Integrand size = 9, antiderivative size = 166 \[ \int \frac {1}{1-x^7} \, dx=\frac {2}{7} \arctan \left (\sec \left (\frac {\pi }{14}\right ) \left (x+\sin \left (\frac {\pi }{14}\right )\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\sec \left (\frac {3 \pi }{14}\right ) \left (x-\sin \left (\frac {3 \pi }{14}\right )\right )\right ) \cos \left (\frac {3 \pi }{14}\right )-\frac {1}{7} \log (1-x)+\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{7}\right )\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\left (x+\cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {208, 648, 632, 210, 642, 31} \[ \int \frac {1}{1-x^7} \, dx=\frac {2}{7} \sin \left (\frac {\pi }{7}\right ) \arctan \left (\csc \left (\frac {\pi }{7}\right ) \left (x+\cos \left (\frac {\pi }{7}\right )\right )\right )+\frac {2}{7} \cos \left (\frac {3 \pi }{14}\right ) \arctan \left (\sec \left (\frac {3 \pi }{14}\right ) \left (x-\sin \left (\frac {3 \pi }{14}\right )\right )\right )+\frac {2}{7} \cos \left (\frac {\pi }{14}\right ) \arctan \left (\sec \left (\frac {\pi }{14}\right ) \left (x+\sin \left (\frac {\pi }{14}\right )\right )\right )-\frac {1}{7} \sin \left (\frac {3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac {3 \pi }{14}\right )+1\right )+\frac {1}{7} \sin \left (\frac {\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac {\pi }{14}\right )+1\right )+\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac {\pi }{7}\right )+1\right )-\frac {1}{7} \log (1-x) \]
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Rule 31
Rule 208
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} \int \frac {1+x \cos \left (\frac {\pi }{7}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{7}\right )} \, dx+\frac {2}{7} \int \frac {1+x \sin \left (\frac {\pi }{14}\right )}{1+x^2+2 x \sin \left (\frac {\pi }{14}\right )} \, dx+\frac {2}{7} \int \frac {1-x \sin \left (\frac {3 \pi }{14}\right )}{1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )} \, dx+\frac {1}{7} \int \frac {1}{1-x} \, dx \\ & = -\frac {1}{7} \log (1-x)+\frac {1}{7} \left (2 \cos ^2\left (\frac {\pi }{14}\right )\right ) \int \frac {1}{1+x^2+2 x \sin \left (\frac {\pi }{14}\right )} \, dx+\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \int \frac {2 x+2 \cos \left (\frac {\pi }{7}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{7}\right )} \, dx+\frac {1}{7} \left (2 \cos ^2\left (\frac {3 \pi }{14}\right )\right ) \int \frac {1}{1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )} \, dx+\frac {1}{7} \sin \left (\frac {\pi }{14}\right ) \int \frac {2 x+2 \sin \left (\frac {\pi }{14}\right )}{1+x^2+2 x \sin \left (\frac {\pi }{14}\right )} \, dx+\frac {1}{7} \left (2 \sin ^2\left (\frac {\pi }{7}\right )\right ) \int \frac {1}{1+x^2+2 x \cos \left (\frac {\pi }{7}\right )} \, dx-\frac {1}{7} \sin \left (\frac {3 \pi }{14}\right ) \int \frac {2 x-2 \sin \left (\frac {3 \pi }{14}\right )}{1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )} \, dx \\ & = -\frac {1}{7} \log (1-x)+\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{7}\right )\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )-\frac {1}{7} \left (4 \cos ^2\left (\frac {\pi }{14}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {\pi }{14}\right )} \, dx,x,2 x+2 \sin \left (\frac {\pi }{14}\right )\right )-\frac {1}{7} \left (4 \cos ^2\left (\frac {3 \pi }{14}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {3 \pi }{14}\right )} \, dx,x,2 x-2 \sin \left (\frac {3 \pi }{14}\right )\right )-\frac {1}{7} \left (4 \sin ^2\left (\frac {\pi }{7}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {\pi }{7}\right )} \, dx,x,2 x+2 \cos \left (\frac {\pi }{7}\right )\right ) \\ & = \frac {2}{7} \tan ^{-1}\left (\sec \left (\frac {\pi }{14}\right ) \left (x+\sin \left (\frac {\pi }{14}\right )\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \tan ^{-1}\left (\sec \left (\frac {3 \pi }{14}\right ) \left (x-\sin \left (\frac {3 \pi }{14}\right )\right )\right ) \cos \left (\frac {3 \pi }{14}\right )-\frac {1}{7} \log (1-x)+\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{7}\right )\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )+\frac {2}{7} \tan ^{-1}\left (\left (x+\cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-x^7} \, dx=\frac {2}{7} \arctan \left (\sec \left (\frac {\pi }{14}\right ) \left (x+\sin \left (\frac {\pi }{14}\right )\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\sec \left (\frac {3 \pi }{14}\right ) \left (x-\sin \left (\frac {3 \pi }{14}\right )\right )\right ) \cos \left (\frac {3 \pi }{14}\right )-\frac {1}{7} \log (1-x)+\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{7}\right )\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\left (x+\cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 11.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.27
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{5}+\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{7}-\frac {\ln \left (-1+x \right )}{7}\) | \(44\) |
default | \(-\frac {\ln \left (-1+x \right )}{7}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{5}+\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{5}+2 \textit {\_R}^{4}+3 \textit {\_R}^{3}+4 \textit {\_R}^{2}+5 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{6 \textit {\_R}^{5}+5 \textit {\_R}^{4}+4 \textit {\_R}^{3}+3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{7}\) | \(89\) |
meijerg | \(-\frac {x \left (\ln \left (1-\left (x^{7}\right )^{\frac {1}{7}}\right )+\cos \left (\frac {2 \pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )-2 \sin \left (\frac {2 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )-\cos \left (\frac {3 \pi }{7}\right ) \ln \left (1+2 \cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )-2 \sin \left (\frac {3 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1+\cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )-\cos \left (\frac {\pi }{7}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )-2 \sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1+\cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}\) | \(188\) |
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Result contains complex when optimal does not.
Time = 3.85 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39 \[ \int \frac {1}{1-x^7} \, dx=\frac {1}{14} \, {\left (\sqrt {-0.7530203962825330? + 0.?e-75 \sqrt {-1}} + 1.801937735804839? + 0.?e-75 \sqrt {-1}\right )} \log \left (2 \, x + \sqrt {-0.7530203962825330? + 0.?e-75 \sqrt {-1}} + 1.801937735804839? + 0.?e-75 \sqrt {-1}\right ) - \frac {1}{14} \, {\left (\sqrt {-0.7530203962825330? + 0.?e-75 \sqrt {-1}} - 1.801937735804839? + 0.?e-75 \sqrt {-1}\right )} \log \left (2 \, x - \sqrt {-0.7530203962825330? + 0.?e-75 \sqrt {-1}} + 1.801937735804839? + 0.?e-75 \sqrt {-1}\right ) + \left (0.03178870485090206? + 0.1392754160259748? \sqrt {-1}\right ) \, \log \left (x + 0.2225209339563144? + 0.9749279121818236? \sqrt {-1}\right ) + \left (0.03178870485090206? - 0.1392754160259748? \sqrt {-1}\right ) \, \log \left (x + 0.2225209339563144? - 0.9749279121818236? \sqrt {-1}\right ) - \left (0.08906997169410479? - 0.11169021178114711? \sqrt {-1}\right ) \, \log \left (x - 0.6234898018587335? + 0.7818314824680299? \sqrt {-1}\right ) - \left (0.08906997169410479? + 0.11169021178114711? \sqrt {-1}\right ) \, \log \left (x - 0.6234898018587335? - 0.7818314824680299? \sqrt {-1}\right ) - \frac {1}{7} \, \log \left (x - 1\right ) \]
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Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.28 \[ \int \frac {1}{1-x^7} \, dx=- \frac {\log {\left (x - 1 \right )}}{7} - \operatorname {RootSum} {\left (117649 t^{6} + 16807 t^{5} + 2401 t^{4} + 343 t^{3} + 49 t^{2} + 7 t + 1, \left ( t \mapsto t \log {\left (- 7 t + x \right )} \right )\right )} \]
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\[ \int \frac {1}{1-x^7} \, dx=\int { -\frac {1}{x^{7} - 1} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.77 \[ \int \frac {1}{1-x^7} \, dx=\frac {1}{7} \, \cos \left (\frac {3}{7} \, \pi \right ) \log \left (x^{2} + 2 \, x \cos \left (\frac {3}{7} \, \pi \right ) + 1\right ) - \frac {1}{7} \, \cos \left (\frac {2}{7} \, \pi \right ) \log \left (x^{2} - 2 \, x \cos \left (\frac {2}{7} \, \pi \right ) + 1\right ) + \frac {1}{7} \, \cos \left (\frac {1}{7} \, \pi \right ) \log \left (x^{2} + 2 \, x \cos \left (\frac {1}{7} \, \pi \right ) + 1\right ) + \frac {2}{7} \, \arctan \left (\frac {x + \cos \left (\frac {3}{7} \, \pi \right )}{\sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right ) + \frac {2}{7} \, \arctan \left (\frac {x - \cos \left (\frac {2}{7} \, \pi \right )}{\sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right ) + \frac {2}{7} \, \arctan \left (\frac {x + \cos \left (\frac {1}{7} \, \pi \right )}{\sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right ) - \frac {1}{7} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 6.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62 \[ \int \frac {1}{1-x^7} \, dx=-\frac {\ln \left (x-1\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\right )}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}}{7} \]
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