Integrand size = 18, antiderivative size = 11 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=-\log (x)+\log \left (1+x^3\right ) \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 78} \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=\log \left (x^3+1\right )-\log (x) \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {-1+2 x}{x (1+x)} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {3}{1+x}\right ) \, dx,x,x^3\right ) \\ & = -\log (x)+\log \left (1+x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=-\log (x)+\log \left (1+x^3\right ) \]
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Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
meijerg | \(-\ln \left (x \right )+\ln \left (x^{3}+1\right )\) | \(12\) |
risch | \(-\ln \left (x \right )+\ln \left (x^{3}+1\right )\) | \(12\) |
default | \(\ln \left (1+x \right )+\ln \left (x^{2}-x +1\right )-\ln \left (x \right )\) | \(19\) |
norman | \(\ln \left (1+x \right )+\ln \left (x^{2}-x +1\right )-\ln \left (x \right )\) | \(19\) |
parallelrisch | \(\ln \left (1+x \right )+\ln \left (x^{2}-x +1\right )-\ln \left (x \right )\) | \(19\) |
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Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=\log \left (x^{3} + 1\right ) - \log \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=- \log {\left (x \right )} + \log {\left (x^{3} + 1 \right )} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=\log \left (x^{3} + 1\right ) - \frac {1}{3} \, \log \left (x^{3}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=\log \left ({\left | x^{3} + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 15.51 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^3}{x \left (1+x^3\right )} \, dx=\ln \left (x^3+1\right )-\ln \left (x\right ) \]
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