Integrand size = 13, antiderivative size = 69 \[ \int \frac {1+x^6}{-1+x^6} \, dx=x+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2 \text {arctanh}(x)}{3}+\frac {1}{6} \log \left (1-x+x^2\right )-\frac {1}{6} \log \left (1+x+x^2\right ) \]
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Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {396, 216, 648, 632, 210, 642, 212} \[ \int \frac {1+x^6}{-1+x^6} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2 \text {arctanh}(x)}{3}+\frac {1}{6} \log \left (x^2-x+1\right )-\frac {1}{6} \log \left (x^2+x+1\right )+x \]
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Rule 210
Rule 212
Rule 216
Rule 396
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = x+2 \int \frac {1}{-1+x^6} \, dx \\ & = x-\frac {2}{3} \int \frac {1}{1-x^2} \, dx-\frac {2}{3} \int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx-\frac {2}{3} \int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx \\ & = x-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {1}{6} \int \frac {1+2 x}{1+x+x^2} \, dx-\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx-\frac {1}{2} \int \frac {1}{1+x+x^2} \, dx \\ & = x-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \log \left (1-x+x^2\right )-\frac {1}{6} \log \left (1+x+x^2\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right ) \\ & = x-\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}(x)+\frac {1}{6} \log \left (1-x+x^2\right )-\frac {1}{6} \log \left (1+x+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {1+x^6}{-1+x^6} \, dx=\frac {1}{6} \left (6 x-2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )+2 \log (1-x)-2 \log (1+x)+\log \left (1-x+x^2\right )-\log \left (1+x+x^2\right )\right ) \]
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Time = 0.83 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97
method | result | size |
default | \(x -\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (x -1\right )}{3}-\frac {\ln \left (x +1\right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(67\) |
risch | \(x -\frac {\ln \left (4 x^{2}+4 x +4\right )}{6}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (4 x^{2}-4 x +4\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}-\frac {\ln \left (x +1\right )}{3}+\frac {\ln \left (x -1\right )}{3}\) | \(73\) |
meijerg | \(\frac {x \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}\right )-\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}\right )+\frac {\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}-\frac {\left (-1\right )^{\frac {5}{6}} \left (6 x \left (-1\right )^{\frac {1}{6}}+\frac {x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}\right )-\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}\right )+\frac {\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{\left (x^{6}\right )^{\frac {1}{6}}}\right )}{6}\) | \(246\) |
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {1+x^6}{-1+x^6} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + x - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \]
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Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.23 \[ \int \frac {1+x^6}{-1+x^6} \, dx=x + \frac {\log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{6} - \frac {\log {\left (x^{2} + x + 1 \right )}}{6} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {1+x^6}{-1+x^6} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + x - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int \frac {1+x^6}{-1+x^6} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + x - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.36 \[ \int \frac {1+x^6}{-1+x^6} \, dx=x+\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3}-\mathrm {atan}\left (\frac {x\,32{}\mathrm {i}}{-32+\sqrt {3}\,32{}\mathrm {i}}-\frac {32\,\sqrt {3}\,x}{-32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}-\frac {1}{3}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,32{}\mathrm {i}}{32+\sqrt {3}\,32{}\mathrm {i}}+\frac {32\,\sqrt {3}\,x}{32+\sqrt {3}\,32{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{3}+\frac {1}{3}{}\mathrm {i}\right ) \]
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