Optimal. Leaf size=109 \[ -\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-x-(-1)^{4/5}\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left (x+(-1)^{2/5}\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (x-(-1)^{3/5}\right ) \]
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Rubi [A] time = 0.0641871, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1586, 2068} \[ -\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-x-(-1)^{4/5}\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left (x+(-1)^{2/5}\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (x-(-1)^{3/5}\right ) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2068
Rubi steps
\begin{align*} \int \frac{1+x}{1+x^5} \, dx &=\int \frac{1}{1-x+x^2-x^3+x^4} \, dx\\ &=\int \left (\frac{-1+(-1)^{4/5}}{5 \left (-1+\sqrt [5]{-1} x\right )}+\frac{-1-(-1)^{3/5}}{5 \left (-1-(-1)^{2/5} x\right )}+\frac{-1+(-1)^{2/5}}{5 \left (-1+(-1)^{3/5} x\right )}+\frac{-1-\sqrt [5]{-1}}{5 \left (-1-(-1)^{4/5} x\right )}\right ) \, dx\\ &=-\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-(-1)^{4/5}-x\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left ((-1)^{2/5}+x\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (-(-1)^{3/5}+x\right )\\ \end{align*}
Mathematica [C] time = 0.0098981, size = 51, normalized size = 0.47 \[ \text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^3+\text{$\#$1}^2-\text{$\#$1}+1\& ,\frac{\log (x-\text{$\#$1})}{4 \text{$\#$1}^3-3 \text{$\#$1}^2+2 \text{$\#$1}-1}\& \right ] \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 173, normalized size = 1.6 \begin{align*} -{\frac{\sqrt{5}\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{10}}+{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{10}}+{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{x^{5} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 8.73671, size = 2955, normalized size = 27.11 \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 = 0.372474, size = 36, normalized size = 0.33 \begin{align*} \operatorname{RootSum}{\left (125 t^{4} - 5 t + 1, \left ( t \mapsto t \log{\left (\frac{375 t^{3}}{11} + \frac{100 t^{2}}{11} + \frac{45 t}{11} + x - \frac{14}{11} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10738, size = 136, normalized size = 1.25 \begin{align*} \frac{1}{5} \, \sqrt{-2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) + \frac{1}{5} \, \sqrt{2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{5} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{10} \, \sqrt{5} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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