Optimal. Leaf size=122 \[ \frac{x^4}{4}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{3}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-4 x-\frac{1}{6} \log (x+1)+\frac{3}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{3}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
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Rubi [A] time = 0.0936454, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {1367, 1502, 1422, 200, 31, 634, 618, 204, 628, 617} \[ \frac{x^4}{4}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{3}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-4 x-\frac{1}{6} \log (x+1)+\frac{3}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{3}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]
Antiderivative was successfully verified.
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Rule 1367
Rule 1502
Rule 1422
Rule 200
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rule 617
Rubi steps
\begin{align*} \int \frac{x^9}{3+4 x^3+x^6} \, dx &=\frac{x^4}{4}-\frac{1}{4} \int \frac{x^3 \left (12+16 x^3\right )}{3+4 x^3+x^6} \, dx\\ &=-4 x+\frac{x^4}{4}+\frac{1}{4} \int \frac{48+52 x^3}{3+4 x^3+x^6} \, dx\\ &=-4 x+\frac{x^4}{4}-\frac{1}{2} \int \frac{1}{1+x^3} \, dx+\frac{27}{2} \int \frac{1}{3+x^3} \, dx\\ &=-4 x+\frac{x^4}{4}-\frac{1}{6} \int \frac{1}{1+x} \, dx-\frac{1}{6} \int \frac{2-x}{1-x+x^2} \, dx+\frac{1}{2} \left (3 \sqrt [3]{3}\right ) \int \frac{1}{\sqrt [3]{3}+x} \, dx+\frac{1}{2} \left (3 \sqrt [3]{3}\right ) \int \frac{2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=-4 x+\frac{x^4}{4}-\frac{1}{6} \log (1+x)+\frac{3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\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{1}{4} \left (3 \sqrt [3]{3}\right ) \int \frac{-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx+\frac{1}{4} \left (9\ 3^{2/3}\right ) \int \frac{1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=-4 x+\frac{x^4}{4}-\frac{1}{6} \log (1+x)+\frac{3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\frac{1}{12} \log \left (1-x+x^2\right )-\frac{3}{4} \sqrt [3]{3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{2} \left (9 \sqrt [3]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{3}}\right )\\ &=-4 x+\frac{x^4}{4}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{3}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )-\frac{1}{6} \log (1+x)+\frac{3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\frac{1}{12} \log \left (1-x+x^2\right )-\frac{3}{4} \sqrt [3]{3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0295371, size = 114, normalized size = 0.93 \[ \frac{1}{12} \left (3 x^4+\log \left (x^2-x+1\right )-9 \sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-48 x-2 \log (x+1)+18 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )-18\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 92, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}}-4\,x+{\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{3\,\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{2}}-{\frac{3\,\sqrt [3]{3}\ln \left ({3}^{2/3}-\sqrt [3]{3}x+{x}^{2} \right ) }{4}}+{\frac{3\,{3}^{5/6}}{2}\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 = 1.67359, size = 124, normalized size = 1.02 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{2} \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{3}{4} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{3}{2} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) - 4 \, x + \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 = 1.55849, size = 305, normalized size = 2.5 \begin{align*} \frac{1}{4} \, x^{4} + \frac{3}{2} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{2}{3} \cdot 3^{\frac{1}{6}} x - \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{3}{4} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{3}{2} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) - 4 \, x + \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 = 0.644529, size = 129, normalized size = 1.06 \begin{align*} \frac{x^{4}}{4} - 4 x - \frac{\log{\left (x + 1 \right )}}{6} + \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{9841}{19692} - \frac{9841 \sqrt{3} i}{19692} + \frac{360 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4}}{547} \right )} + \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{9841}{19692} + \frac{360 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4}}{547} + \frac{9841 \sqrt{3} i}{19692} \right )} + \operatorname{RootSum}{\left (8 t^{3} - 81, \left ( t \mapsto t \log{\left (\frac{360 t^{4}}{547} - \frac{9841 t}{1641} + 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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