Optimal. Leaf size=111 \[ -\frac {x}{12 \left (2+3 x^4\right )}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}-\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{96 \sqrt [4]{6}}+\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{96 \sqrt [4]{6}} \]
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Rubi [A]
time = 0.05, antiderivative size = 111, 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 = {294, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac {\text {ArcTan}\left (\sqrt [4]{6} x+1\right )}{48 \sqrt [4]{6}}-\frac {x}{12 \left (3 x^4+2\right )}-\frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{96 \sqrt [4]{6}}+\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{96 \sqrt [4]{6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 294
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^4}{\left (2+3 x^4\right )^2} \, dx &=-\frac {x}{12 \left (2+3 x^4\right )}+\frac {1}{12} \int \frac {1}{2+3 x^4} \, dx\\ &=-\frac {x}{12 \left (2+3 x^4\right )}+\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{24 \sqrt {2}}+\frac {\int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{24 \sqrt {2}}\\ &=-\frac {x}{12 \left (2+3 x^4\right )}+\frac {\int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{48 \sqrt {6}}+\frac {\int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{48 \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}{96 \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}{96 \sqrt [4]{6}}\\ &=-\frac {x}{12 \left (2+3 x^4\right )}-\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{96 \sqrt [4]{6}}+\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{96 \sqrt [4]{6}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}\\ &=-\frac {x}{12 \left (2+3 x^4\right )}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{48 \sqrt [4]{6}}-\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{96 \sqrt [4]{6}}+\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{96 \sqrt [4]{6}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 105, normalized size = 0.95 \begin {gather*} \frac {1}{576} \left (-\frac {48 x}{2+3 x^4}-2\ 6^{3/4} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )-6^{3/4} \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+6^{3/4} \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 104, normalized size = 0.94
method | result | size |
risch | \(-\frac {x}{36 \left (x^{4}+\frac {2}{3}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{144}\) | \(35\) |
default | \(-\frac {x}{36 \left (x^{4}+\frac {2}{3}\right )}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{576}\) | \(104\) |
meijerg | \(\frac {24^{\frac {3}{4}} \left (-\frac {x 2^{\frac {3}{4}} 3^{\frac {1}{4}}}{2 \left (1+\frac {3 x^{4}}{2}\right )}+\frac {x 2^{\frac {3}{4}} 3^{\frac {1}{4}} \left (-\frac {6^{\frac {3}{4}} \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{6 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {8^{\frac {1}{4}} 3^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{3 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {6^{\frac {3}{4}} \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{6 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {8^{\frac {1}{4}} 3^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{3 \left (x^{4}\right )^{\frac {1}{4}}}\right )}{8}\right )}{288}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 133, normalized size = 1.20 \begin {gather*} \frac {1}{288} \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}{288} \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}{576} \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}{576} \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}{12 \, {\left (3 \, x^{4} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs.
\(2 (80) = 160\).
time = 0.39, size = 202, normalized size = 1.82 \begin {gather*} -\frac {4 \cdot 24^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \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 ) + 4 \cdot 24^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \arctan \left (-\frac {1}{2} \cdot 24^{\frac {1}{4}} \sqrt {2} x + \frac {1}{48} \cdot 24^{\frac {1}{4}} \sqrt {2} \sqrt {-48 \cdot 24^{\frac {3}{4}} \sqrt {2} x + 576 \, x^{2} + 192 \, \sqrt {6}} + 1\right ) - 24^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (48 \cdot 24^{\frac {3}{4}} \sqrt {2} x + 576 \, x^{2} + 192 \, \sqrt {6}\right ) + 24^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{4} + 2\right )} \log \left (-48 \cdot 24^{\frac {3}{4}} \sqrt {2} x + 576 \, x^{2} + 192 \, \sqrt {6}\right ) + 192 \, x}{2304 \, {\left (3 \, x^{4} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 95, normalized size = 0.86 \begin {gather*} - \frac {x}{36 x^{4} + 24} - \frac {6^{\frac {3}{4}} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{576} + \frac {6^{\frac {3}{4}} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{576} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{288} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{288} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 107, normalized size = 0.96 \begin {gather*} \frac {1}{288} \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}{288} \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}{576} \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}{576} \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}{12 \, {\left (3 \, x^{4} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 45, normalized size = 0.41 \begin {gather*} -\frac {x}{36\,\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}{288}+\frac {1}{288}{}\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}{288}-\frac {1}{288}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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