Optimal. Leaf size=78 \[ -\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\sqrt [3]{2} x+2^{2/3} x^2\right )}{6 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {206, 31, 648,
631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{2} x+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{-1+2 x^3} \, dx &=\frac {1}{3} \int \frac {1}{-1+\sqrt [3]{2} x} \, dx+\frac {1}{3} \int \frac {-2-\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx\\ &=\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {1}{2} \int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx-\frac {\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx}{6 \sqrt [3]{2}}\\ &=\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\sqrt [3]{2} x+2^{2/3} x^2\right )}{6 \sqrt [3]{2}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{2} x\right )}{\sqrt [3]{2}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\sqrt [3]{2} x+2^{2/3} x^2\right )}{6 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 66, normalized size = 0.85 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{2} x}{\sqrt {3}}\right )-2 \log \left (1-\sqrt [3]{2} x\right )+\log \left (1+\sqrt [3]{2} x+2^{2/3} x^2\right )}{6 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 58, normalized size = 0.74
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{3}-1\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{2}}\right )}{6}\) | \(24\) |
default | \(\frac {2^{\frac {2}{3}} \ln \left (x -\frac {2^{\frac {2}{3}}}{2}\right )}{6}-\frac {2^{\frac {2}{3}} \ln \left (x^{2}+\frac {2^{\frac {2}{3}} x}{2}+\frac {2^{\frac {1}{3}}}{2}\right )}{12}-\frac {\arctan \left (\frac {\left (1+2 \,2^{\frac {1}{3}} x \right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6}\) | \(58\) |
meijerg | \(\frac {2^{\frac {2}{3}} x \left (\ln \left (1-2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2+2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{6 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.65, size = 66, normalized size = 0.85 \begin {gather*} -\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \cdot 2^{\frac {2}{3}} x + 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} x + 1\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {1}{2} \cdot 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.73, size = 63, normalized size = 0.81 \begin {gather*} -\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} {\left (2 \cdot 2^{\frac {2}{3}} x + 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2 \, x^{2} + 2^{\frac {2}{3}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (2 \, x - 2^{\frac {2}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 78, normalized size = 1.00 \begin {gather*} \frac {2^{\frac {2}{3}} \log {\left (x - \frac {2^{\frac {2}{3}}}{2} \right )}}{6} - \frac {2^{\frac {2}{3}} \log {\left (x^{2} + \frac {2^{\frac {2}{3}} x}{2} + \frac {\sqrt [3]{2}}{2} \right )}}{12} - \frac {2^{\frac {2}{3}} \sqrt {3} \operatorname {atan}{\left (\frac {2 \cdot \sqrt [3]{2} \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 57, normalized size = 0.73 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (2 \, x + \left (\frac {1}{2}\right )^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (x^{2} + \left (\frac {1}{2}\right )^{\frac {1}{3}} x + \left (\frac {1}{2}\right )^{\frac {2}{3}}\right ) + \frac {1}{3} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {1}{2}\right )^{\frac {1}{3}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 72, normalized size = 0.92 \begin {gather*} \frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}}{2}\right )}{6}+\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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