\(\int \frac {x}{1-x} \, dx\) [100]

Optimal result

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

[Out]

-ln(1-x)-x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x}{1-x} \, dx=-x-\log (1-x) \]

[In]

Int[x/(1 - x),x]

[Out]

-x - Log[1 - x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (-1+\frac {1}{1-x}\right ) \, dx \\ & = -x-\log (1-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {x}{1-x} \, dx=-x-\log (1-x) \]

[In]

Integrate[x/(1 - x),x]

[Out]

-x - Log[1 - x]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
default	\(-x -\ln \left (-1+x \right )\)	\(11\)
norman	\(-x -\ln \left (-1+x \right )\)	\(11\)
risch	\(-x -\ln \left (-1+x \right )\)	\(11\)
parallelrisch	\(-x -\ln \left (-1+x \right )\)	\(11\)
meijerg	\(-\ln \left (1-x \right )-x\)	\(13\)

[In]

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

[Out]

-x-ln(-1+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x}{1-x} \, dx=-x - \log \left (x - 1\right ) \]

[In]

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

[Out]

-x - log(x - 1)

Sympy [A] (verification not implemented)

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

[In]

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

[Out]

-x - log(x - 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x}{1-x} \, dx=-x - \log \left (x - 1\right ) \]

[In]

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

[Out]

-x - log(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {x}{1-x} \, dx=-x - \log \left ({\left | x - 1 \right |}\right ) \]

[In]

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

[Out]

-x - log(abs(x - 1))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x}{1-x} \, dx=-x-\ln \left (x-1\right ) \]

[In]

int(-x/(x - 1),x)

[Out]

- x - log(x - 1)