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

Optimal. Leaf size=27 \[ \frac{x+1}{2 \left (1-(x+1)^2\right )}+\frac{1}{2} \tanh ^{-1}(x+1) \]

[Out]

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

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Rubi [A]  time = 0.0165959, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{x+1}{2 \left (1-(x+1)^2\right )}+\frac{1}{2} \tanh ^{-1}(x+1) \]

Antiderivative was successfully verified.

[In]  Int[(1 - (1 + x)^2)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.37929, size = 17, normalized size = 0.63 \[ \frac{x + 1}{2 \left (- \left (x + 1\right )^{2} + 1\right )} + \frac{\operatorname{atanh}{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + 1)/(2*(-(x + 1)**2 + 1)) + atanh(x + 1)/2

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Mathematica [A]  time = 0.0295108, size = 26, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 (x+1)}{x (x+2)}-\log (x)+\log (x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 24, normalized size = 0.9 \[ -{\frac{1}{4\,x}}-{\frac{1}{8+4\,x}}-{\frac{\ln \left ( x \right ) }{4}}+{\frac{\ln \left ( 2+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-(1+x)^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.797471, size = 34, normalized size = 1.26 \[ -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \, \log \left (x + 2\right ) - \frac{1}{4} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273721, size = 53, normalized size = 1.96 \[ \frac{{\left (x^{2} + 2 \, x\right )} \log \left (x + 2\right ) -{\left (x^{2} + 2 \, x\right )} \log \left (x\right ) - 2 \, x - 2}{4 \,{\left (x^{2} + 2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.236917, size = 22, normalized size = 0.81 \[ - \frac{x + 1}{2 x^{2} + 4 x} - \frac{\log{\left (x \right )}}{4} + \frac{\log{\left (x + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-(1+x)**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.261958, size = 36, normalized size = 1.33 \[ -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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