Optimal. Leaf size=32 \[ \frac {1}{3} (a-c) \log \left (x^2+x+1\right )-\frac {1}{3} (2 a+c) \log (1-x) \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1875, 31, 628} \[ \frac {1}{3} (a-c) \log \left (x^2+x+1\right )-\frac {1}{3} (2 a+c) \log (1-x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 628
Rule 1875
Rubi steps
\begin {align*} \int \frac {a+a x+c x^2}{1-x^3} \, dx &=\frac {1}{3} \int \frac {a-c+(2 a-2 c) x}{1+x+x^2} \, dx+\frac {1}{3} (2 a+c) \int \frac {1}{1-x} \, dx\\ &=-\frac {1}{3} (2 a+c) \log (1-x)+\frac {1}{3} (a-c) \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.97 \[ \frac {1}{3} \left ((a-c) \log \left (x^2+x+1\right )-(2 a+c) \log (1-x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 26, normalized size = 0.81 \[ \frac {1}{3} \, {\left (a - c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (2 \, a + c\right )} \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 0.84 \[ \frac {1}{3} \, {\left (a - c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (2 \, a + c\right )} \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.12 \[ -\frac {2 a \ln \left (x -1\right )}{3}+\frac {a \ln \left (x^{2}+x +1\right )}{3}-\frac {c \ln \left (x -1\right )}{3}-\frac {c \ln \left (x^{2}+x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 26, normalized size = 0.81 \[ \frac {1}{3} \, {\left (a - c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (2 \, a + c\right )} \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.78, size = 35, normalized size = 1.09 \[ \frac {a\,\ln \left (x^2+x+1\right )}{3}-\frac {c\,\ln \left (x-1\right )}{3}-\frac {2\,a\,\ln \left (x-1\right )}{3}-\frac {c\,\ln \left (x^2+x+1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 24, normalized size = 0.75 \[ \frac {\left (a - c\right ) \log {\left (x^{2} + x + 1 \right )}}{3} - \frac {\left (2 a + c\right ) \log {\left (x - 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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