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

Optimal. Leaf size=25 \[ \frac{1}{1-x}-\frac{1}{x}-2 \log (1-x)+2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0204274, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{1-x}-\frac{1}{x}-2 \log (1-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/((-1 + x)^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.53029, size = 19, normalized size = 0.76 \[ 2 \log{\left (x \right )} - 2 \log{\left (- x + 1 \right )} + \frac{1}{- x + 1} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0187926, size = 25, normalized size = 1. \[ -\frac{1}{x-1}-\frac{1}{x}-2 \log (1-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((-1 + x)^2*x^2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 24, normalized size = 1. \[ -{x}^{-1}+2\,\ln \left ( x \right ) - \left ( -1+x \right ) ^{-1}-2\,\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+x)^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.38095, size = 36, normalized size = 1.44 \[ -\frac{2 \, x - 1}{x^{2} - x} - 2 \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.198302, size = 54, normalized size = 2.16 \[ -\frac{2 \,{\left (x^{2} - x\right )} \log \left (x - 1\right ) - 2 \,{\left (x^{2} - x\right )} \log \left (x\right ) + 2 \, x - 1}{x^{2} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.11697, size = 20, normalized size = 0.8 \[ - \frac{2 x - 1}{x^{2} - x} + 2 \log{\left (x \right )} - 2 \log{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-1+x)**2/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207611, size = 41, normalized size = 1.64 \[ -\frac{1}{x - 1} + \frac{1}{\frac{1}{x - 1} + 1} + 2 \,{\rm ln}\left ({\left | -\frac{1}{x - 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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