Integrand size = 20, antiderivative size = 55 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\frac {(a-b) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} (a+b+c) \log (1-x)+\frac {1}{6} (a+b-2 c) \log \left (1+x+x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1889, 31, 648, 632, 210, 642} \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\frac {(a-b) \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (x^2+x+1\right ) (a+b-2 c)-\frac {1}{3} \log (1-x) (a+b+c) \]
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Rule 31
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1889
Rubi steps \begin{align*} \text {integral}& = \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) \text {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*}
Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\frac {1}{6} \left (2 \sqrt {3} (a-b) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-2 (a+b) \log (1-x)+(a+b) \log \left (1+x+x^2\right )-2 c \log \left (1-x^3\right )\right ) \]
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Time = 1.50 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00
method | result | size |
default | \(\left (-\frac {c}{3}-\frac {b}{3}-\frac {a}{3}\right ) \ln \left (-1+x \right )+\frac {\left (a +b -2 c \right ) \ln \left (x^{2}+x +1\right )}{6}+\frac {2 \left (\frac {3 a}{2}-\frac {3 b}{2}\right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}\) | \(55\) |
risch | \(-\frac {\ln \left (-1+x \right ) c}{3}-\frac {\ln \left (-1+x \right ) b}{3}-\frac {\ln \left (-1+x \right ) a}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (-a -b +2 c \right ) \textit {\_Z} +a^{2}-a b -a c +b^{2}-b c +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-\textit {\_R} a -a c +b^{2}\right ) x -\textit {\_R}^{2}-2 c \textit {\_R} +a b -c^{2}\right )\right )}{3}\) | \(101\) |
meijerg | \(-\frac {c \ln \left (-x^{3}+1\right )}{3}-\frac {b \,x^{2} \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}-\frac {a x \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(138\) |
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Time = 0.37 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\frac {1}{3} \, \sqrt {3} {\left (a - b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, {\left (a + b - 2 \, c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (a + b + c\right )} \log \left (x - 1\right ) \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 323, normalized size of antiderivative = 5.87 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=- \frac {\left (a + b + c\right ) \log {\left (x + \frac {a^{2} c - a^{2} \left (a + b + c\right ) - 2 a b^{2} + b c^{2} - 2 b c \left (a + b + c\right ) + b \left (a + b + c\right )^{2}}{a^{3} - b^{3}} \right )}}{3} - \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) \log {\left (x + \frac {a^{2} c - 3 a^{2} \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) - 2 a b^{2} + b c^{2} - 6 b c \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) + 9 b \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} - \frac {\sqrt {3} i \left (a - b\right )}{6}\right )^{2}}{a^{3} - b^{3}} \right )} - \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) \log {\left (x + \frac {a^{2} c - 3 a^{2} \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) - 2 a b^{2} + b c^{2} - 6 b c \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right ) + 9 b \left (- \frac {a}{6} - \frac {b}{6} + \frac {c}{3} + \frac {\sqrt {3} i \left (a - b\right )}{6}\right )^{2}}{a^{3} - b^{3}} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\frac {1}{3} \, \sqrt {3} {\left (a - b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, {\left (a + b - 2 \, c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (a + b + c\right )} \log \left (x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\frac {1}{3} \, {\left (\sqrt {3} a - \sqrt {3} b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, {\left (a + b - 2 \, c\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, {\left (a + b + c\right )} \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 10.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.58 \[ \int \frac {a+b x+c x^2}{1-x^3} \, dx=\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {a}{6}+\frac {b}{6}-\frac {c}{3}-\frac {\sqrt {3}\,a\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {a}{6}+\frac {b}{6}-\frac {c}{3}+\frac {\sqrt {3}\,a\,1{}\mathrm {i}}{6}-\frac {\sqrt {3}\,b\,1{}\mathrm {i}}{6}\right )-\ln \left (x-1\right )\,\left (\frac {a}{3}+\frac {b}{3}+\frac {c}{3}\right ) \]
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