\(\int \frac {x}{-1+x^2} \, dx\) [454]

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int \frac {x}{-1+x^2} \, dx=\frac {1}{2} \log \left (1-x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {266} \[ \int \frac {x}{-1+x^2} \, dx=\frac {1}{2} \log \left (1-x^2\right ) \]

[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*} \text {integral}& = \frac {1}{2} \log \left (1-x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x}{-1+x^2} \, dx=\frac {1}{2} \log \left (-1+x^2\right ) \]

[In]

Integrate[x/(-1 + x^2),x]

[Out]

Log[-1 + x^2]/2

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75

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	\(\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(14\)
norman	\(\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(14\)
parallelrisch	\(\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x}{-1+x^2} \, dx=\frac {1}{2} \, \log \left (x^{2} - 1\right ) \]

[In]

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

[Out]

1/2*log(x^2 - 1)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {x}{-1+x^2} \, dx=\frac {\log {\left (x^{2} - 1 \right )}}{2} \]

[In]

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

[Out]

log(x**2 - 1)/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x}{-1+x^2} \, dx=\frac {1}{2} \, \log \left (x^{2} - 1\right ) \]

[In]

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

[Out]

1/2*log(x^2 - 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {x}{-1+x^2} \, dx=\frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]

[In]

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

[Out]

1/2*log(abs(x^2 - 1))

Mupad [B] (verification not implemented)

Time = 9.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x}{-1+x^2} \, dx=\frac {\ln \left (x^2-1\right )}{2} \]

[In]

int(x/(x^2 - 1),x)

[Out]

log(x^2 - 1)/2