Integrand size = 13, antiderivative size = 63 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=-\frac {1}{a^3 x}+\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^4}+\frac {\log (a+x)}{3 a^4}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^4} \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^4}+\frac {\log (a+x)}{3 a^4}-\frac {1}{a^3 x}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^4} \]
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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}{a^3 x}-\frac {\int \frac {x}{a^3+x^3} \, dx}{a^3} \\ & = -\frac {1}{a^3 x}+\frac {\int \frac {1}{a+x} \, dx}{3 a^4}-\frac {\int \frac {a+x}{a^2-a x+x^2} \, dx}{3 a^4} \\ & = -\frac {1}{a^3 x}+\frac {\log (a+x)}{3 a^4}-\frac {\int \frac {-a+2 x}{a^2-a x+x^2} \, dx}{6 a^4}-\frac {\int \frac {1}{a^2-a x+x^2} \, dx}{2 a^3} \\ & = -\frac {1}{a^3 x}+\frac {\log (a+x)}{3 a^4}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^4}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a^4} \\ & = -\frac {1}{a^3 x}+\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^4}+\frac {\log (a+x)}{3 a^4}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=-\frac {6 a+2 \sqrt {3} x \arctan \left (\frac {-a+2 x}{\sqrt {3} a}\right )-2 x \log (a+x)+x \log \left (a^2-a x+x^2\right )}{6 a^4 x} \]
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Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {1}{a^{3} x}+\frac {-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (-a +2 x \right ) \sqrt {3}}{3 a}\right )}{3 a^{4}}+\frac {\ln \left (a +x \right )}{3 a^{4}}\) | \(60\) |
risch | \(-\frac {1}{a^{3} x}-\frac {\ln \left (4 a^{2}-4 a x +4 x^{2}\right )}{6 a^{4}}+\frac {\arctan \left (\frac {\left (a -2 x \right ) \sqrt {3}}{3 a}\right ) \sqrt {3}}{3 a^{4}}+\frac {\ln \left (-a -x \right )}{3 a^{4}}\) | \(66\) |
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Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) + x \log \left (a^{2} - a x + x^{2}\right ) - 2 \, x \log \left (a + x\right ) + 6 \, a}{6 \, a^{4} x} \]
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Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=- \frac {1}{a^{3} x} + \frac {\frac {\log {\left (a + x \right )}}{3} + \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right )^{2} + x \right )} + \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right )^{2} + x \right )}}{a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{4}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{4}} + \frac {\log \left (a + x\right )}{3 \, a^{4}} - \frac {1}{a^{3} x} \]
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Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{4}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{4}} + \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{4}} - \frac {1}{a^{3} x} \]
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Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx=\frac {\ln \left (a+x\right )}{3\,a^4}-\frac {1}{a^3\,x}+\frac {\ln \left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^4}{4}+x\,a^3\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^4}-\frac {\ln \left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^4}{4}+x\,a^3\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^4} \]
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