Integrand size = 18, antiderivative size = 44 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=4 x+\frac {4 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \log (1+x)+\frac {1}{3} \log \left (1-x+x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1901, 1874, 31, 648, 632, 210, 642} \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=\frac {4 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x^2-x+1\right )+4 x-\frac {2}{3} \log (x+1) \]
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Rule 31
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1874
Rule 1901
Rubi steps \begin{align*} \text {integral}& = \int \left (4-\frac {3+x}{1+x^3}\right ) \, dx \\ & = 4 x-\int \frac {3+x}{1+x^3} \, dx \\ & = 4 x-\frac {1}{3} \int \frac {7-2 x}{1-x+x^2} \, dx-\frac {2}{3} \int \frac {1}{1+x} \, dx \\ & = 4 x-\frac {2}{3} \log (1+x)+\frac {1}{3} \int \frac {-1+2 x}{1-x+x^2} \, dx-2 \int \frac {1}{1-x+x^2} \, dx \\ & = 4 x-\frac {2}{3} \log (1+x)+\frac {1}{3} \log \left (1-x+x^2\right )+4 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right ) \\ & = 4 x+\frac {4 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \log (1+x)+\frac {1}{3} \log \left (1-x+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=4 x-\frac {4 \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \log (1+x)+\frac {1}{3} \log \left (1-x+x^2\right ) \]
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Time = 1.45 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86
method | result | size |
default | \(4 x -\frac {2 \ln \left (1+x \right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{3}\) | \(38\) |
risch | \(4 x -\frac {2 \ln \left (1+x \right )}{3}+\frac {\ln \left (16 x^{2}-16 x +16\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (-2+4 x \right ) \sqrt {3}}{6}\right )}{3}\) | \(40\) |
meijerg | \(4 x -\frac {4 x \left (\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}-\frac {\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3}+\frac {x^{2} \ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}-\frac {x^{2} \ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{6 \left (x^{3}\right )^{\frac {2}{3}}}-\frac {x^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x \ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}-\frac {x \ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{6 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {x \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(226\) |
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Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 4 \, x + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=4 x - \frac {2 \log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{3} - \frac {4 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 4 \, x + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 4 \, x + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 9.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11 \[ \int \frac {1-x+4 x^3}{1+x^3} \, dx=4\,x-\frac {2\,\ln \left (x+1\right )}{3}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right ) \]
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