Optimal. Leaf size=54 \[ -\frac {1}{6} (a-b) \log \left (x^2-x+1\right )+\frac {1}{3} (a-b) \log (x+1)-\frac {(a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {800, 634, 618, 204, 628} \begin {gather*} -\frac {1}{6} (a-b) \log \left (x^2-x+1\right )+\frac {1}{3} (a-b) \log (x+1)-\frac {(a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 800
Rubi steps
\begin {align*} \int \frac {a+b x}{(1+x) \left (1-x+x^2\right )} \, dx &=\int \left (\frac {a-b}{3 (1+x)}+\frac {2 a+b-(a-b) x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac {1}{3} (a-b) \log (1+x)+\frac {1}{3} \int \frac {2 a+b-(a-b) x}{1-x+x^2} \, dx\\ &=\frac {1}{3} (a-b) \log (1+x)+\frac {1}{6} (-a+b) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{2} (a+b) \int \frac {1}{1-x+x^2} \, dx\\ &=\frac {1}{3} (a-b) \log (1+x)-\frac {1}{6} (a-b) \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) \log (1+x)-\frac {1}{6} (a-b) \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.91 \begin {gather*} \frac {1}{6} (a-b) \left (2 \log (x+1)-\log \left (x^2-x+1\right )\right )+\frac {(a+b) \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(1+x) \left (1-x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 47, normalized size = 0.87 \begin {gather*} \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\right )} \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, {\left (a - b\right )} \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.98 \begin {gather*} \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\right )} \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, {\left (a - b\right )} \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 74, normalized size = 1.37 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 47, normalized size = 0.87 \begin {gather*} \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\right )} \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, {\left (a - b\right )} \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 78, normalized size = 1.44 \begin {gather*} \ln \left (x+1\right )\,\left (\frac {a}{3}-\frac {b}{3}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {b}{6}-\frac {a}{6}+\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 {\sqrt {3}\,a\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.43, size = 201, normalized size = 3.72 \begin {gather*} \frac {\left (a - b\right ) \log {\left (x + \frac {a^{2} \left (a - b\right ) + 2 a b^{2} + b \left (a - b\right )^{2}}{a^{3} + b^{3}} \right )}}{3} + \left (- \frac {a}{6} + \frac {b}{6} - \frac {\sqrt {3} i \left (a + b\right )}{6}\right ) \log {\left (x + \frac {3 a^{2} \left (- \frac {a}{6} + \frac {b}{6} - \frac {\sqrt {3} i \left (a + b\right )}{6}\right ) + 2 a b^{2} + 9 b \left (- \frac {a}{6} + \frac {b}{6} - \frac {\sqrt {3} i \left (a + b\right )}{6}\right )^{2}}{a^{3} + b^{3}} \right )} + \left (- \frac {a}{6} + \frac {b}{6} + \frac {\sqrt {3} i \left (a + b\right )}{6}\right ) \log {\left (x + \frac {3 a^{2} \left (- \frac {a}{6} + \frac {b}{6} + \frac {\sqrt {3} i \left (a + b\right )}{6}\right ) + 2 a b^{2} + 9 b \left (- \frac {a}{6} + \frac {b}{6} + \frac {\sqrt {3} i \left (a + b\right )}{6}\right )^{2}}{a^{3} + b^{3}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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