Integrand size = 13, antiderivative size = 73 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}-\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^7}-\frac {\log (a+x)}{3 a^7}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^7}-\frac {\log (a+x)}{3 a^7}+\frac {1}{a^6 x}-\frac {1}{4 a^3 x^4}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7} \]
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 a^3 x^4}-\frac {\int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx}{a^3} \\ & = -\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}+\frac {\int \frac {x}{a^3+x^3} \, dx}{a^6} \\ & = -\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}-\frac {\int \frac {1}{a+x} \, dx}{3 a^7}+\frac {\int \frac {a+x}{a^2-a x+x^2} \, dx}{3 a^7} \\ & = -\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}-\frac {\log (a+x)}{3 a^7}+\frac {\int \frac {-a+2 x}{a^2-a x+x^2} \, dx}{6 a^7}+\frac {\int \frac {1}{a^2-a x+x^2} \, dx}{2 a^6} \\ & = -\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}-\frac {\log (a+x)}{3 a^7}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a^7} \\ & = -\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}-\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^7}-\frac {\log (a+x)}{3 a^7}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}+\frac {\arctan \left (\frac {-a+2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^7}-\frac {\log (a+x)}{3 a^7}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7} \]
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Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {1}{4 a^{3} x^{4}}+\frac {1}{a^{6} x}+\frac {\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 x \right ) \sqrt {3}}{3 a}\right )}{3 a^{7}}-\frac {\ln \left (a +x \right )}{3 a^{7}}\) | \(66\) |
risch | \(\frac {\frac {x^{3}}{a^{6}}-\frac {1}{4 a^{3}}}{x^{4}}-\frac {\ln \left (a +x \right )}{3 a^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{14} \textit {\_Z}^{2}-a^{7} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{21}-3\right ) x +a^{15} \textit {\_R}^{2}\right )\right )}{3}\) | \(72\) |
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Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {4 \, \sqrt {3} x^{4} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) + 2 \, x^{4} \log \left (a^{2} - a x + x^{2}\right ) - 4 \, x^{4} \log \left (a + x\right ) - 3 \, a^{4} + 12 \, a x^{3}}{12 \, a^{7} x^{4}} \]
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Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {- a^{3} + 4 x^{3}}{4 a^{6} x^{4}} + \frac {- \frac {\log {\left (a + x \right )}}{3} + \left (\frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (\frac {1}{6} - \frac {\sqrt {3} i}{6}\right )^{2} + x \right )} + \left (\frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (\frac {1}{6} + \frac {\sqrt {3} i}{6}\right )^{2} + x \right )}}{a^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac {\log \left (a + x\right )}{3 \, a^{7}} - \frac {a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{7}} - \frac {a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \]
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Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {\frac {1}{4\,a^3}-\frac {x^3}{a^6}}{x^4}-\frac {\ln \left (a+x\right )}{3\,a^7}-\frac {\ln \left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^7}{4}+x\,a^6\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^7}+\frac {\ln \left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^7}{4}+x\,a^6\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^7} \]
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