Optimal. Leaf size=126 \[ -\frac {1}{15 x^5}+\frac {2}{9 x^2}-\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108\ 3^{2/3}}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{54\ 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}} \]
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Rubi [A] time = 0.10, antiderivative size = 126, normalized size of antiderivative = 1.00, 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 = {1368, 1504, 1422, 200, 31, 634, 618, 204, 628, 617} \[ \frac {2}{9 x^2}-\frac {1}{15 x^5}-\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108\ 3^{2/3}}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{54\ 3^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 618
Rule 628
Rule 634
Rule 1368
Rule 1422
Rule 1504
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx &=-\frac {1}{15 x^5}+\frac {1}{15} \int \frac {-20-5 x^3}{x^3 \left (3+4 x^3+x^6\right )} \, dx\\ &=-\frac {1}{15 x^5}+\frac {2}{9 x^2}-\frac {1}{90} \int \frac {-130-40 x^3}{3+4 x^3+x^6} \, dx\\ &=-\frac {1}{15 x^5}+\frac {2}{9 x^2}-\frac {1}{18} \int \frac {1}{3+x^3} \, dx+\frac {1}{2} \int \frac {1}{1+x^3} \, dx\\ &=-\frac {1}{15 x^5}+\frac {2}{9 x^2}+\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}{54\ 3^{2/3}}-\frac {\int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{54\ 3^{2/3}}\\ &=-\frac {1}{15 x^5}+\frac {2}{9 x^2}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{54\ 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}{108\ 3^{2/3}}-\frac {\int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{36 \sqrt [3]{3}}\\ &=-\frac {1}{15 x^5}+\frac {2}{9 x^2}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{54\ 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 )}{108\ 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 )}{18\ 3^{2/3}}\\ &=-\frac {1}{15 x^5}+\frac {2}{9 x^2}-\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 )}{54 \sqrt [6]{3}}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{54\ 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 )}{108\ 3^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 118, normalized size = 0.94 \[ \frac {-\frac {108}{x^5}+\frac {360}{x^2}-135 \log \left (x^2-x+1\right )+5 \sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+270 \log (x+1)-10 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+10\ 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+270 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{1620} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 153, normalized size = 1.21 \[ \frac {30 \cdot 9^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - 5 \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x + 3 \, x^{2} + 3 \cdot 9^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}}\right ) + 10 \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \, x\right ) + 810 \, \sqrt {3} x^{5} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 405 \, x^{5} \log \left (x^{2} - x + 1\right ) + 810 \, x^{5} \log \left (x + 1\right ) + 1080 \, x^{3} - 324}{4860 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 98, normalized size = 0.78 \[ -\frac {1}{162} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{324} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{162} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {10 \, x^{3} - 3}{45 \, x^{5}} - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 0.75 \[ -\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{162}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (x +3^{\frac {1}{3}}\right )}{162}+\frac {3^{\frac {1}{3}} \ln \left (x^{2}-3^{\frac {1}{3}} x +3^{\frac {2}{3}}\right )}{324}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {2}{9 x^{2}}-\frac {1}{15 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.66, size = 96, normalized size = 0.76 \[ -\frac {1}{162} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{324} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{162} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {10 \, x^{3} - 3}{45 \, x^{5}} - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 121, normalized size = 0.96 \[ \frac {\ln \left (x+1\right )}{6}-\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{162}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {\frac {2\,x^3}{9}-\frac {1}{15}}{x^5}+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}\,\ln \left (x-{\left (-1\right )}^{1/3}\,3^{1/3}\right )}{162}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (x+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}+\frac {{\left (-1\right )}^{1/3}\,3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{324} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.78, size = 136, normalized size = 1.08 \[ \frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {88573}{6562} - \frac {88573 \sqrt {3} i}{6562} + \frac {119042784 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{3281} \right )} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {88573}{6562} + \frac {119042784 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{3281} + \frac {88573 \sqrt {3} i}{6562} \right )} + \operatorname {RootSum} {\left (1417176 t^{3} + 1, \left (t \mapsto t \log {\left (\frac {119042784 t^{4}}{3281} - \frac {531438 t}{3281} + x \right )} \right )\right )} + \frac {10 x^{3} - 3}{45 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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