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

Optimal. Leaf size=12 \[ -x-\log (1-x) \]

[Out]

-ln(1-x)-x

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} -x-\log (1-x) \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \text {Integral} &=\int \left (-1+\frac {1}{1-x}\right ) \, dx\\ &=-x-\log (1-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -x-\log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x),x]

[Out]

-x - Log[1 - x]

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Maple [A]
time = 0.02, size = 11, normalized size = 0.92

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x-ln(x-1)

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Maxima [A]
time = 0.32, size = 10, normalized size = 0.83 \begin {gather*} -x - \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(x - 1)

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Fricas [A]
time = 0.57, size = 10, normalized size = 0.83 \begin {gather*} -x - \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(x - 1)

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Sympy [A]
time = 0.02, size = 7, normalized size = 0.58 \begin {gather*} - x - \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(x - 1)

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Giac [A]
time = 0.46, size = 11, normalized size = 0.92 \begin {gather*} -x - \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(abs(x - 1))

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Mupad [B]
time = 0.03, size = 10, normalized size = 0.83 \begin {gather*} -x-\ln \left (x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/(x - 1),x)

[Out]

- x - log(x - 1)

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