Optimal. Leaf size=29 \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0534352, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 + x^4)/(-1 + x - x^2 + x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} + 1}{x^{3} - x^{2} + x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**4+1)/(x**3-x**2+x-1),x)
[Out]
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Mathematica [A] time = 0.0111239, size = 29, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^4)/(-1 + x - x^2 + x^3),x]
[Out]
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Maple [A] time = 0.009, size = 24, normalized size = 0.8 \[ x+{\frac{{x}^{2}}{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-\arctan \left ( x \right ) +\ln \left ( -1+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^4+1)/(x^3-x^2+x-1),x)
[Out]
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Maxima [A] time = 1.53022, size = 31, normalized size = 1.07 \[ \frac{1}{2} \, x^{2} + x - \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 1)/(x^3 - x^2 + x - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209742, size = 31, normalized size = 1.07 \[ \frac{1}{2} \, x^{2} + x - \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 1)/(x^3 - x^2 + x - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.123373, size = 22, normalized size = 0.76 \[ \frac{x^{2}}{2} + x + \log{\left (x - 1 \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**4+1)/(x**3-x**2+x-1),x)
[Out]
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GIAC/XCAS [A] time = 0.200354, size = 32, normalized size = 1.1 \[ \frac{1}{2} \, x^{2} + x - \arctan \left (x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 1)/(x^3 - x^2 + x - 1),x, algorithm="giac")
[Out]