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 = 0.221749, antiderivative size = 185, normalized size of antiderivative = 1., 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 = {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 293
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
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{\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 )}{\sqrt{5}}-\frac{\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 )}{\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*}
Mathematica [A] time = 0.0737115, size = 144, normalized size = 0.78 \[ \frac{1}{20} \left (\left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )-\left (\sqrt{5}-1\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-4 \log (x+1)-2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )-2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.014, size = 156, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+x \right ) }{5}}-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}+{\frac{2\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}-{\frac{2\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48074, size = 167, normalized size = 0.9 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 11.7926, size = 3717, normalized size = 20.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.738041, size = 36, normalized size = 0.19 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18451, size = 151, normalized size = 0.82 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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