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

Optimal. Leaf size=34 \[ \frac{3}{2 (1-x)}-\frac{5}{4} \log (1-x)+2 \log (x)-\frac{3}{4} \log (x+1) \]

[Out]

3/(2*(1 - x)) - (5*Log[1 - x])/4 + 2*Log[x] - (3*Log[1 + x])/4

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Rubi [A]  time = 0.0894976, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{3}{2 (1-x)}-\frac{5}{4} \log (1-x)+2 \log (x)-\frac{3}{4} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

3/(2*(1 - x)) - (5*Log[1 - x])/4 + 2*Log[x] - (3*Log[1 + x])/4

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0253209, size = 32, normalized size = 0.94 \[ -\frac{3}{2 (x-1)}-\frac{5}{4} \log (1-x)+2 \log (x)-\frac{3}{4} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-3/(2*(-1 + x)) - (5*Log[1 - x])/4 + 2*Log[x] - (3*Log[1 + x])/4

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Maple [A]  time = 0.013, size = 25, normalized size = 0.7 \[ -{\frac{3}{2\,x-2}}-{\frac{5\,\ln \left ( -1+x \right ) }{4}}-{\frac{3\,\ln \left ( 1+x \right ) }{4}}+2\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2)/(-1+x)^2/x/(1+x),x)

[Out]

-3/2/(-1+x)-5/4*ln(-1+x)-3/4*ln(1+x)+2*ln(x)

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Maxima [A]  time = 0.804361, size = 32, normalized size = 0.94 \[ -\frac{3}{2 \,{\left (x - 1\right )}} - \frac{3}{4} \, \log \left (x + 1\right ) - \frac{5}{4} \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2/(x - 1) - 3/4*log(x + 1) - 5/4*log(x - 1) + 2*log(x)

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Fricas [A]  time = 0.253541, size = 46, normalized size = 1.35 \[ -\frac{3 \,{\left (x - 1\right )} \log \left (x + 1\right ) + 5 \,{\left (x - 1\right )} \log \left (x - 1\right ) - 8 \,{\left (x - 1\right )} \log \left (x\right ) + 6}{4 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*(x - 1)*log(x + 1) + 5*(x - 1)*log(x - 1) - 8*(x - 1)*log(x) + 6)/(x - 1
)

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Sympy [A]  time = 0.336332, size = 27, normalized size = 0.79 \[ 2 \log{\left (x \right )} - \frac{5 \log{\left (x - 1 \right )}}{4} - \frac{3 \log{\left (x + 1 \right )}}{4} - \frac{3}{2 x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x) - 5*log(x - 1)/4 - 3*log(x + 1)/4 - 3/(2*x - 2)

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GIAC/XCAS [A]  time = 0.261878, size = 46, normalized size = 1.35 \[ -\frac{3}{2 \,{\left (x - 1\right )}} + 2 \,{\rm ln}\left ({\left | -\frac{1}{x - 1} - 1 \right |}\right ) - \frac{3}{4} \,{\rm ln}\left ({\left | -\frac{2}{x - 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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