Optimal. Leaf size=112 \[ -\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12\ 3^{2/3}}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{6\ 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{6 \sqrt [6]{3}} \]
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Rubi [A] time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1347, 200, 31, 634, 618, 204, 628, 617} \[ -\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12\ 3^{2/3}}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{6\ 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{6 \sqrt [6]{3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 618
Rule 628
Rule 634
Rule 1347
Rubi steps
\begin {align*} \int \frac {1}{3+4 x^3+x^6} \, dx &=\frac {1}{2} \int \frac {1}{1+x^3} \, dx-\frac {1}{2} \int \frac {1}{3+x^3} \, dx\\ &=\frac {1}{6} \int \frac {1}{1+x} \, dx+\frac {1}{6} \int \frac {2-x}{1-x+x^2} \, dx-\frac {\int \frac {1}{\sqrt [3]{3}+x} \, dx}{6\ 3^{2/3}}-\frac {\int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{6\ 3^{2/3}}\\ &=\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{6\ 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}{12\ 3^{2/3}}-\frac {\int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{4 \sqrt [3]{3}}\\ &=\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{6\ 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 )}{12\ 3^{2/3}}-\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 )}{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 )}{6 \sqrt [6]{3}}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{6\ 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 )}{12\ 3^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 107, normalized size = 0.96 \[ \frac {1}{36} \left (-3 \log \left (x^2-x+1\right )+\sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+6 \log (x+1)-2 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+2\ 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+6 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 124, normalized size = 1.11 \[ \frac {1}{18} \cdot 9^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{108} \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x + 3 \, x^{2} + 3 \cdot 9^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}}\right ) + \frac {1}{54} \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \, x\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 86, normalized size = 0.77 \[ -\frac {1}{18} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{36} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.75 \[ -\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{18}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (x +3^{\frac {1}{3}}\right )}{18}+\frac {3^{\frac {1}{3}} \ln \left (x^{2}-3^{\frac {1}{3}} x +3^{\frac {2}{3}}\right )}{36}-\frac {\ln \left (x^{2}-x +1\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 84, normalized size = 0.75 \[ -\frac {1}{18} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{36} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 110, normalized size = 0.98 \[ \frac {\ln \left (x+1\right )}{6}-\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{18}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}\,\ln \left (x-{\left (-1\right )}^{1/3}\,3^{1/3}\right )}{18}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (x+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}+\frac {{\left (-1\right )}^{1/3}\,3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.82, size = 124, normalized size = 1.11 \[ \frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {13}{10} - \frac {13 \sqrt {3} i}{10} + \frac {23328 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{5} \right )} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {13}{10} + \frac {23328 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{5} + \frac {13 \sqrt {3} i}{10} \right )} + \operatorname {RootSum} {\left (1944 t^{3} + 1, \left (t \mapsto t \log {\left (\frac {23328 t^{4}}{5} - \frac {78 t}{5} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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