Optimal. Leaf size=185 \[ \frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )-\frac {1}{5} \log (x+1)+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right ) \]
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Rubi [A] time = 0.27, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {293, 634, 618, 204, 628, 31} \[ \frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) x+1\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )-\frac {1}{5} \log (x+1)+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} x+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 293
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^3}{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 {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 (5-\sqrt {5}\right ) \int \frac {1}{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}{20} \left (5+\sqrt {5}\right ) \int \frac {1}{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 {1}{10} \left (-5+\sqrt {5}\right ) \operatorname {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} \left (5+\sqrt {5}\right ) \operatorname {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}{5} \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}{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 = 0.10, size = 144, normalized size = 0.78 \[ \frac {1}{20} \left (-\left (\sqrt {5}-1\right ) \log \left (x^2+\frac {1}{2} \left (\sqrt {5}-1\right ) x+1\right )+\left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )-4 \log (x+1)-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {-4 x+\sqrt {5}+1}{\sqrt {10-2 \sqrt {5}}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {4 x+\sqrt {5}-1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 3.08, size = 905, normalized size = 4.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 112, normalized size = 0.61 \[ \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 216, normalized size = 1.17 \[ \frac {\arctan \left (\frac {4 x -\sqrt {5}-1}{\sqrt {10-2 \sqrt {5}}}\right )}{\sqrt {10-2 \sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {4 x -\sqrt {5}-1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\arctan \left (\frac {4 x +\sqrt {5}-1}{\sqrt {10+2 \sqrt {5}}}\right )}{\sqrt {10+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctan \left (\frac {4 x +\sqrt {5}-1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\ln \left (x +1\right )}{5}+\frac {\sqrt {5}\, \ln \left (2 x^{2}-\sqrt {5}\, x -x +2\right )}{20}+\frac {\ln \left (2 x^{2}-\sqrt {5}\, x -x +2\right )}{20}-\frac {\sqrt {5}\, \ln \left (2 x^{2}+\sqrt {5}\, x -x +2\right )}{20}+\frac {\ln \left (2 x^{2}+\sqrt {5}\, x -x +2\right )}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.28, size = 144, normalized size = 0.78 \[ \frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} + \frac {{\left (\sqrt {5} + 3\right )} \log \left (2 \, x^{2} - x {\left (\sqrt {5} + 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} + 1\right )}} + \frac {{\left (\sqrt {5} - 3\right )} \log \left (2 \, x^{2} + x {\left (\sqrt {5} - 1\right )} + 2\right )}{10 \, {\left (\sqrt {5} - 1\right )}} - \frac {1}{5} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 196, normalized size = 1.06 \[ \ln \left (25\,x\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-5\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-\frac {\ln \left (x+1\right )}{5}-\ln \left (25\,x\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )+5\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{20}+\frac {\sqrt {5}}{20}-\frac {1}{20}\right )+\ln \left (25\,x\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-5\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )+\ln \left (25\,x\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right )-5\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{20}+\frac {1}{20}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 36, normalized size = 0.19 \[ - \frac {\log {\left (x + 1 \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left (t \mapsto t \log {\left (625 t^{4} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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