Optimal. Leaf size=192 \[ -\frac {1}{2 x^2}+\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+2 \sqrt {\frac {2}{5+\sqrt {5}}} x\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right )-\frac {1}{5} \log (1+x)+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2\right )+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {331, 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}{2 x^2}+\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 331
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^5\right )} \, dx &=-\frac {1}{2 x^2}-\int \frac {x^2}{1+x^5} \, dx\\ &=-\frac {1}{2 x^2}-\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}{2 x^2}-\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}{2 x^2}-\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}}\\ &=-\frac {1}{2 x^2}-\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 = 0.07, size = 149, normalized size = 0.78 \begin {gather*} \frac {1}{20} \left (-\frac {10}{x^2}+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 = 158, normalized size = 0.82
method | result | size |
risch | \(-\frac {1}{2 x^{2}}-\frac {\ln \left (x +1\right )}{5}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{2}+x \right )\right )}{5}\) | \(43\) |
default | \(-\frac {\ln \left (x +1\right )}{5}-\frac {1}{2 x^{2}}+\frac {\left (\sqrt {5}+1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{20}+\frac {2 \left (\sqrt {5}+1-\frac {\left (\sqrt {5}+1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{20}-\frac {2 \left (\sqrt {5}-1-\frac {\left (-\sqrt {5}-1\right ) \left (\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(158\) |
meijerg | \(-\frac {1}{2 x^{2}}-\frac {x^{3} \left (\frac {\ln \left (1+\left (x^{5}\right )^{\frac {1}{5}}\right )}{\left (x^{5}\right )^{\frac {3}{5}}}+\frac {\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{\left (x^{5}\right )^{\frac {3}{5}}}+\frac {2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{\left (x^{5}\right )^{\frac {3}{5}}}-\frac {\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )}{\left (x^{5}\right )^{\frac {3}{5}}}-\frac {2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )}{\left (x^{5}\right )^{\frac {3}{5}}}\right )}{5}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 129, 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}{2 \, x^{2}} - \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. 692 vs.
\(2 (127) = 254\).
time = 1.07, size = 692, normalized size = 3.60 \begin {gather*} -\frac {2 \, x^{2} {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} \log \left (\frac {1}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} + x\right ) + 2 \, x^{2} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} \log \left (\frac {1}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} + x\right ) + 8 \, x^{2} \log \left (x + 1\right ) - {\left (x^{2} {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} + x^{2} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} + 4 \, \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} x^{2} + 4 \, x^{2}\right )} \log \left (-\frac {1}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} + \frac {1}{2} \, \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} {\left (\sqrt {5} + 1\right )} + 2 \, x - 1\right ) - {\left (x^{2} {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} + x^{2} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - 4 \, \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} x^{2} + 4 \, x^{2}\right )} \log \left (-\frac {1}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {-\frac {3}{16} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{8} \, {\left (\sqrt {5} + 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} + 3\right )} {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )} - \frac {3}{16} \, {\left (\sqrt {5} - 10 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - 1\right )}^{2} - \frac {1}{2} \, \sqrt {5} - 5 \, \sqrt {-\frac {1}{50} \, \sqrt {5} - \frac {1}{10}} - \frac {5}{2}} {\left (\sqrt {5} + 1\right )} + 2 \, x - 1\right ) + 20}{40 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 42, normalized size = 0.22 \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 (25 t^{2} + x \right )} \right )\right )} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 132, normalized size = 0.69 \begin {gather*} \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}{20} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{20} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{2 \, x^{2}} + \frac {1}{20} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\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 = 1.70, size = 200, normalized size = 1.04 \begin {gather*} \ln \left (1-\frac {x\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}-\sqrt {5}+1\right )}^3}{64}\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 (\frac {x\,{\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}+\sqrt {5}-1\right )}^3}{64}+1\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\frac {1}{2\,x^2}+\ln \left (1-\frac {x\,{\left (\sqrt {5}+\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{64}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )+\ln \left (1-\frac {x\,{\left (\sqrt {5}-\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}^3}{64}\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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