Integrand size = 16, antiderivative size = 119 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{6 x^2}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18 \sqrt [6]{3}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 3^{2/3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{36\ 3^{2/3}} \]
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Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1382, 1436, 206, 31, 648, 632, 210, 642, 631} \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18 \sqrt [6]{3}}-\frac {1}{6 x^2}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{36\ 3^{2/3}}-\frac {1}{6} \log (x+1)+\frac {\log \left (x+\sqrt [3]{3}\right )}{18\ 3^{2/3}} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 632
Rule 642
Rule 648
Rule 1382
Rule 1436
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6 x^2}+\frac {1}{6} \int \frac {-8-2 x^3}{3+4 x^3+x^6} \, dx \\ & = -\frac {1}{6 x^2}+\frac {1}{6} \int \frac {1}{3+x^3} \, dx-\frac {1}{2} \int \frac {1}{1+x^3} \, dx \\ & = -\frac {1}{6 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}{18\ 3^{2/3}}+\frac {\int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{18\ 3^{2/3}} \\ & = -\frac {1}{6 x^2}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 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}{36\ 3^{2/3}}+\frac {\int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{12 \sqrt [3]{3}} \\ & = -\frac {1}{6 x^2}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 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 )}{36\ 3^{2/3}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{3}}\right )}{6\ 3^{2/3}} \\ & = -\frac {1}{6 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 )}{18 \sqrt [6]{3}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{18\ 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 )}{36\ 3^{2/3}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=\frac {1}{108} \left (-\frac {18}{x^2}-2\ 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-18 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-18 \log (1+x)+2 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )+9 \log \left (1-x+x^2\right )-\sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {1}{6 x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (x +3 \textit {\_R} \right )\right )}{18}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x +1\right )}{6}\) | \(59\) |
default | \(-\frac {1}{6 x^{2}}-\frac {\ln \left (x +1\right )}{6}+\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{54}-\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{108}+\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{54}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(89\) |
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Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=\frac {6 \cdot 9^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - 9^{\frac {2}{3}} x^{2} \log \left (3 \, x^{2} - 9^{\frac {2}{3}} x + 3 \cdot 9^{\frac {1}{3}}\right ) + 2 \cdot 9^{\frac {2}{3}} x^{2} \log \left (3 \, x + 9^{\frac {2}{3}}\right ) - 54 \, \sqrt {3} x^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 27 \, x^{2} \log \left (x^{2} - x + 1\right ) - 54 \, x^{2} \log \left (x + 1\right ) - 54}{324 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=- \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {1093}{244} - \frac {1093 \sqrt {3} i}{244} + \frac {787320 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{61} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {1093}{244} + \frac {787320 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{61} + \frac {1093 \sqrt {3} i}{244} \right )} + \operatorname {RootSum} {\left (52488 t^{3} - 1, \left ( t \mapsto t \log {\left (\frac {787320 t^{4}}{61} + \frac {3279 t}{61} + x \right )} \right )\right )} - \frac {1}{6 x^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=\frac {1}{54} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{108} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{54} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6 \, x^{2}} + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]
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Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=\frac {1}{54} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{108} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{54} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{6 \, x^{2}} + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 8.46 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 \left (3+4 x^3+x^6\right )} \, dx=\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{54}-\frac {\ln \left (x+1\right )}{6}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {1}{6\,x^2}-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{108}+\frac {3^{5/6}\,1{}\mathrm {i}}{108}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{108}-\frac {3^{5/6}\,1{}\mathrm {i}}{108}\right ) \]
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