Optimal. Leaf size=56 \[ -\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a}-\frac {\log (a+x)}{3 a}+\frac {\log \left (a^2-a x+x^2\right )}{6 a} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {298, 31, 648,
631, 210, 642} \begin {gather*} \frac {\log \left (a^2-a x+x^2\right )}{6 a}-\frac {\text {ArcTan}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a}-\frac {\log (a+x)}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x}{a^3+x^3} \, dx &=-\frac {\int \frac {1}{a+x} \, dx}{3 a}+\frac {\int \frac {a+x}{a^2-a x+x^2} \, dx}{3 a}\\ &=-\frac {\log (a+x)}{3 a}+\frac {1}{2} \int \frac {1}{a^2-a x+x^2} \, dx+\frac {\int \frac {-a+2 x}{a^2-a x+x^2} \, dx}{6 a}\\ &=-\frac {\log (a+x)}{3 a}+\frac {\log \left (a^2-a x+x^2\right )}{6 a}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a}\\ &=-\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a}-\frac {\log (a+x)}{3 a}+\frac {\log \left (a^2-a x+x^2\right )}{6 a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 50, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {-a+2 x}{\sqrt {3} a}\right )-2 \log (a+x)+\log \left (a^2-a x+x^2\right )}{6 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 51, normalized size = 0.91
method | result | size |
risch | \(-\frac {\ln \left (a +x \right )}{3 a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{2} a^{2}-a \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (a^{2} \textit {\_R} -a +x \right )\right )}{3}\) | \(43\) |
default | \(\frac {\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 x -a \right ) \sqrt {3}}{3 a}\right )}{3 a}-\frac {\ln \left (a +x \right )}{3 a}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.54, size = 49, normalized size = 0.88 \begin {gather*} \frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a} - \frac {\log \left (a + x\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 43, normalized size = 0.77 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.04, size = 71, normalized size = 1.27 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 50, normalized size = 0.89 \begin {gather*} \frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a} - \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 68, normalized size = 1.21 \begin {gather*} -\frac {\ln \left (a+x\right )}{3\,a}-\frac {\ln \left (x+\frac {a\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a}+\frac {\ln \left (x+\frac {a\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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