3.356 \(\int \frac{x^2}{(-1+x) \left (1+2 x+x^2\right )} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{2 (x+1)}+\frac{1}{4} \log (1-x)+\frac{3}{4} \log (x+1) \]

[Out]

1/(2*(1 + x)) + Log[1 - x]/4 + (3*Log[1 + x])/4

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Rubi [A]  time = 0.0363008, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{2 (x+1)}+\frac{1}{4} \log (1-x)+\frac{3}{4} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

1/(2*(1 + x)) + Log[1 - x]/4 + (3*Log[1 + x])/4

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Rubi in Sympy [A]  time = 6.78452, size = 20, normalized size = 0.71 \[ \frac{\log{\left (- x + 1 \right )}}{4} + \frac{3 \log{\left (x + 1 \right )}}{4} + \frac{1}{2 \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(-x + 1)/4 + 3*log(x + 1)/4 + 1/(2*(x + 1))

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Mathematica [A]  time = 0.0212229, size = 22, normalized size = 0.79 \[ \frac{1}{4} \left (\frac{2}{x+1}+\log (x-1)+3 \log (x+1)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2/(1 + x) + Log[-1 + x] + 3*Log[1 + x])/4

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Maple [A]  time = 0.011, size = 21, normalized size = 0.8 \[{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{1}{2+2\,x}}+{\frac{3\,\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-1+x)/(x^2+2*x+1),x)

[Out]

1/4*ln(-1+x)+1/2/(1+x)+3/4*ln(1+x)

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Maxima [A]  time = 0.802438, size = 27, normalized size = 0.96 \[ \frac{1}{2 \,{\left (x + 1\right )}} + \frac{3}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x + 1) + 3/4*log(x + 1) + 1/4*log(x - 1)

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Fricas [A]  time = 0.253225, size = 35, normalized size = 1.25 \[ \frac{3 \,{\left (x + 1\right )} \log \left (x + 1\right ) +{\left (x + 1\right )} \log \left (x - 1\right ) + 2}{4 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*(x + 1)*log(x + 1) + (x + 1)*log(x - 1) + 2)/(x + 1)

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Sympy [A]  time = 0.23563, size = 20, normalized size = 0.71 \[ \frac{\log{\left (x - 1 \right )}}{4} + \frac{3 \log{\left (x + 1 \right )}}{4} + \frac{1}{2 x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x - 1)/4 + 3*log(x + 1)/4 + 1/(2*x + 2)

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GIAC/XCAS [A]  time = 0.260459, size = 30, normalized size = 1.07 \[ \frac{1}{2 \,{\left (x + 1\right )}} + \frac{3}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(x + 1) + 3/4*ln(abs(x + 1)) + 1/4*ln(abs(x - 1))