3.107 \(\int \frac{1+x^4}{-1+x-x^2+x^3} \, dx\)

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]

x + x^2/2 - ArcTan[x] + Log[1 - x] - Log[1 + x^2]/2

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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]

x + x^2/2 - ArcTan[x] + Log[1 - x] - Log[1 + x^2]/2

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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]

Integral((x**4 + 1)/(x**3 - x**2 + x - 1), x)

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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]

x + x^2/2 - ArcTan[x] + Log[1 - x] - Log[1 + x^2]/2

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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]

x+1/2*x^2-1/2*ln(x^2+1)-arctan(x)+ln(-1+x)

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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]

1/2*x^2 + x - arctan(x) - 1/2*log(x^2 + 1) + log(x - 1)

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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]

1/2*x^2 + x - arctan(x) - 1/2*log(x^2 + 1) + log(x - 1)

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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]

x**2/2 + x + log(x - 1) - log(x**2 + 1)/2 - atan(x)

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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]

1/2*x^2 + x - arctan(x) - 1/2*ln(x^2 + 1) + ln(abs(x - 1))