Optimal. Leaf size=39 \[ -\frac{1}{6} \log \left (x^6-x^3+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{\sqrt{3}}+\log (x) \]
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Rubi [A] time = 0.0563145, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1474, 800, 634, 618, 204, 628} \[ -\frac{1}{6} \log \left (x^6-x^3+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{\sqrt{3}}+\log (x) \]
Antiderivative was successfully verified.
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Rule 1474
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1+x^3}{x \left (1-x^3+x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1+x}{x \left (1-x+x^2\right )} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{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} \operatorname{Subst}\left (\int \frac{2-x}{1-x+x^2} \, dx,x,x^3\right )\\ &=\log (x)-\frac{1}{6} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,x^3\right )+\frac{1}{2} \operatorname{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 )-\operatorname{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*}
Mathematica [C] time = 0.0138819, size = 55, normalized size = 1.41 \[ \log (x)-\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\& ,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\& \right ] \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ({x}^{6}-{x}^{3}+1 \right ) }{6}}+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4859, size = 51, normalized size = 1.31 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47567, size = 107, normalized size = 2.74 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.14287, size = 41, normalized size = 1.05 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12803, size = 47, normalized size = 1.21 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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