Integrand size = 13, antiderivative size = 201 \[ \int \frac {x^3}{a^5+x^5} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a}-\frac {\log (a+x)}{5 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a} \]
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Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {299, 648, 632, 210, 642, 31} \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a}-\frac {\log (a+x)}{5 a} \]
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Rule 31
Rule 210
Rule 299
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {2 \int \frac {\frac {1}{4} \left (1+\sqrt {5}\right ) a-\frac {1}{4} \left (-1+\sqrt {5}\right ) x}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{5 a}+\frac {2 \int \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) a-\frac {1}{4} \left (-1-\sqrt {5}\right ) x}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{5 a}-\frac {\int \frac {1}{a+x} \, dx}{5 a} \\ & = -\frac {\log (a+x)}{5 a}+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {1}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {1}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {-\frac {1}{2} \left (1-\sqrt {5}\right ) a+2 x}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{20 a}+\frac {\left (1+\sqrt {5}\right ) \int \frac {-\frac {1}{2} \left (1+\sqrt {5}\right ) a+2 x}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{20 a} \\ & = -\frac {\log (a+x)}{5 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a}+\frac {1}{10} \left (-5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5-\sqrt {5}\right ) a^2-x^2} \, dx,x,-\frac {1}{2} \left (1+\sqrt {5}\right ) a+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 ) a^2-x^2} \, dx,x,-\frac {1}{2} \left (1-\sqrt {5}\right ) a+2 x\right ) \\ & = -\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a}-\frac {\log (a+x)}{5 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {\left (-1+\sqrt {5}\right ) a+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )+2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {-\left (\left (1+\sqrt {5}\right ) a\right )+4 x}{\sqrt {10-2 \sqrt {5}} a}\right )-4 \log (a+x)+\log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )-\sqrt {5} \log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )+\log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )+\sqrt {5} \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.36
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{4}-a^{3} \textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3} a^{4}-\textit {\_R}^{2} a^{3}+a^{2} \textit {\_R} -a +x \right )\right )}{5}-\frac {\ln \left (a +x \right )}{5 a}\) | \(73\) |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}+3 \textit {\_R}^{2} a -2 a^{2} \textit {\_R} +a^{3}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 a^{2} \textit {\_R} -a^{3}}}{5 a}-\frac {\ln \left (a +x \right )}{5 a}\) | \(97\) |
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Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 17865, normalized size of antiderivative = 88.88 \[ \int \frac {x^3}{a^5+x^5} \, dx=\text {Too large to display} \]
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Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.19 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {- \frac {\log {\left (a + x \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log {\left (625 t^{4} a + x \right )} \right )\right )}}{a} \]
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Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {-2 \, \sqrt {5} + 10}} + \frac {{\left (\sqrt {5} + 3\right )} \log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} + 1\right )}} + \frac {{\left (\sqrt {5} - 3\right )} \log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} - 1\right )}} - \frac {\log \left (a + x\right )}{5 \, a} \]
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Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{10 \, a} + \frac {\sqrt {-2 \, \sqrt {5} + 10} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{10 \, a} + \frac {\sqrt {5} \log \left (a^{2} - \frac {1}{2} \, {\left (\sqrt {5} a + a\right )} x + x^{2}\right )}{20 \, a} - \frac {\sqrt {5} \log \left (a^{2} + \frac {1}{2} \, {\left (\sqrt {5} a - a\right )} x + x^{2}\right )}{20 \, a} + \frac {\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a} - \frac {\log \left ({\left | a + x \right |}\right )}{5 \, a} \]
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Time = 0.43 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{a^5+x^5} \, dx=\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}-\frac {\ln \left (5\,a^{10}+\frac {5\,x\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )\,a^9}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a}-\frac {\ln \left (a+x\right )}{5\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a} \]
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