Optimal. Leaf size=163 \[ \frac {1}{20} \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {\sqrt {10-2 \sqrt {5}} x}{2 \left (1-x^2\right )}\right )+\frac {1}{20} \sqrt {10+2 \sqrt {5}} \tan ^{-1}\left (\frac {\sqrt {10+2 \sqrt {5}} x}{2 \left (1-x^2\right )}\right )+\frac {1}{5} \tanh ^{-1}(x)+\frac {1}{20} \left (1-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) x}{2 \left (1+x^2\right )}\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\left (1+\sqrt {5}\right ) x}{2 \left (1+x^2\right )}\right ) \]
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Rubi [A]
time = 0.16, antiderivative size = 325, normalized size of antiderivative = 1.99, number of steps
used = 10, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {216, 648, 632,
210, 642, 212} \begin {gather*} -\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (-4 x+\sqrt {5}+1\right )\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\left (\frac {4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 x+\sqrt {5}+1\right )\right )-\frac {1}{40} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )+\frac {1}{40} \left (1-\sqrt {5}\right ) \log \left (x^2+\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )-\frac {1}{40} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{40} \left (1+\sqrt {5}\right ) \log \left (x^2+\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{5} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{1-x^{10}} \, dx &=\frac {1}{5} \int \frac {1-\frac {1}{4} \left (-1+\sqrt {5}\right ) x}{1-\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{5} \int \frac {1+\frac {1}{4} \left (-1+\sqrt {5}\right ) x}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{5} \int \frac {1-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{5} \int \frac {1+\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{5} \int \frac {1}{1-x^2} \, dx\\ &=\frac {1}{5} \tanh ^{-1}(x)+\frac {1}{40} \left (-1-\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (1-\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (5-\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (5-\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (-1+\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (1+\sqrt {5}\right ) \int \frac {\frac {1}{2} \left (1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (5+\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx+\frac {1}{40} \left (5+\sqrt {5}\right ) \int \frac {1}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx\\ &=\frac {1}{5} \tanh ^{-1}(x)-\frac {1}{40} \left (1+\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )+\frac {1}{40} \left (1-\sqrt {5}\right ) \log \left (2+x-\sqrt {5} x+2 x^2\right )-\frac {1}{40} \left (1-\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )+\frac {1}{40} \left (1+\sqrt {5}\right ) \log \left (2+x+\sqrt {5} x+2 x^2\right )+\frac {1}{20} \left (-5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1-\sqrt {5}\right )+2 x\right )+\frac {1}{20} \left (-5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1+\sqrt {5}\right )+2 x\right )-\frac {1}{20} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1-\sqrt {5}\right )+2 x\right )-\frac {1}{20} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (-1+\sqrt {5}\right )+2 x\right )\\ &=-\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}-4 x\right )\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 x\right )\right )+\frac {1}{5} \tanh ^{-1}(x)-\frac {1}{40} \left (1+\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )+\frac {1}{40} \left (1-\sqrt {5}\right ) \log \left (2+x-\sqrt {5} x+2 x^2\right )-\frac {1}{40} \left (1-\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )+\frac {1}{40} \left (1+\sqrt {5}\right ) \log \left (2+x+\sqrt {5} x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 289, normalized size = 1.77 \begin {gather*} \frac {1}{40} \left (-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}+4 x}{\sqrt {10-2 \sqrt {5}}}\right )-4 \log (1-x)+4 \log (1+x)-\left (-1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )+\left (-1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )-\left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )+\left (1+\sqrt {5}\right ) \log \left (\frac {1}{2} \left (2+x+\sqrt {5} x+2 x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs.
\(2(121)=242\).
time = 0.20, size = 288, normalized size = 1.77
method | result | size |
risch | \(-\frac {\ln \left (x -1\right )}{10}+\frac {\ln \left (x +1\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{10}\) | \(66\) |
meijerg | \(-\frac {x \left (\ln \left (1-\left (x^{10}\right )^{\frac {1}{10}}\right )-\ln \left (1+\left (x^{10}\right )^{\frac {1}{10}}\right )+\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )-\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )\right )}{10 \left (x^{10}\right )^{\frac {1}{10}}}\) | \(254\) |
default | \(-\frac {\ln \left (x -1\right )}{10}-\frac {\left (-\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}+x +2\right )}{40}-\frac {\left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {1+4 x +\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (-\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}+x +2\right )}{40}+\frac {\left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1+4 x -\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {\ln \left (x +1\right )}{10}+\frac {\left (\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{40}+\frac {\left (4-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{40}-\frac {\left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(288\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1039 vs.
\(2 (113) = 226\).
time = 1.17, size = 1039, normalized size = 6.37 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.33, size = 70, normalized size = 0.43 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{10} + \frac {\log {\left (x + 1 \right )}}{10} - \operatorname {RootSum} {\left (10000 t^{4} - 1000 t^{3} + 100 t^{2} - 10 t + 1, \left ( t \mapsto t \log {\left (- 10 t + x \right )} \right )\right )} - \operatorname {RootSum} {\left (10000 t^{4} + 1000 t^{3} + 100 t^{2} + 10 t + 1, \left ( t \mapsto t \log {\left (- 10 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.09, size = 223, normalized size = 1.37 \begin {gather*} \frac {1}{40} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {1}{40} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{40} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{40} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} + 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{10} \, \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 = 1.29, size = 852, normalized size = 5.23 \begin {gather*} -\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )\,1{}\mathrm {i}+\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )-\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}\,1{}\mathrm {i}}{20}-\frac {\sqrt {5}\,1{}\mathrm {i}}{20}+\frac {1}{20}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}+\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )-\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}\,1{}\mathrm {i}}{20}+\frac {\sqrt {5}\,1{}\mathrm {i}}{20}-\frac {1}{20}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (\frac {5\,\sqrt {5}}{2}-10\,x+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}-\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (\frac {5\,\sqrt {5}}{2}-10\,x+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )+\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {5}\,1{}\mathrm {i}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}\,1{}\mathrm {i}}{20}+\frac {1}{20}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}-\frac {5}{2}\right )\,1{}\mathrm {i}+\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}-\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}-\frac {5}{2}\right )-\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}-\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {5}\,1{}\mathrm {i}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}\,1{}\mathrm {i}}{20}+\frac {1}{20}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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