3.32 \(\int \frac{1}{x \left (x-x^3\right )} \, dx\)

Optimal. Leaf size=8 \[ \tanh ^{-1}(x)-\frac{1}{x} \]

[Out]

-x^(-1) + ArcTanh[x]

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Rubi [A]  time = 0.0157326, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \tanh ^{-1}(x)-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(x - x^3)),x]

[Out]

-x^(-1) + ArcTanh[x]

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Rubi in Sympy [A]  time = 4.24567, size = 5, normalized size = 0.62 \[ \operatorname{atanh}{\left (x \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-x**3+x),x)

[Out]

atanh(x) - 1/x

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Mathematica [B]  time = 0.0048887, size = 24, normalized size = 3. \[ -\frac{1}{x}-\frac{1}{2} \log (1-x)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(x - x^3)),x]

[Out]

-x^(-1) - Log[1 - x]/2 + Log[1 + x]/2

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Maple [B]  time = 0.01, size = 19, normalized size = 2.4 \[ -{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}-{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^3+x),x)

[Out]

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

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Maxima [A]  time = 1.3911, size = 24, normalized size = 3. \[ -\frac{1}{x} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204814, size = 27, normalized size = 3.38 \[ \frac{x \log \left (x + 1\right ) - x \log \left (x - 1\right ) - 2}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.212866, size = 15, normalized size = 1.88 \[ - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-x**3+x),x)

[Out]

-log(x - 1)/2 + log(x + 1)/2 - 1/x

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GIAC/XCAS [A]  time = 0.215718, size = 27, normalized size = 3.38 \[ -\frac{1}{x} + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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