Optimal. Leaf size=131 \[ \frac {x}{8 \left (3 x^4+2\right )}-\frac {3^{3/4} \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{64 \sqrt [4]{2}}-\frac {3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac {3^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}} \]
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Rubi [A] time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {199, 211, 1165, 628, 1162, 617, 204} \[ \frac {x}{8 \left (3 x^4+2\right )}-\frac {3^{3/4} \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{64 \sqrt [4]{2}}-\frac {3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac {3^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{\left (2+3 x^4\right )^2} \, dx &=\frac {x}{8 \left (2+3 x^4\right )}+\frac {3}{8} \int \frac {1}{2+3 x^4} \, dx\\ &=\frac {x}{8 \left (2+3 x^4\right )}+\frac {3 \int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{16 \sqrt {2}}+\frac {3 \int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{16 \sqrt {2}}\\ &=\frac {x}{8 \left (2+3 x^4\right )}+\frac {1}{32} \sqrt {\frac {3}{2}} \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{32} \sqrt {\frac {3}{2}} \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac {3^{3/4} \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}{64 \sqrt [4]{2}}-\frac {3^{3/4} \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}{64 \sqrt [4]{2}}\\ &=\frac {x}{8 \left (2+3 x^4\right )}-\frac {3^{3/4} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac {3^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}\\ &=\frac {x}{8 \left (2+3 x^4\right )}-\frac {3^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}+\frac {3^{3/4} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{32 \sqrt [4]{2}}-\frac {3^{3/4} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}+\frac {3^{3/4} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{64 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 105, normalized size = 0.80 \[ \frac {1}{128} \left (\frac {16 x}{3 x^4+2}-6^{3/4} \log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} \log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 248, normalized size = 1.89 \[ -\frac {4 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac {1}{18} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} x + \frac {1}{108} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} \sqrt {3 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 36 \, x^{2} + 12 \, \sqrt {3} \sqrt {2}} - 1\right ) + 4 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac {1}{18} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} x + \frac {1}{108} \cdot 27^{\frac {3}{4}} 8^{\frac {1}{4}} \sqrt {2} \sqrt {-3 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 36 \, x^{2} + 12 \, \sqrt {3} \sqrt {2}} + 1\right ) - 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (3 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 36 \, x^{2} + 12 \, \sqrt {3} \sqrt {2}\right ) + 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (-3 \cdot 27^{\frac {1}{4}} 8^{\frac {3}{4}} \sqrt {2} x + 36 \, x^{2} + 12 \, \sqrt {3} \sqrt {2}\right ) - 64 \, x}{512 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 107, normalized size = 0.82 \[ \frac {1}{64} \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{64} \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{128} \cdot 6^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{128} \cdot 6^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {x}{8 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 123, normalized size = 0.94 \[ \frac {x}{24 x^{4}+16}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{64}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{64}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 133, normalized size = 1.02 \[ \frac {1}{64} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{64} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{128} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{128} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {x}{8 \, {\left (3 \, x^{4} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 44, normalized size = 0.34 \[ \frac {x}{24\,\left (x^4+\frac {2}{3}\right )}+6^{3/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{64}+\frac {1}{64}{}\mathrm {i}\right )+6^{3/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{64}-\frac {1}{64}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 95, normalized size = 0.73 \[ \frac {x}{24 x^{4} + 16} - \frac {6^{\frac {3}{4}} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{128} + \frac {6^{\frac {3}{4}} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{128} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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