Optimal. Leaf size=112 \[ \frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4\ 3^{2/3}}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{2\ 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}} \]
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Rubi [A] time = 0.0693467, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1374, 200, 31, 634, 617, 204, 628, 618} \[ \frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4\ 3^{2/3}}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{2\ 3^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}} \]
Antiderivative was successfully verified.
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Rule 1374
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 618
Rubi steps
\begin{align*} \int \frac{x^3}{3+4 x^3+x^6} \, dx &=-\left (\frac{1}{2} \int \frac{1}{1+x^3} \, dx\right )+\frac{3}{2} \int \frac{1}{3+x^3} \, dx\\ &=-\left (\frac{1}{6} \int \frac{1}{1+x} \, dx\right )-\frac{1}{6} \int \frac{2-x}{1-x+x^2} \, dx+\frac{\int \frac{1}{\sqrt [3]{3}+x} \, dx}{2\ 3^{2/3}}+\frac{\int \frac{2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{2\ 3^{2/3}}\\ &=-\frac{1}{6} \log (1+x)+\frac{\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac{1}{12} \int \frac{-1+2 x}{1-x+x^2} \, dx-\frac{1}{4} \int \frac{1}{1-x+x^2} \, dx-\frac{\int \frac{-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{4\ 3^{2/3}}+\frac{1}{4} 3^{2/3} \int \frac{1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=-\frac{1}{6} \log (1+x)+\frac{\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac{1}{12} \log \left (1-x+x^2\right )-\frac{\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4\ 3^{2/3}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{2} \sqrt [3]{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{3}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}}-\frac{1}{6} \log (1+x)+\frac{\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac{1}{12} \log \left (1-x+x^2\right )-\frac{\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4\ 3^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0241865, size = 106, normalized size = 0.95 \[ \frac{1}{12} \left (\log \left (x^2-x+1\right )-\sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-2 \log (x+1)+2 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )-2\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 84, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{6}}-{\frac{\sqrt [3]{3}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{12}}+{\frac{{3}^{{\frac{5}{6}}}}{6}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56599, size = 113, normalized size = 1.01 \begin{align*} \frac{1}{6} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{12} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54202, size = 342, normalized size = 3.05 \begin{align*} \frac{1}{6} \cdot 9^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{1}{27} \cdot 9^{\frac{1}{6}}{\left (2 \cdot 9^{\frac{2}{3}} \sqrt{3} x - 3 \cdot 9^{\frac{1}{3}} \sqrt{3}\right )}\right ) - \frac{1}{36} \cdot 9^{\frac{2}{3}} \log \left (3 \, x^{2} - 9^{\frac{2}{3}} x + 3 \cdot 9^{\frac{1}{3}}\right ) + \frac{1}{18} \cdot 9^{\frac{2}{3}} \log \left (3 \, x + 9^{\frac{2}{3}}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.65432, size = 110, normalized size = 0.98 \begin{align*} - \frac{\log{\left (x + 1 \right )}}{6} + \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{1}{4} + 648 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4} + \frac{\sqrt{3} i}{4} \right )} + \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{1}{4} + 648 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4} - \frac{\sqrt{3} i}{4} \right )} + \operatorname{RootSum}{\left (72 t^{3} - 1, \left ( t \mapsto t \log{\left (648 t^{4} - 3 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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