Optimal. Leaf size=165 \[ \frac {2}{7} \tan ^{-1}\left (x \sec \left (\frac {\pi }{14}\right )-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \tan ^{-1}\left (x \sec \left (\frac {3 \pi }{14}\right )+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\frac {1}{7} \log (1+x)-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{7}\right )\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-\frac {2}{7} \tan ^{-1}\left (\cot \left (\frac {\pi }{7}\right )-x \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {207, 648, 632,
210, 642, 31} \begin {gather*} \frac {2}{7} \cos \left (\frac {3 \pi }{14}\right ) \text {ArcTan}\left (x \sec \left (\frac {3 \pi }{14}\right )+\tan \left (\frac {3 \pi }{14}\right )\right )+\frac {2}{7} \cos \left (\frac {\pi }{14}\right ) \text {ArcTan}\left (x \sec \left (\frac {\pi }{14}\right )-\tan \left (\frac {\pi }{14}\right )\right )-\frac {2}{7} \sin \left (\frac {\pi }{7}\right ) \text {ArcTan}\left (\cot \left (\frac {\pi }{7}\right )-x \csc \left (\frac {\pi }{7}\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 (x+1) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 31
Rule 207
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{1+x^7} \, dx &=\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*}
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Mathematica [A]
time = 0.00, size = 166, normalized size = 1.01 \begin {gather*} \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 {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.18, size = 97, normalized size = 0.59
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{5}+\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{7}+\frac {\ln \left (x +1\right )}{7}\) | \(38\) |
default | \(\frac {\ln \left (x +1\right )}{7}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{5}+\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{5}+2 \textit {\_R}^{4}-3 \textit {\_R}^{3}+4 \textit {\_R}^{2}-5 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{6 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{7}\) | \(97\) |
meijerg | \(\frac {x \ln \left (1+\left (x^{7}\right )^{\frac {1}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}-\frac {x \cos \left (\frac {\pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}-\frac {x \cos \left (\frac {3 \pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {3 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {x \cos \left (\frac {2 \pi }{7}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {2 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1+\cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 3.41, size = 65, normalized size = 0.39 \begin {gather*} \frac {1}{14} \, {\left (\sqrt {-2.445041867912629? + 0.?e-37 \sqrt {-1}} + 1.246979603717467? + 0.?e-36 \sqrt {-1}\right )} \log \left (2 \, x + \sqrt {-2.445041867912629? + 0.?e-37 \sqrt {-1}} + 1.246979603717467? + 0.?e-36 \sqrt {-1}\right ) - \frac {1}{14} \, {\left (\sqrt {-2.445041867912629? + 0.?e-37 \sqrt {-1}} - 1.246979603717467? + 0.?e-36 \sqrt {-1}\right )} \log \left (2 \, x - \sqrt {-2.445041867912629? + 0.?e-37 \sqrt {-1}} + 1.246979603717467? + 0.?e-36 \sqrt {-1}\right ) + \frac {1}{7} \, \log \left (x + 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 (0.1287098382717742? - 0.06198339130250831? \sqrt {-1}\right ) \, \log \left (x - 0.9009688679024191? + 0.4338837391175581? \sqrt {-1}\right ) - \left (0.1287098382717742? + 0.06198339130250831? \sqrt {-1}\right ) \, \log \left (x - 0.9009688679024191? - 0.4338837391175582? \sqrt {-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 44, normalized size = 0.27 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 129, normalized size = 0.78 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.34, size = 103, normalized size = 0.62 \begin {gather*} \frac {\ln \left (x+1\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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