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

Optimal. Leaf size=24 \[ \frac{1}{x+1}+\frac{3}{2} \log (1-x)-\frac{1}{2} \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0433183, antiderivative size = 24, 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 \[ \frac{1}{x+1}+\frac{3}{2} \log (1-x)-\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.94529, size = 19, normalized size = 0.79 \[ \frac{3 \log{\left (- x + 1 \right )}}{2} - \frac{\log{\left (x + 1 \right )}}{2} + \frac{1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0164353, size = 22, normalized size = 0.92 \[ \frac{1}{x+1}+\frac{3}{2} \log (x-1)-\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 19, normalized size = 0.8 \[ \left ( 1+x \right ) ^{-1}-{\frac{\ln \left ( 1+x \right ) }{2}}+{\frac{3\,\ln \left ( -1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.38366, size = 24, normalized size = 1. \[ \frac{1}{x + 1} - \frac{1}{2} \, \log \left (x + 1\right ) + \frac{3}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.196349, size = 35, normalized size = 1.46 \[ -\frac{{\left (x + 1\right )} \log \left (x + 1\right ) - 3 \,{\left (x + 1\right )} \log \left (x - 1\right ) - 2}{2 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.120675, size = 19, normalized size = 0.79 \[ \frac{3 \log{\left (x - 1 \right )}}{2} - \frac{\log{\left (x + 1 \right )}}{2} + \frac{1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210527, size = 32, normalized size = 1.33 \[ \frac{1}{x + 1} +{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | -\frac{2}{x + 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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