Integrand size = 9, antiderivative size = 185 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+2 \sqrt {\frac {2}{5+\sqrt {5}}} x\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )+\frac {1}{5} \log (1+x)-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \]
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Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {1607, 299, 648, 632, 210, 642, 31} \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{5} \log (x+1) \]
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Rule 31
Rule 210
Rule 299
Rule 632
Rule 642
Rule 648
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{1+x^5} \, dx \\ & = \frac {2}{5} \int \frac {\frac {1}{4} \left (-1-\sqrt {5}\right )-\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 {\frac {1}{4} \left (-1+\sqrt {5}\right )-\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 \\ & = \frac {1}{5} \log (1+x)+\frac {\int \frac {1}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx}{2 \sqrt {5}}-\frac {\int \frac {1}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx}{2 \sqrt {5}}+\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 (-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}{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 {\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 )}{\sqrt {5}}-\frac {\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 )}{\sqrt {5}} \\ & = \sqrt {\frac {2}{5 \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 = 0.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=\frac {1}{20} \left (-2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )-2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+4 \log (1+x)-\left (1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )+\left (-1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.18
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{2}+x \right )\right )}{5}+\frac {\ln \left (1+x \right )}{5}\) | \(34\) |
default | \(\frac {\ln \left (1+x \right )}{5}-\frac {\left (-\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (-\sqrt {5}+1-\frac {\left (-\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (-\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (-\sqrt {5}-1-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}\) | \(165\) |
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Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (122) = 244\).
Time = 0.90 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.44 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=-\frac {1}{20} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )} \log \left (\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + x\right ) + \frac {1}{20} \, {\left (\sqrt {5} + 2 \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - 1\right )} \log \left (-\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} - \frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \frac {1}{2} \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} {\left (\sqrt {5} - 1\right )} + 2 \, x - 1\right ) + \frac {1}{20} \, {\left (\sqrt {5} - 2 \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} - 1\right )} \log \left (-\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} - \frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {-\frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {1}{8} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {3}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} + \frac {1}{2} \, \sqrt {5} - \frac {5}{2}} {\left (\sqrt {5} - 1\right )} + 2 \, x - 1\right ) + \frac {1}{20} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} \log \left (\frac {1}{16} \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + x\right ) + \frac {1}{5} \, \log \left (x + 1\right ) \]
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Time = 0.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=\frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left ( t \mapsto t \log {\left (25 t^{2} + x \right )} \right )\right )} \]
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Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=-\frac {2 \, \sqrt {5} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} + \frac {\log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{5 \, {\left (\sqrt {5} + 1\right )}} - \frac {\log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{5 \, {\left (\sqrt {5} - 1\right )}} + \frac {1}{5} \, \log \left (x + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=\frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 9.64 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\frac {1}{x^2}+x^3} \, dx=\frac {\ln \left (x+1\right )}{5}-\ln \left (1-\frac {x\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}-\sqrt {5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left (\frac {x\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}+\sqrt {5}-1\right )}^3}{64}+1\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\ln \left (1-\frac {x\,{\left (\sqrt {5}+\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-\ln \left (1-\frac {x\,{\left (\sqrt {5}-\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right ) \]
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