Integrand size = 11, antiderivative size = 186 \[ \int \frac {x^5}{1+x^5} \, dx=x-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+2 \sqrt {\frac {2}{5+\sqrt {5}}} x\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )-\frac {1}{5} \log (1+x)+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \]
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Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {327, 207, 648, 632, 210, 642, 31} \[ \int \frac {x^5}{1+x^5} \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+x-\frac {1}{5} \log (x+1) \]
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Rule 31
Rule 207
Rule 210
Rule 327
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = x-\int \frac {1}{1+x^5} \, dx \\ & = x-\frac {2}{5} \int \frac {1-\frac {1}{4} \left (1-\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx-\frac {2}{5} \int \frac {1-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx-\frac {1}{5} \int \frac {1}{1+x} \, dx \\ & = x-\frac {1}{5} \log (1+x)-\frac {1}{20} \left (-1-\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx-\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx-\frac {1}{20} \left (-1+\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx-\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx \\ & = x-\frac {1}{5} \log (1+x)+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )-\frac {1}{10} \left (-5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x\right ) \\ & = x+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}-4 x\right )\right )-\frac {1}{5} \log (1+x)+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{1+x^5} \, dx=\frac {1}{20} \left (20 x+2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )-2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-4 \log (1+x)-\left (-1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )+\left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.45 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.21
method | result | size |
risch | \(x -\frac {\ln \left (1+x \right )}{5}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (x -\textit {\_R} \right )\right )}{5}\) | \(39\) |
default | \(x -\frac {\ln \left (1+x \right )}{5}-\frac {\left (-\sqrt {5}-1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (4-\frac {\left (-\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (-\sqrt {5}+1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (-4-\frac {\left (-\sqrt {5}+1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}\) | \(151\) |
meijerg | \(x -\frac {x \left (\frac {\ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{\left (x^{5}\right )^{\frac {1}{5}}}-\frac {\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{\left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{\left (x^{5}\right )^{\frac {1}{5}}}+\frac {\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{\left (x^{5}\right )^{\frac {1}{5}}}+\frac {2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{\left (x^{5}\right )^{\frac {1}{5}}}\right )}{5}\) | \(153\) |
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Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (123) = 246\).
Time = 0.94 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.70 \[ \int \frac {x^5}{1+x^5} \, dx=\frac {1}{20} \, {\left (\sqrt {5} - 2 \, \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} + 1\right )} \log \left (2 \, x - \frac {1}{2} \, \sqrt {5} + \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} - \frac {1}{2}\right ) + \frac {1}{20} \, {\left (\sqrt {5} + 2 \, \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} + 1\right )} \log \left (2 \, x - \frac {1}{2} \, \sqrt {5} - \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} - \frac {1}{2}\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} \log \left (x + \frac {1}{4} \, \sqrt {5} + \frac {5}{2} \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {1}{4}\right ) - \frac {1}{20} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} \log \left (x + \frac {1}{4} \, \sqrt {5} - \frac {5}{2} \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {1}{4}\right ) + x - \frac {1}{5} \, \log \left (x + 1\right ) \]
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Time = 0.49 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.19 \[ \int \frac {x^5}{1+x^5} \, dx=x - \frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log {\left (- 5 t + x \right )} \right )\right )} \]
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.78 \[ \int \frac {x^5}{1+x^5} \, dx=-\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} - \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} + x + \frac {{\left (\sqrt {5} + 3\right )} \log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} + 1\right )}} + \frac {{\left (\sqrt {5} - 3\right )} \log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} - 1\right )}} - \frac {1}{5} \, \log \left (x + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{1+x^5} \, dx=-\frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {1}{20} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + x + \frac {1}{20} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) - \frac {1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{1+x^5} \, dx=x-\frac {\ln \left (x+1\right )}{5}+\ln \left (x-\frac {\sqrt {5}}{4}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )+\ln \left (x-\frac {\sqrt {5}}{4}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )+\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{4}+\frac {\sqrt {5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{4}+\frac {\sqrt {5}}{4}-\frac {1}{4}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right ) \]
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