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

Optimal. Leaf size=12 \[ \frac{1}{2} \log \left (1-x^2\right ) \]

[Out]

Log[1 - x^2]/2

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Rubi [A]  time = 0.0018597, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {260} \[ \frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^2),x]

[Out]

Log[1 - x^2]/2

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x}{-1+x^2} \, dx &=\frac{1}{2} \log \left (1-x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.001481, size = 10, normalized size = 0.83 \[ \frac{1}{2} \log \left (x^2-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^2),x]

[Out]

Log[-1 + x^2]/2

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Maple [A]  time = 0.001, size = 14, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2-1),x)

[Out]

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

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Maxima [A]  time = 0.917003, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{2} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-1),x, algorithm="maxima")

[Out]

1/2*log(x^2 - 1)

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Fricas [A]  time = 2.01126, size = 24, normalized size = 2. \begin{align*} \frac{1}{2} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-1),x, algorithm="fricas")

[Out]

1/2*log(x^2 - 1)

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Sympy [A]  time = 0.071522, size = 7, normalized size = 0.58 \begin{align*} \frac{\log{\left (x^{2} - 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2-1),x)

[Out]

log(x**2 - 1)/2

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Giac [A]  time = 1.05911, size = 12, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-1),x, algorithm="giac")

[Out]

1/2*log(abs(x^2 - 1))