Optimal. Leaf size=43 \[ -\frac {1}{24} \log \left (x^2+2 x+4\right )+\frac {1}{12} \log (2-x)-\frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {200, 31, 634, 618, 204, 628} \[ -\frac {1}{24} \log \left (x^2+2 x+4\right )+\frac {1}{12} \log (2-x)-\frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{-8+x^3} \, dx &=\frac {1}{12} \int \frac {1}{-2+x} \, dx+\frac {1}{12} \int \frac {-4-x}{4+2 x+x^2} \, dx\\ &=\frac {1}{12} \log (2-x)-\frac {1}{24} \int \frac {2+2 x}{4+2 x+x^2} \, dx-\frac {1}{4} \int \frac {1}{4+2 x+x^2} \, dx\\ &=\frac {1}{12} \log (2-x)-\frac {1}{24} \log \left (4+2 x+x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,2+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{12} \log (2-x)-\frac {1}{24} \log \left (4+2 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ -\frac {1}{24} \log \left (x^2+2 x+4\right )+\frac {1}{12} \log (2-x)-\frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 32, normalized size = 0.74 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 33, normalized size = 0.77 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac {1}{12} \, \log \left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.81 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +2\right ) \sqrt {3}}{6}\right )}{12}+\frac {\ln \left (x -2\right )}{12}-\frac {\ln \left (x^{2}+2 x +4\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 32, normalized size = 0.74 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 46, normalized size = 1.07 \[ \frac {\ln \left (x-2\right )}{12}+\ln \left (x+1-\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )-\ln \left (x+1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 41, normalized size = 0.95 \[ \frac {\log {\left (x - 2 \right )}}{12} - \frac {\log {\left (x^{2} + 2 x + 4 \right )}}{24} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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