Optimal. Leaf size=40 \[ \frac {5}{6} \log \left (x^2+x+1\right )-\frac {2}{3} \log (1-x)+\sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {1875, 31, 634, 618, 204, 628} \[ \frac {5}{6} \log \left (x^2+x+1\right )-\frac {2}{3} \log (1-x)+\sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1875
Rubi steps
\begin {align*} \int \frac {-3+x^2}{-1+x^3} \, dx &=-\left (\frac {1}{3} \int \frac {-7-5 x}{1+x+x^2} \, dx\right )+\frac {2}{3} \int \frac {1}{1-x} \, dx\\ &=-\frac {2}{3} \log (1-x)+\frac {5}{6} \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {3}{2} \int \frac {1}{1+x+x^2} \, dx\\ &=-\frac {2}{3} \log (1-x)+\frac {5}{6} \log \left (1+x+x^2\right )-3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\frac {2}{3} \log (1-x)+\frac {5}{6} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 1.25 \[ \frac {1}{3} \log \left (1-x^3\right )+\frac {1}{2} \log \left (x^2+x+1\right )-\log (1-x)+\sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 31, normalized size = 0.78 \[ \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {5}{6} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 32, normalized size = 0.80 \[ \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {5}{6} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{3} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.80 \[ \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )-\frac {2 \ln \left (x -1\right )}{3}+\frac {5 \ln \left (x^{2}+x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 31, normalized size = 0.78 \[ \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {5}{6} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 46, normalized size = 1.15 \[ -\frac {2\,\ln \left (x-1\right )}{3}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {5}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {5}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 42, normalized size = 1.05 \[ - \frac {2 \log {\left (x - 1 \right )}}{3} + \frac {5 \log {\left (x^{2} + x + 1 \right )}}{6} + \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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