Optimal. Leaf size=63 \[ \frac {\log (a+x)}{3 a^4}+\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^4}-\frac {1}{a^3 x}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^4} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {325, 292, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^2-a x+x^2\right )}{6 a^4}-\frac {1}{a^3 x}+\frac {\log (a+x)}{3 a^4}+\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^4} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 325
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^3+x^3\right )} \, dx &=-\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 {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a^4}\\ &=-\frac {1}{a^3 x}+\frac {\tan ^{-1}\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*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 0.95 \[ -\frac {x \log \left (a^2-a x+x^2\right )-2 x \log (a+x)+2 \sqrt {3} x \tan ^{-1}\left (\frac {2 x-a}{\sqrt {3} a}\right )+6 a}{6 a^4 x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 66, normalized size = 1.05 \[ \frac {\log (a+x)}{3 a^4}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^4}-\frac {1}{a^3 x}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 53, normalized size = 0.84 \[ -\frac {2 \, \sqrt {3} x \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) + x \log \left (a^{2} - a x + x^{2}\right ) - 2 \, x \log \left (a + x\right ) + 6 \, a}{6 \, a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 58, normalized size = 0.92 \[ -\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{4}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{4}} + \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{4}} - \frac {1}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 60, normalized size = 0.95
method | result | size |
default | \(\frac {-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 x -a \right ) \sqrt {3}}{3 a}\right )}{3 a^{4}}-\frac {1}{a^{3} x}+\frac {\ln \left (a +x \right )}{3 a^{4}}\) | \(60\) |
risch | \(-\frac {1}{a^{3} x}+\frac {\ln \left (-a -x \right )}{3 a^{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{8} \textit {\_Z}^{2}+a^{4} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{12}+3\right ) x -a^{9} \textit {\_R}^{2}\right )\right )}{3}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 57, normalized size = 0.90 \[ -\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{4}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{4}} + \frac {\log \left (a + x\right )}{3 \, a^{4}} - \frac {1}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 88, normalized size = 1.40 \[ \frac {\ln \left (a+x\right )}{3\,a^4}-\frac {1}{a^3\,x}+\frac {\ln \left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^4}{4}+x\,a^3\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^4}-\frac {\ln \left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^4}{4}+x\,a^3\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.17, size = 83, normalized size = 1.32 \[ - \frac {1}{a^{3} x} + \frac {\frac {\log {\left (a + x \right )}}{3} + \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right )^{2} + x \right )} + \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right )^{2} + x \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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