Optimal. Leaf size=124 \[ -\frac {1}{2 x}+\frac {\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}+\frac {\sqrt [4]{3} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 124, 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 = {331, 303,
1176, 631, 210, 1179, 642} \begin {gather*} \frac {\sqrt [4]{3} \text {ArcTan}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3} \text {ArcTan}\left (\sqrt [4]{6} x+1\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 2^{3/4}}+\frac {\sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 2^{3/4}}-\frac {1}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (2+3 x^4\right )} \, dx &=-\frac {1}{2 x}-\frac {3}{2} \int \frac {x^2}{2+3 x^4} \, dx\\ &=-\frac {1}{2 x}+\frac {1}{4} \sqrt {3} \int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx-\frac {1}{4} \sqrt {3} \int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx\\ &=-\frac {1}{2 x}-\frac {1}{8} \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac {1}{8} \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac {\sqrt [4]{3} \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}{8\ 2^{3/4}}-\frac {\sqrt [4]{3} \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}{8\ 2^{3/4}}\\ &=-\frac {1}{2 x}-\frac {\sqrt [4]{3} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}+\frac {\sqrt [4]{3} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}-\frac {\sqrt [4]{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}+\frac {\sqrt [4]{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4\ 2^{3/4}}\\ &=-\frac {1}{2 x}+\frac {\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3} \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}+\frac {\sqrt [4]{3} \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 101, normalized size = 0.81 \begin {gather*} -\frac {8-2 \sqrt [4]{6} x \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} x \tan ^{-1}\left (1+\sqrt [4]{6} x\right )+\sqrt [4]{6} x \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )-\sqrt [4]{6} x \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )}{16 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 99, normalized size = 0.80
method | result | size |
risch | \(-\frac {1}{2 x}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (54 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-18 \textit {\_R}^{3}+x \right )\right )}{8}\) | \(30\) |
default | \(-\frac {1}{2 x}-\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{96}\) | \(99\) |
meijerg | \(\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} \left (-\frac {4 \,2^{\frac {1}{4}} 3^{\frac {3}{4}}}{3 x}-\frac {x^{3} 2^{\frac {1}{4}} 3^{\frac {3}{4}} \left (\frac {2^{\frac {1}{4}} 27^{\frac {3}{4}} \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {2 \,2^{\frac {1}{4}} 27^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}-\frac {2^{\frac {1}{4}} 27^{\frac {3}{4}} \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {2 \,2^{\frac {1}{4}} 27^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}\right )}{2}\right )}{16}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 126, normalized size = 1.02 \begin {gather*} -\frac {1}{8} \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}{8} \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}{16} \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}{16} \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}{2 \, x} \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. 180 vs.
\(2 (85) = 170\).
time = 0.36, size = 180, normalized size = 1.45 \begin {gather*} \frac {4 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} x \arctan \left (-3^{\frac {1}{4}} 2^{\frac {1}{4}} x + \frac {1}{3} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \sqrt {3 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 9 \, x^{2} + 3 \, \sqrt {3} \sqrt {2}} - 1\right ) + 4 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} x \arctan \left (-3^{\frac {1}{4}} 2^{\frac {1}{4}} x + \frac {1}{3} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \sqrt {-3 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 9 \, x^{2} + 3 \, \sqrt {3} \sqrt {2}} + 1\right ) + 3^{\frac {1}{4}} 2^{\frac {1}{4}} x \log \left (3 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 9 \, x^{2} + 3 \, \sqrt {3} \sqrt {2}\right ) - 3^{\frac {1}{4}} 2^{\frac {1}{4}} x \log \left (-3 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 9 \, x^{2} + 3 \, \sqrt {3} \sqrt {2}\right ) - 8}{16 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 92, normalized size = 0.74 \begin {gather*} - \frac {\sqrt [4]{6} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{16} + \frac {\sqrt [4]{6} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{16} - \frac {\sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{8} - \frac {\sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{8} - \frac {1}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 100, normalized size = 0.81 \begin {gather*} -\frac {1}{8} \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}{8} \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}{16} \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}{16} \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}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 38, normalized size = 0.31 \begin {gather*} -\frac {1}{2\,x}+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\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}{8}-\frac {1}{8}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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