Integrand size = 7, antiderivative size = 74 \[ \int \frac {1}{-2+x^3} \, dx=-\frac {\arctan \left (\frac {1+2^{2/3} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-x\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\sqrt [3]{2} x+x^2\right )}{6\ 2^{2/3}} \]
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Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {206, 31, 648, 631, 210, 642} \[ \int \frac {1}{-2+x^3} \, dx=-\frac {\arctan \left (\frac {2^{2/3} x+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (x^2+\sqrt [3]{2} x+2^{2/3}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-x\right )}{3\ 2^{2/3}} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{-\sqrt [3]{2}+x} \, dx}{3\ 2^{2/3}}+\frac {\int \frac {-2 \sqrt [3]{2}-x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx}{3\ 2^{2/3}} \\ & = \frac {\log \left (\sqrt [3]{2}-x\right )}{3\ 2^{2/3}}-\frac {\int \frac {\sqrt [3]{2}+2 x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx}{6\ 2^{2/3}}-\frac {\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx}{2 \sqrt [3]{2}} \\ & = \frac {\log \left (\sqrt [3]{2}-x\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\sqrt [3]{2} x+x^2\right )}{6\ 2^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} x\right )}{2^{2/3}} \\ & = -\frac {\arctan \left (\frac {1+2^{2/3} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-x\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\sqrt [3]{2} x+x^2\right )}{6\ 2^{2/3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {1}{-2+x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+2^{2/3} x}{\sqrt {3}}\right )-2 \log \left (2-2^{2/3} x\right )+\log \left (2+2^{2/3} x+\sqrt [3]{2} x^2\right )}{6\ 2^{2/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.30
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3}\) | \(22\) |
default | \(\frac {2^{\frac {1}{3}} \ln \left (x -2^{\frac {1}{3}}\right )}{6}-\frac {\ln \left (2^{\frac {2}{3}}+2^{\frac {1}{3}} x +x^{2}\right ) 2^{\frac {1}{3}}}{12}-\frac {\arctan \left (\frac {\left (1+2^{\frac {2}{3}} x \right ) \sqrt {3}}{3}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}\) | \(54\) |
meijerg | \(\frac {2^{\frac {1}{3}} x \left (\ln \left (1-\frac {2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2}\right )-\frac {\ln \left (1+\frac {2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{2}+\frac {2^{\frac {1}{3}} \left (x^{3}\right )^{\frac {2}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}{4+2^{\frac {2}{3}} \left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{6 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(84\) |
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Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {1}{-2+x^3} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {2}{3}} \sqrt {3} x + 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (2 \, x^{2} + 4^{\frac {2}{3}} x + 2 \cdot 4^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (2 \, x - 4^{\frac {2}{3}}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {1}{-2+x^3} \, dx=\frac {\sqrt [3]{2} \log {\left (x - \sqrt [3]{2} \right )}}{6} - \frac {\sqrt [3]{2} \log {\left (x^{2} + \sqrt [3]{2} x + 2^{\frac {2}{3}} \right )}}{12} - \frac {\sqrt [3]{2} \sqrt {3} \operatorname {atan}{\left (\frac {2^{\frac {2}{3}} \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76 \[ \int \frac {1}{-2+x^3} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x + 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (x^{2} + 2^{\frac {1}{3}} x + 2^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left (x - 2^{\frac {1}{3}}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {1}{-2+x^3} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x + 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (x^{2} + 2^{\frac {1}{3}} x + 2^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | x - 2^{\frac {1}{3}} \right |}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {1}{-2+x^3} \, dx=\frac {2^{1/3}\,\ln \left (x-2^{1/3}\right )}{6}+\frac {2^{1/3}\,\ln \left (x-\frac {2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{1/3}\,\ln \left (x+\frac {2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]
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