Integrand size = 9, antiderivative size = 56 \[ \int \frac {1}{a^3+x^3} \, dx=-\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2}+\frac {\log (a+x)}{3 a^2}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {206, 31, 648, 631, 210, 642} \[ \int \frac {1}{a^3+x^3} \, dx=-\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}+\frac {\log (a+x)}{3 a^2} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{a+x} \, dx}{3 a^2}+\frac {\int \frac {2 a-x}{a^2-a x+x^2} \, dx}{3 a^2} \\ & = \frac {\log (a+x)}{3 a^2}-\frac {\int \frac {-a+2 x}{a^2-a x+x^2} \, dx}{6 a^2}+\frac {\int \frac {1}{a^2-a x+x^2} \, dx}{2 a} \\ & = \frac {\log (a+x)}{3 a^2}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a^2} \\ & = -\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2}+\frac {\log (a+x)}{3 a^2}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a^3+x^3} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {-a+2 x}{\sqrt {3} a}\right )+2 \log (a+x)-\log \left (a^2-a x+x^2\right )}{6 a^2} \]
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Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 x \right ) \sqrt {3}}{3 a}\right )}{3 a^{2}}+\frac {\ln \left (a +x \right )}{3 a^{2}}\) | \(51\) |
risch | \(-\frac {\ln \left (4 a^{2}-4 a x +4 x^{2}\right )}{6 a^{2}}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-a +2 x \right ) \sqrt {3}}{3 a}\right )}{3 a^{2}}+\frac {\ln \left (a +x \right )}{3 a^{2}}\) | \(56\) |
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none
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {1}{a^3+x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left (a^{2} - a x + x^{2}\right ) + 2 \, \log \left (a + x\right )}{6 \, a^{2}} \]
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Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int \frac {1}{a^3+x^3} \, dx=\frac {\frac {\log {\left (a + x \right )}}{3} + \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (3 a \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) + x \right )} + \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (3 a \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) + x \right )}}{a^{2}} \]
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none
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {1}{a^3+x^3} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{2}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{2}} + \frac {\log \left (a + x\right )}{3 \, a^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a^3+x^3} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{2}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{2}} + \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{2}} \]
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Time = 0.46 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a^3+x^3} \, dx=\frac {\ln \left (a+x\right )}{3\,a^2}+\frac {\ln \left (x+\frac {a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^2}-\frac {\ln \left (x-\frac {a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^2} \]
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