Optimal. Leaf size=55 \[ \frac {1}{6} \log \left (x^2+x+1\right ) (a+b-2 c)-\frac {1}{3} \log (1-x) (a+b+c)+\frac {(a-b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1875, 31, 634, 618, 204, 628} \[ \frac {1}{6} \log \left (x^2+x+1\right ) (a+b-2 c)-\frac {1}{3} \log (1-x) (a+b+c)+\frac {(a-b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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 {a+b x+c x^2}{1-x^3} \, dx &=\frac {1}{3} \int \frac {2 a-b-c+(a+b-2 c) x}{1+x+x^2} \, dx+\frac {1}{3} (a+b+c) \int \frac {1}{1-x} \, dx\\ &=-\frac {1}{3} (a+b+c) \log (1-x)+\frac {1}{2} (a-b) \int \frac {1}{1+x+x^2} \, dx+\frac {1}{6} (a+b-2 c) \int \frac {1+2 x}{1+x+x^2} \, dx\\ &=-\frac {1}{3} (a+b+c) \log (1-x)+\frac {1}{6} (a+b-2 c) \log \left (1+x+x^2\right )+(-a+b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {(a-b) \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} (a+b+c) \log (1-x)+\frac {1}{6} (a+b-2 c) \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 1.13 \[ \frac {1}{6} \left ((a+b) \log \left (x^2+x+1\right )-2 (a+b) \log (1-x)+2 \sqrt {3} (a-b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )-2 c \log \left (1-x^3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 47, normalized size = 0.85 \[ \frac {1}{3} \, \sqrt {3} {\left (a - b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, {\left (a + b - 2 \, c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (a + b + c\right )} \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 52, normalized size = 0.95 \[ \frac {1}{3} \, {\left (\sqrt {3} a - \sqrt {3} b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, {\left (a + b - 2 \, c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (a + b + c\right )} \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 87, normalized size = 1.58 \[ \frac {\sqrt {3}\, a \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}-\frac {a \ln \left (x -1\right )}{3}+\frac {a \ln \left (x^{2}+x +1\right )}{6}-\frac {\sqrt {3}\, b \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}-\frac {b \ln \left (x -1\right )}{3}+\frac {b \ln \left (x^{2}+x +1\right )}{6}-\frac {c \ln \left (x -1\right )}{3}-\frac {c \ln \left (x^{2}+x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 47, normalized size = 0.85 \[ \frac {1}{3} \, \sqrt {3} {\left (a - b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, {\left (a + b - 2 \, c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (a + b + c\right )} \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.95, size = 87, normalized size = 1.58 \[ \ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {a}{6}+\frac {b}{6}-\frac {c}{3}-\frac {\sqrt {3}\,a\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {a}{6}+\frac {b}{6}-\frac {c}{3}+\frac {\sqrt {3}\,a\,1{}\mathrm {i}}{6}-\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{6}\right )-\ln \left (x-1\right )\,\left (\frac {a}{3}+\frac {b}{3}+\frac {c}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.89, size = 323, normalized size = 5.87 \[ - \frac {\left (a + b + c\right ) \log {\left (x + \frac {a^{2} c - a^{2} \left (a + b + c\right ) - 2 a b^{2} + b c^{2} - 2 b c \left (a + b + c\right ) + b \left (a + b + c\right )^{2}}{a^{3} - b^{3}} \right )}}{3} - \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) \log {\left (x + \frac {a^{2} c - 3 a^{2} \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) - 2 a b^{2} + b c^{2} - 6 b c \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) + 9 b \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right )^{2}}{a^{3} - b^{3}} \right )} - \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) \log {\left (x + \frac {a^{2} c - 3 a^{2} \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) - 2 a b^{2} + b c^{2} - 6 b c \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) + 9 b \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right )^{2}}{a^{3} - b^{3}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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