Optimal. Leaf size=65 \[ -\frac {\log (a+x)}{3 a^5}+\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^5}-\frac {1}{2 a^3 x^2}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^5} \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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, 200, 31, 634, 617, 204, 628} \[ -\frac {1}{2 a^3 x^2}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^5}-\frac {\log (a+x)}{3 a^5}+\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^5} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 325
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a^3+x^3\right )} \, dx &=-\frac {1}{2 a^3 x^2}-\frac {\int \frac {1}{a^3+x^3} \, dx}{a^3}\\ &=-\frac {1}{2 a^3 x^2}-\frac {\int \frac {1}{a+x} \, dx}{3 a^5}-\frac {\int \frac {2 a-x}{a^2-a x+x^2} \, dx}{3 a^5}\\ &=-\frac {1}{2 a^3 x^2}-\frac {\log (a+x)}{3 a^5}+\frac {\int \frac {-a+2 x}{a^2-a x+x^2} \, dx}{6 a^5}-\frac {\int \frac {1}{a^2-a x+x^2} \, dx}{2 a^4}\\ &=-\frac {1}{2 a^3 x^2}-\frac {\log (a+x)}{3 a^5}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^5}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a^5}\\ &=-\frac {1}{2 a^3 x^2}+\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^5}-\frac {\log (a+x)}{3 a^5}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 1.05 \[ -\frac {\log (a+x)}{3 a^5}-\frac {\tan ^{-1}\left (\frac {2 x-a}{\sqrt {3} a}\right )}{\sqrt {3} a^5}-\frac {1}{2 a^3 x^2}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^5} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 68, normalized size = 1.05 \[ -\frac {\log (a+x)}{3 a^5}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^5}-\frac {1}{2 a^3 x^2}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 62, normalized size = 0.95 \[ -\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) - x^{2} \log \left (a^{2} - a x + x^{2}\right ) + 2 \, x^{2} \log \left (a + x\right ) + 3 \, a^{2}}{6 \, a^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 58, normalized size = 0.89 \[ -\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{5}} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{5}} - \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{5}} - \frac {1}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 60, normalized size = 0.92
method | result | size |
default | \(\frac {\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 x -a \right ) \sqrt {3}}{3 a}\right )}{3 a^{5}}-\frac {\ln \left (a +x \right )}{3 a^{5}}-\frac {1}{2 a^{3} x^{2}}\) | \(60\) |
risch | \(-\frac {1}{2 a^{3} x^{2}}+\frac {\ln \left (4 a^{2}-4 a x +4 x^{2}\right )}{6 a^{5}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -a \right ) \sqrt {3}}{3 a}\right )}{3 a^{5}}-\frac {\ln \left (a +x \right )}{3 a^{5}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 57, normalized size = 0.88 \[ -\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{5}} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{5}} - \frac {\log \left (a + x\right )}{3 \, a^{5}} - \frac {1}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 86, normalized size = 1.32 \[ -\frac {\ln \left (a+x\right )}{3\,a^5}-\frac {1}{2\,a^3\,x^2}-\frac {\ln \left (\frac {3\,a^7\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+3\,a^6\,x\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^5}+\frac {\ln \left (\frac {3\,a^7\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-3\,a^6\,x\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.20, size = 80, normalized size = 1.23 \[ - \frac {1}{2 a^{3} x^{2}} + \frac {- \frac {\log {\left (a + x \right )}}{3} + \left (\frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (- 3 a \left (\frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) + x \right )} + \left (\frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (- 3 a \left (\frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) + x \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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