Optimal. Leaf size=119 \[ -\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{36 \sqrt [3]{3}}-\frac {1}{3 x}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{18 \sqrt [3]{3}}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{6\ 3^{5/6}} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1368, 1510, 292, 31, 634, 618, 204, 628, 617} \[ -\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{36 \sqrt [3]{3}}-\frac {1}{3 x}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{18 \sqrt [3]{3}}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{6\ 3^{5/6}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 618
Rule 628
Rule 634
Rule 1368
Rule 1510
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (3+4 x^3+x^6\right )} \, dx &=-\frac {1}{3 x}+\frac {1}{3} \int \frac {x \left (-4-x^3\right )}{3+4 x^3+x^6} \, dx\\ &=-\frac {1}{3 x}+\frac {1}{6} \int \frac {x}{3+x^3} \, dx-\frac {1}{2} \int \frac {x}{1+x^3} \, dx\\ &=-\frac {1}{3 x}+\frac {1}{6} \int \frac {1}{1+x} \, dx-\frac {1}{6} \int \frac {1+x}{1-x+x^2} \, dx-\frac {\int \frac {1}{\sqrt [3]{3}+x} \, dx}{18 \sqrt [3]{3}}+\frac {\int \frac {\sqrt [3]{3}+x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{18 \sqrt [3]{3}}\\ &=-\frac {1}{3 x}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{18 \sqrt [3]{3}}-\frac {1}{12} \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{12} \int \frac {1}{3^{2/3}-\sqrt [3]{3} 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}{36 \sqrt [3]{3}}\\ &=-\frac {1}{3 x}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{18 \sqrt [3]{3}}-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{36 \sqrt [3]{3}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{3}}\right )}{6 \sqrt [3]{3}}\\ &=-\frac {1}{3 x}+\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 )}{6\ 3^{5/6}}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{18 \sqrt [3]{3}}-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{36 \sqrt [3]{3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 118, normalized size = 0.99 \[ -\frac {9 x \log \left (x^2-x+1\right )-3^{2/3} x \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-18 x \log (x+1)+2\ 3^{2/3} x \log \left (3^{2/3} x+3\right )+6 \sqrt [6]{3} x \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+18 \sqrt {3} x \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )+36}{108 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 117, normalized size = 0.98 \[ -\frac {3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x \log \left (-3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} x + x^{2} - 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}}\right ) - 2 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x \log \left (3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + x\right ) + 18 \, \sqrt {3} x \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 6 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, \left (-1\right )^{\frac {1}{3}} x + 3^{\frac {1}{3}}\right )}\right ) + 9 \, x \log \left (x^{2} - x + 1\right ) - 18 \, x \log \left (x + 1\right ) + 36}{108 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 91, normalized size = 0.76 \[ \frac {1}{108} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{54} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{18} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{3 \, x} - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 0.75 \[ \frac {3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{18}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {3^{\frac {2}{3}} \ln \left (x +3^{\frac {1}{3}}\right )}{54}+\frac {3^{\frac {2}{3}} \ln \left (x^{2}-3^{\frac {1}{3}} x +3^{\frac {2}{3}}\right )}{108}-\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {1}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 89, normalized size = 0.75 \[ \frac {1}{108} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{54} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{18} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{3 \, x} - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 119, normalized size = 1.00 \[ \frac {\ln \left (x+1\right )}{6}-\frac {3^{2/3}\,\ln \left (x+3^{1/3}\right )}{54}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {1}{3\,x}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (x-\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}-\frac {{\left (-1\right )}^{1/6}\,3^{5/6}}{2}+\frac {3^{1/3}}{2}\right )\,\left (3^{2/3}+3^{1/6}\,3{}\mathrm {i}\right )}{108}+\frac {{\left (-1\right )}^{1/3}\,3^{2/3}\,\ln \left (x+{\left (-1\right )}^{2/3}\,3^{1/3}\right )}{54} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.83, size = 139, normalized size = 1.17 \[ \frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {8188128 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{41} + \frac {39384 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{41} \right )} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {39384 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{41} - \frac {8188128 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{41} \right )} + \operatorname {RootSum} {\left (17496 t^{3} + 1, \left (t \mapsto t \log {\left (- \frac {8188128 t^{5}}{41} + \frac {39384 t^{2}}{41} + x \right )} \right )\right )} - \frac {1}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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