Optimal. Leaf size=109 \[ -\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (\sqrt [5]{-1}-x\right )+\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (-x-(-1)^{4/5}\right )+\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left (x+(-1)^{2/5}\right )-\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (x-(-1)^{3/5}\right ) \]
[Out]
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Rubi [F] time = 0.048902, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{1+x}{1+x^5},x\right ) \]
Verification is Not applicable to the result.
[In] Int[(1 + x)/(1 + x^5),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} - x^{3} + x^{2} - x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)/(x**5+1),x)
[Out]
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Mathematica [C] time = 0.0173895, 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.
[In] Integrate[(1 + x)/(1 + x^5),x]
[Out]
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Maple [B] time = 0.033, size = 173, normalized size = 1.6 \[ -{\frac{\sqrt{5}\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{10}}+{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{10}}+{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)/(x^5+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 1}{x^{5} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)/(x^5 + 1),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)/(x^5 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.798626, size = 36, normalized size = 0.33 \[ \operatorname{RootSum}{\left (125 t^{4} - 5 t + 1, \left ( t \mapsto t \log{\left (\frac{375 t^{3}}{11} + \frac{100 t^{2}}{11} + \frac{45 t}{11} + x - \frac{14}{11} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)/(x**5+1),x)
[Out]
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GIAC/XCAS [A] time = 0.220089, size = 136, normalized size = 1.25 \[ \frac{1}{5} \, \sqrt{-2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) + \frac{1}{5} \, \sqrt{2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{5}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{10} \, \sqrt{5}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)/(x^5 + 1),x, algorithm="giac")
[Out]