\right ) - 54^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (9 \, x^{2} - 3 \cdot 54^{\frac {1}{4}} \sqrt {2} x + 3 \, \sqrt {6}\right ) - 432 \, x^{3}}{3456 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 109, normalized size = 0.96 \[ \frac {x^{3}}{8 \, {\left (3 \, x^{4} + 2\right )}} + \frac {1}{96} \cdot 6^{\frac {1}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{96} \cdot 6^{\frac {1}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{192} \cdot 6^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {1}{192} \cdot 6^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 125, normalized size = 1.11 \[ \frac {x^{3}}{24 x^{4}+16}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{576}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{576}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{1152} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.17, size = 135, normalized size = 1.19 \[ \frac {x^{3}}{8 \, {\left (3 \, x^{4} + 2\right )}} + \frac {1}{96} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{96} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) - \frac {1}{192} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{192} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 46, normalized size = 0.41 \[ \frac {x^3}{24\,\left (x^4+\frac {2}{3}\right )}+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{96}-\frac {1}{96}{}\mathrm {i}\right )+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{96}+\frac {1}{96}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 97, normalized size = 0.86 \[ \frac {x^{3}}{24 x^{4} + 16} + \frac {\sqrt [4]{6} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{192} - \frac {\sqrt [4]{6} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{192} + \frac {\sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{96} + \frac {\sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{96} \]
Verification of antiderivative is not currently implemented for this CAS.
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