Optimal. Leaf size=185 \[ -\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+2 \sqrt {\frac {2}{5+\sqrt {5}}} x\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )-\frac {1}{5} \log (1+x)+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \]
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Rubi [A]
time = 0.16, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {299, 648, 632,
210, 642, 31} \begin {gather*} -\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )-\frac {1}{5} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 299
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x}{1+x^5} \, dx &=\frac {2}{5} \int \frac {\frac {1}{4} \left (1-\sqrt {5}\right )-\frac {1}{4} \left (-1-\sqrt {5}\right ) x}{1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx+\frac {2}{5} \int \frac {\frac {1}{4} \left (1+\sqrt {5}\right )-\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} \, dx\\ &=-\frac {1}{5} \log (1+x)+\frac {\int \frac {1}{1+\frac {1}{2} \left (-1-\sqrt {5}\right ) x+x^2} \, dx}{2 \sqrt {5}}-\frac {\int \frac {1}{1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2} \, dx}{2 \sqrt {5}}+\frac {1}{20} \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}{20} \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}{5} \log (1+x)+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (2-x+\sqrt {5} x+2 x^2\right )+\frac {\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 )}{\sqrt {5}}-\frac {\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 )}{\sqrt {5}}\\ &=\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \tan ^{-1}\left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{5} \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} \log (1+x)+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (2-x-\sqrt {5} x+2 x^2\right )+\frac {1}{20} \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 = 0.03, size = 144, normalized size = 0.78 \begin {gather*} \frac {1}{20} \left (-2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-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 )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 157, normalized size = 0.85
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}+x \right )\right )}{5}-\frac {\ln \left (x +1\right )}{5}\) | \(40\) |
default | \(-\frac {\ln \left (x +1\right )}{5}+\frac {\left (\sqrt {5}+1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}+1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (-\sqrt {5}-1-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(157\) |
meijerg | \(-\frac {x^{2} \ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {x^{2} \cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}+\frac {2 x^{2} \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}+\frac {x^{2} \cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {2 x^{2} \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 124, normalized size = 0.67 \begin {gather*} -\frac {2 \, \sqrt {5} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} - \frac {\log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{5 \, {\left (\sqrt {5} + 1\right )}} + \frac {\log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{5 \, {\left (\sqrt {5} - 1\right )}} - \frac {1}{5} \, \log \left (x + 1\right ) \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. 980 vs.
\(2 (122) = 244\).
time = 1.12, size = 980, normalized size = 5.30 \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 = 0.50, size = 36, normalized size = 0.19 \begin {gather*} - \frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log {\left (- 125 t^{3} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.49, size = 112, normalized size = 0.61 \begin {gather*} -\frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {1}{5} \, \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 = 0.26, size = 200, normalized size = 1.08 \begin {gather*} \ln \left (x-125\,{\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )}^3\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-\frac {\ln \left (x+1\right )}{5}-\ln \left (x+125\,{\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )}^3\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )+\ln \left (x-125\,{\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )}^3\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )+\ln \left (x-125\,{\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )}^3\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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