Optimal. Leaf size=28 \[ \frac {4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1885, 1600,
632, 210, 266} \begin {gather*} \frac {4 \text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 266
Rule 632
Rule 1600
Rule 1885
Rubi steps
\begin {align*} \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx &=3 \int \frac {x^2}{-1+x^3} \, dx+\int \frac {-2+2 x}{-1+x^3} \, dx\\ &=\log \left (1-x^3\right )+\int \frac {1}{\frac {1}{2}+\frac {x}{2}+\frac {x^2}{2}} \, dx\\ &=\log \left (1-x^3\right )-2 \text {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,\frac {1}{2}+x\right )\\ &=\frac {4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} \frac {4 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 29, normalized size = 1.04
method | result | size |
default | \(\ln \left (-1+x \right )+\ln \left (x^{2}+x +1\right )+\frac {4 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(29\) |
risch | \(\ln \left (-1+x \right )+\ln \left (16 x^{2}+16 x +16\right )+\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2+4 x \right ) \sqrt {3}}{6}\right )}{3}\) | \(33\) |
meijerg | \(-\frac {2 x \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}+\ln \left (-x^{3}+1\right )+\frac {2 x^{2} \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.47, size = 28, normalized size = 1.00 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 28, normalized size = 1.00 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 3, normalized size = 0.11 \begin {gather*} \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 29, normalized size = 1.04 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 57, normalized size = 2.04 \begin {gather*} \ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (x-1\right )-\frac {\sqrt {3}\,\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}}{3}+\frac {\sqrt {3}\,\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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