Integrand size = 7, antiderivative size = 138 \[ \int \frac {1}{2+x^6} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac {\arctan \left (\sqrt {3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac {\arctan \left (\sqrt {3}+2^{5/6} x\right )}{6\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}} \]
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Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {215, 648, 632, 210, 642, 209} \[ \int \frac {1}{2+x^6} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac {\arctan \left (\sqrt {3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac {\arctan \left (2^{5/6} x+\sqrt {3}\right )}{6\ 2^{5/6}}-\frac {\log \left (x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}\right )}{4\ 2^{5/6} \sqrt {3}}+\frac {\log \left (x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}\right )}{4\ 2^{5/6} \sqrt {3}} \]
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Rule 209
Rule 210
Rule 215
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt [6]{2}-\frac {\sqrt {3} x}{2}}{\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2} \, dx}{3\ 2^{5/6}}+\frac {\int \frac {\sqrt [6]{2}+\frac {\sqrt {3} x}{2}}{\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2} \, dx}{3\ 2^{5/6}}+\frac {\int \frac {1}{\sqrt [3]{2}+x^2} \, dx}{3\ 2^{2/3}} \\ & = \frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac {\int \frac {1}{\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2} \, dx}{12\ 2^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2} \, dx}{12\ 2^{2/3}}-\frac {\int \frac {-\sqrt [6]{2} \sqrt {3}+2 x}{\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2} \, dx}{4\ 2^{5/6} \sqrt {3}}+\frac {\int \frac {\sqrt [6]{2} \sqrt {3}+2 x}{\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2} \, dx}{4\ 2^{5/6} \sqrt {3}} \\ & = \frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2^{5/6} x}{\sqrt {3}}\right )}{6\ 2^{5/6} \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2^{5/6} x}{\sqrt {3}}\right )}{6\ 2^{5/6} \sqrt {3}} \\ & = \frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac {\arctan \left (\sqrt {3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac {\arctan \left (\sqrt {3}+2^{5/6} x\right )}{6\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83 \[ \int \frac {1}{2+x^6} \, dx=\frac {4 \arctan \left (\frac {x}{\sqrt [6]{2}}\right )-2 \arctan \left (\sqrt {3}-2^{5/6} x\right )+2 \arctan \left (\sqrt {3}+2^{5/6} x\right )-\sqrt {3} \log \left (2-2^{5/6} \sqrt {3} x+2^{2/3} x^2\right )+\sqrt {3} \log \left (2+2^{5/6} \sqrt {3} x+2^{2/3} x^2\right )}{12\ 2^{5/6}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.16
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6}\) | \(22\) |
default | \(\frac {\arctan \left (\frac {x 2^{\frac {5}{6}}}{2}\right ) 2^{\frac {1}{6}}}{6}+\frac {\arctan \left (x 2^{\frac {5}{6}}-\sqrt {3}\right ) 2^{\frac {1}{6}}}{12}+\frac {\arctan \left (x 2^{\frac {5}{6}}+\sqrt {3}\right ) 2^{\frac {1}{6}}}{12}-\frac {\ln \left (2^{\frac {1}{3}}+x^{2}-2^{\frac {1}{6}} x \sqrt {3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{24}+\frac {\ln \left (2^{\frac {1}{3}}+x^{2}+2^{\frac {1}{6}} x \sqrt {3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{24}\) | \(95\) |
meijerg | \(\frac {2^{\frac {1}{6}} \left (-\frac {x \sqrt {3}\, \ln \left (1-\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4-\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )}{\left (x^{6}\right )^{\frac {1}{6}}}+\frac {2 x \arctan \left (\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )}{\left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \sqrt {3}\, \ln \left (1+\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4+\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )}{\left (x^{6}\right )^{\frac {1}{6}}}\right )}{12}\) | \(170\) |
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Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.19 \[ \int \frac {1}{2+x^6} \, dx=\frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} + 32 \, x\right ) - \frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (-32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} + 32 \, x\right ) + \frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} + 32 \, x\right ) - \frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (-32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} + 32 \, x\right ) + \frac {1}{192} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} \log \left (32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} + 16 \, x\right ) - \frac {1}{192} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} \log \left (-32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} + 16 \, x\right ) \]
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Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.10 \[ \int \frac {1}{2+x^6} \, dx=\operatorname {RootSum} {\left (1492992 t^{6} + 1, \left ( t \mapsto t \log {\left (12 t + x \right )} \right )\right )} \]
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2+x^6} \, dx=\frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x + \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x - \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2+x^6} \, dx=\frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x + \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x - \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} x\right ) \]
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Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.98 \[ \int \frac {1}{2+x^6} \, dx=\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{5/6}\,x}{2}\right )}{6}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{1/6}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (\sqrt {3}-\mathrm {i}\right )\,1{}\mathrm {i}}{12}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{1/6}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{12} \]
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