Optimal. Leaf size=138 \[ \frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}} \]
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Rubi [A]
time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {216, 648, 632,
210, 642, 212} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2^{5/6} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}+\frac {\log \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}-\frac {\log \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}{12\ 2^{5/6}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{-2+x^6} \, dx &=-\frac {\int \frac {\sqrt [6]{2}-\frac {x}{2}}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx}{3\ 2^{5/6}}-\frac {\int \frac {\sqrt [6]{2}+\frac {x}{2}}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx}{3\ 2^{5/6}}-\frac {\int \frac {1}{\sqrt [3]{2}-x^2} \, dx}{3\ 2^{2/3}}\\ &=-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac {\int \frac {-\sqrt [6]{2}+2 x}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx}{12\ 2^{5/6}}-\frac {\int \frac {\sqrt [6]{2}+2 x}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx}{12\ 2^{5/6}}-\frac {\int \frac {1}{\sqrt [3]{2}-\sqrt [6]{2} x+x^2} \, dx}{4\ 2^{2/3}}-\frac {\int \frac {1}{\sqrt [3]{2}+\sqrt [6]{2} x+x^2} \, dx}{4\ 2^{2/3}}\\ &=-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{5/6} x\right )}{2\ 2^{5/6}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{5/6} x\right )}{2\ 2^{5/6}}\\ &=\frac {\tan ^{-1}\left (\frac {1-2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 122, normalized size = 0.88 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2^{5/6} x}{\sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1+2^{5/6} x}{\sqrt {3}}\right )-2 \log \left (2-2^{5/6} x\right )+2 \log \left (2+2^{5/6} x\right )-\log \left (2-2^{5/6} x+2^{2/3} x^2\right )+\log \left (2+2^{5/6} x+2^{2/3} x^2\right )}{12\ 2^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 111, normalized size = 0.80
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{5}}\right )}{6}\) | \(22\) |
default | \(-\frac {\ln \left (2^{\frac {1}{3}}+2^{\frac {1}{6}} x +x^{2}\right ) 2^{\frac {1}{6}}}{24}-\frac {\arctan \left (\frac {\sqrt {3}}{3}+\frac {2^{\frac {5}{6}} x \sqrt {3}}{3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{12}-\frac {2^{\frac {1}{6}} \ln \left (x +2^{\frac {1}{6}}\right )}{12}+\frac {\ln \left (2^{\frac {1}{3}}-2^{\frac {1}{6}} x +x^{2}\right ) 2^{\frac {1}{6}}}{24}-\frac {\arctan \left (-\frac {\sqrt {3}}{3}+\frac {2^{\frac {5}{6}} x \sqrt {3}}{3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{12}+\frac {2^{\frac {1}{6}} \ln \left (x -2^{\frac {1}{6}}\right )}{12}\) | \(111\) |
meijerg | \(\frac {2^{\frac {1}{6}} x \left (\ln \left (1-\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )-\ln \left (1+\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )+\frac {\ln \left (1-\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4-2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4+2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.74, size = 112, normalized size = 0.81 \begin {gather*} -\frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x + 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x - 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} + 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} - 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left (x + 2^{\frac {1}{6}}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left (x - 2^{\frac {1}{6}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 176, normalized size = 1.28 \begin {gather*} \frac {1}{96} \cdot 32^{\frac {5}{6}} \sqrt {3} \arctan \left (-\frac {1}{3} \cdot 32^{\frac {1}{6}} \sqrt {3} x + \frac {1}{12} \cdot 32^{\frac {1}{6}} \sqrt {3} \sqrt {16 \, x^{2} + 32^{\frac {5}{6}} x + 8 \cdot 4^{\frac {2}{3}}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{96} \cdot 32^{\frac {5}{6}} \sqrt {3} \arctan \left (-\frac {1}{3} \cdot 32^{\frac {1}{6}} \sqrt {3} x + \frac {1}{12} \cdot 32^{\frac {1}{6}} \sqrt {3} \sqrt {16 \, x^{2} - 32^{\frac {5}{6}} x + 8 \cdot 4^{\frac {2}{3}}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{384} \cdot 32^{\frac {5}{6}} \log \left (1024 \, x^{2} + 64 \cdot 32^{\frac {5}{6}} x + 512 \cdot 4^{\frac {2}{3}}\right ) + \frac {1}{384} \cdot 32^{\frac {5}{6}} \log \left (1024 \, x^{2} - 64 \cdot 32^{\frac {5}{6}} x + 512 \cdot 4^{\frac {2}{3}}\right ) - \frac {1}{192} \cdot 32^{\frac {5}{6}} \log \left (16 \, x + 32^{\frac {5}{6}}\right ) + \frac {1}{192} \cdot 32^{\frac {5}{6}} \log \left (16 \, x - 32^{\frac {5}{6}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 14, normalized size = 0.10 \begin {gather*} \operatorname {RootSum} {\left (1492992 t^{6} - 1, \left ( t \mapsto t \log {\left (- 12 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 114, normalized size = 0.83 \begin {gather*} -\frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x + 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x - 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} + 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} - 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left ({\left | x + 2^{\frac {1}{6}} \right |}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left ({\left | x - 2^{\frac {1}{6}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 140, normalized size = 1.01 \begin {gather*} -\frac {2^{1/6}\,\mathrm {atanh}\left (\frac {2^{5/6}\,x}{2}\right )}{6}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x\,1{}\mathrm {i}}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{1/6}\,\sqrt {3}\,x}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{12}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x\,1{}\mathrm {i}}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{1/6}\,\sqrt {3}\,x}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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