3.268 \(\int \frac{5+3 x}{1-x-x^2+x^3} \, dx\)

Optimal. Leaf size=12 \[ \frac{4}{1-x}+\tanh ^{-1}(x) \]

[Out]

4/(1 - x) + ArcTanh[x]

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Rubi [A]  time = 0.0402657, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{4}{1-x}+\tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(5 + 3*x)/(1 - x - x^2 + x^3),x]

[Out]

4/(1 - x) + ArcTanh[x]

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Rubi in Sympy [A]  time = 15.4377, size = 7, normalized size = 0.58 \[ \operatorname{atanh}{\left (x \right )} + \frac{4}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5+3*x)/(x**3-x**2-x+1),x)

[Out]

atanh(x) + 4/(-x + 1)

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Mathematica [A]  time = 0.0158097, size = 24, normalized size = 2. \[ -\frac{4}{x-1}-\frac{1}{2} \log (x-1)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + 3*x)/(1 - x - x^2 + x^3),x]

[Out]

-4/(-1 + x) - Log[-1 + x]/2 + Log[1 + x]/2

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Maple [A]  time = 0.011, size = 21, normalized size = 1.8 \[ -4\, \left ( -1+x \right ) ^{-1}-{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5+3*x)/(x^3-x^2-x+1),x)

[Out]

-4/(-1+x)-1/2*ln(-1+x)+1/2*ln(1+x)

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Maxima [A]  time = 0.789688, size = 27, normalized size = 2.25 \[ -\frac{4}{x - 1} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 5)/(x^3 - x^2 - x + 1),x, algorithm="maxima")

[Out]

-4/(x - 1) + 1/2*log(x + 1) - 1/2*log(x - 1)

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Fricas [A]  time = 0.248448, size = 35, normalized size = 2.92 \[ \frac{{\left (x - 1\right )} \log \left (x + 1\right ) -{\left (x - 1\right )} \log \left (x - 1\right ) - 8}{2 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 5)/(x^3 - x^2 - x + 1),x, algorithm="fricas")

[Out]

1/2*((x - 1)*log(x + 1) - (x - 1)*log(x - 1) - 8)/(x - 1)

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Sympy [A]  time = 0.186911, size = 17, normalized size = 1.42 \[ - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} - \frac{4}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5+3*x)/(x**3-x**2-x+1),x)

[Out]

-log(x - 1)/2 + log(x + 1)/2 - 4/(x - 1)

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GIAC/XCAS [A]  time = 0.264683, size = 30, normalized size = 2.5 \[ -\frac{4}{x - 1} + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 5)/(x^3 - x^2 - x + 1),x, algorithm="giac")

[Out]

-4/(x - 1) + 1/2*ln(abs(x + 1)) - 1/2*ln(abs(x - 1))