Integrand size = 21, antiderivative size = 39 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {1-2 x^3}{\sqrt {3}}\right )}{\sqrt {3}}+\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right ) \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1488, 814, 648, 632, 210, 642} \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {1-2 x^3}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^6-x^3+1\right )+\log (x) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 1488
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1+x}{x \left (1-x+x^2\right )} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{x}+\frac {2-x}{1-x+x^2}\right ) \, dx,x,x^3\right ) \\ & = \log (x)+\frac {1}{3} \text {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,x^3\right ) \\ & = \log (x)-\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^3\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^3\right ) \\ & = \log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^3\right ) \\ & = \frac {\tan ^{-1}\left (\frac {-1+2 x^3}{\sqrt {3}}\right )}{\sqrt {3}}+\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.41 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=\log (x)-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-1+2 \text {$\#$1}^3}\&\right ] \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\ln \left (x \right )-\frac {\ln \left (x^{6}-x^{3}+1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{3}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{3}\) | \(33\) |
default | \(-\frac {\ln \left (x^{6}-x^{3}+1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{3}+\ln \left (x \right )\) | \(35\) |
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \left (x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=\log {\left (x \right )} - \frac {\log {\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{3}}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \frac {1}{3} \, \log \left (x^{3}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \left ({\left | x \right |}\right ) \]
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Time = 8.86 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^3}{x \left (1-x^3+x^6\right )} \, dx=\ln \left (x\right )-\frac {\ln \left (x^6-x^3+1\right )}{6}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^3}{3}\right )}{3} \]
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