3.3.52 \(\int \frac {x}{-1+x^2} \, dx\) [252]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {266} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^2),x]

[Out]

Log[1 - x^2]/2

Rule 266

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \log \left (-1+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^2),x]

[Out]

Log[-1 + x^2]/2

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Mathics [A]
time = 1.63, size = 8, normalized size = 0.67 \begin {gather*} \frac {\text {Log}\left [-1+x^2\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x/(-1 + x^2),x]')

[Out]

Log[-1 + x ^ 2] / 2

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Maple [A]
time = 0.05, size = 14, normalized size = 1.17

method	result	size
derivativedivides	\(\frac {\ln \left (x^{2}-1\right )}{2}\)	\(9\)
risch	\(\frac {\ln \left (x^{2}-1\right )}{2}\)	\(9\)
meijerg	\(\frac {\ln \left (-x^{2}+1\right )}{2}\)	\(11\)
default	\(\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(14\)
norman	\(\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2-1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}

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 = 0.32, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}

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 = 0.03, size = 7, normalized size = 0.58 \begin {gather*} \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}

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 = 0.00, size = 10, normalized size = 0.83 \begin {gather*} \frac {\ln \left |x^{2}-1\right |}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(x^2 - 1))

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Mupad [B]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 - 1),x)

[Out]

log(x^2 - 1)/2

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