Optimal. Leaf size=38 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0489702, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 + x^2 + x^3 + x^5)^(-1),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} + x^{3} + x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**5+x**3+x**2+1),x)
[Out]
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Mathematica [A] time = 0.0107985, size = 38, normalized size = 1. \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^2 + x^3 + x^5)^(-1),x]
[Out]
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Maple [A] time = 0.01, size = 31, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^5+x^3+x^2+1),x)
[Out]
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Maxima [A] time = 0.909293, size = 41, normalized size = 1.08 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^5 + x^3 + x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25712, size = 41, normalized size = 1.08 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^5 + x^3 + x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.318706, size = 29, normalized size = 0.76 \[ \frac{\log{\left (x + 1 \right )}}{6} + \frac{\log{\left (x^{2} + 1 \right )}}{4} - \frac{\log{\left (x^{2} - x + 1 \right )}}{3} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**5+x**3+x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.26338, size = 42, normalized size = 1.11 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^5 + x^3 + x^2 + 1),x, algorithm="giac")
[Out]