Integrand size = 23, antiderivative size = 31 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=-\frac {1}{3 x^3}+\frac {2 \arctan \left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1488, 814, 632, 210} \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=\frac {2 \arctan \left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3 x^3} \]
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Rule 210
Rule 632
Rule 814
Rule 1488
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1-x}{x^2 \left (1-x+x^2\right )} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {1}{-1+x-x^2}\right ) \, dx,x,x^3\right ) \\ & = -\frac {1}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+x-x^2} \, dx,x,x^3\right ) \\ & = -\frac {1}{3 x^3}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 x^3\right ) \\ & = -\frac {1}{3 x^3}+\frac {2 \tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{-1+2 \text {$\#$1}^3}\&\right ] \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{9}-\frac {1}{3 x^{3}}\) | \(25\) |
risch | \(-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{9}-\frac {1}{3 x^{3}}\) | \(25\) |
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Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) + 3}{9 \, x^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=- \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{3}}{3} - \frac {\sqrt {3}}{3} \right )}}{9} - \frac {1}{3 x^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{3 \, x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{3 \, x^{3}} \]
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Time = 10.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {1-x^3}{x^4 \left (1-x^3+x^6\right )} \, dx=\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^3}{3}\right )}{9}-\frac {1}{3\,x^3} \]
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