3.343 \(\int \frac{x}{(1-x) (1+x)^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2 (x+1)}+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

1/(2*(1 + x)) + ArcTanh[x]/2

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Rubi [A]  time = 0.0292138, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2 (x+1)}+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x/((1 - x)*(1 + x)^2),x]

[Out]

1/(2*(1 + x)) + ArcTanh[x]/2

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Rubi in Sympy [A]  time = 4.71427, size = 10, normalized size = 0.62 \[ \frac{\operatorname{atanh}{\left (x \right )}}{2} + \frac{1}{2 \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(x)/2 + 1/(2*(x + 1))

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Mathematica [A]  time = 0.013501, size = 24, normalized size = 1.5 \[ \frac{1}{4} \left (\frac{2}{x+1}-\log (1-x)+\log (x+1)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 21, normalized size = 1.3 \[ -{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{1}{2+2\,x}}+{\frac{\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1-x)/(1+x)^2,x)

[Out]

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

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Maxima [A]  time = 0.830957, size = 27, normalized size = 1.69 \[ \frac{1}{2 \,{\left (x + 1\right )}} + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24334, size = 35, normalized size = 2.19 \[ \frac{{\left (x + 1\right )} \log \left (x + 1\right ) -{\left (x + 1\right )} \log \left (x - 1\right ) + 2}{4 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.223792, size = 19, normalized size = 1.19 \[ - \frac{\log{\left (x - 1 \right )}}{4} + \frac{\log{\left (x + 1 \right )}}{4} + \frac{1}{2 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1-x)/(1+x)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.25892, size = 28, normalized size = 1.75 \[ \frac{1}{2 \,{\left (x + 1\right )}} - \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{2}{x + 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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