Optimal. Leaf size=79 \[ \frac {x (a+b x)}{3 \left (x^3+1\right )}-\frac {1}{18} (2 a-b) \log \left (x^2-x+1\right )+\frac {1}{9} (2 a-b) \log (x+1)-\frac {(2 a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {809, 1855, 1860, 31, 634, 618, 204, 628} \begin {gather*} \frac {x (a+b x)}{3 \left (x^3+1\right )}-\frac {1}{18} (2 a-b) \log \left (x^2-x+1\right )+\frac {1}{9} (2 a-b) \log (x+1)-\frac {(2 a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 809
Rule 1855
Rule 1860
Rubi steps
\begin {align*} \int \frac {a+b x}{(1+x)^2 \left (1-x+x^2\right )^2} \, dx &=\int \frac {a+b x}{\left (1+x^3\right )^2} \, dx\\ &=\frac {x (a+b x)}{3 \left (1+x^3\right )}-\frac {1}{3} \int \frac {-2 a-b x}{1+x^3} \, dx\\ &=\frac {x (a+b x)}{3 \left (1+x^3\right )}-\frac {1}{9} \int \frac {-4 a-b+(2 a-b) x}{1-x+x^2} \, dx-\frac {1}{9} (-2 a+b) \int \frac {1}{1+x} \, dx\\ &=\frac {x (a+b x)}{3 \left (1+x^3\right )}+\frac {1}{9} (2 a-b) \log (1+x)-\frac {1}{6} (-2 a-b) \int \frac {1}{1-x+x^2} \, dx-\frac {1}{18} (2 a-b) \int \frac {-1+2 x}{1-x+x^2} \, dx\\ &=\frac {x (a+b x)}{3 \left (1+x^3\right )}+\frac {1}{9} (2 a-b) \log (1+x)-\frac {1}{18} (2 a-b) \log \left (1-x+x^2\right )-\frac {1}{3} (2 a+b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=\frac {x (a+b x)}{3 \left (1+x^3\right )}-\frac {(2 a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{9} (2 a-b) \log (1+x)-\frac {1}{18} (2 a-b) \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.91 \begin {gather*} \frac {1}{18} \left (\frac {6 x (a+b x)}{x^3+1}+(b-2 a) \log \left (x^2-x+1\right )+2 (2 a-b) \log (x+1)+2 \sqrt {3} (2 a+b) \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \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)^2 \left (1-x+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 103, normalized size = 1.30 \begin {gather*} \frac {6 \, b x^{2} + 2 \, \sqrt {3} {\left ({\left (2 \, a + b\right )} x^{3} + 2 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 6 \, a x - {\left ({\left (2 \, a - b\right )} x^{3} + 2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) + 2 \, {\left ({\left (2 \, a - b\right )} x^{3} + 2 \, a - b\right )} \log \left (x + 1\right )}{18 \, {\left (x^{3} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 101, normalized size = 1.28 \begin {gather*} \frac {1}{9} \, \sqrt {3} {\left (2 \, a + b\right )} \arctan \left (-\sqrt {3} {\left (\frac {2}{x + 1} - 1\right )}\right ) - \frac {1}{18} \, {\left (2 \, a - b\right )} \log \left (-\frac {3}{x + 1} + \frac {3}{{\left (x + 1\right )}^{2}} + 1\right ) - \frac {a}{9 \, {\left (x + 1\right )}} + \frac {b}{9 \, {\left (x + 1\right )}} - \frac {b + \frac {a - b}{x + 1}}{9 \, {\left (\frac {3}{x + 1} - \frac {3}{{\left (x + 1\right )}^{2}} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 116, normalized size = 1.47 \begin {gather*} \frac {2 \sqrt {3}\, a \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}+\frac {2 a \ln \left (x +1\right )}{9}-\frac {a \ln \left (x^{2}-x +1\right )}{9}+\frac {\sqrt {3}\, b \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}-\frac {b \ln \left (x +1\right )}{9}+\frac {b \ln \left (x^{2}-x +1\right )}{18}-\frac {a}{9 \left (x +1\right )}+\frac {b}{9 x +9}-\frac {-a +b +\left (-a -2 b \right ) x}{9 \left (x^{2}-x +1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 71, normalized size = 0.90 \begin {gather*} \frac {1}{9} \, \sqrt {3} {\left (2 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{18} \, {\left (2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) + \frac {1}{9} \, {\left (2 \, a - b\right )} \log \left (x + 1\right ) + \frac {b x^{2} + a x}{3 \, {\left (x^{3} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.29, size = 97, normalized size = 1.23 \begin {gather*} \frac {\frac {b\,x^2}{3}+\frac {a\,x}{3}}{x^3+1}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {a}{9}-\frac {b}{18}+\frac {\sqrt {3}\,a\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{18}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {b}{18}-\frac {a}{9}+\frac {\sqrt {3}\,a\,1{}\mathrm {i}}{9}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{18}\right )+\ln \left (x+1\right )\,\left (\frac {2\,a}{9}-\frac {b}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.59, size = 238, normalized size = 3.01 \begin {gather*} \frac {\left (2 a - b\right ) \log {\left (x + \frac {4 a^{2} \left (2 a - b\right ) + 4 a b^{2} + b \left (2 a - b\right )^{2}}{8 a^{3} + b^{3}} \right )}}{9} + \left (- \frac {a}{9} + \frac {b}{18} - \frac {\sqrt {3} i \left (2 a + b\right )}{18}\right ) \log {\left (x + \frac {36 a^{2} \left (- \frac {a}{9} + \frac {b}{18} - \frac {\sqrt {3} i \left (2 a + b\right )}{18}\right ) + 4 a b^{2} + 81 b \left (- \frac {a}{9} + \frac {b}{18} - \frac {\sqrt {3} i \left (2 a + b\right )}{18}\right )^{2}}{8 a^{3} + b^{3}} \right )} + \left (- \frac {a}{9} + \frac {b}{18} + \frac {\sqrt {3} i \left (2 a + b\right )}{18}\right ) \log {\left (x + \frac {36 a^{2} \left (- \frac {a}{9} + \frac {b}{18} + \frac {\sqrt {3} i \left (2 a + b\right )}{18}\right ) + 4 a b^{2} + 81 b \left (- \frac {a}{9} + \frac {b}{18} + \frac {\sqrt {3} i \left (2 a + b\right )}{18}\right )^{2}}{8 a^{3} + b^{3}} \right )} + \frac {a x + b x^{2}}{3 x^{3} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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