3.274 \(\int \log (-\frac {x}{1+x}) \, dx\)

Optimal. Leaf size=18 \[ x \log \left (-\frac {x}{x+1}\right )-\log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2486, 31} \[ x \log \left (-\frac {x}{x+1}\right )-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[Log[-(x/(1 + x))],x]

[Out]

x*Log[-(x/(1 + x))] - Log[1 + x]

Rule 31

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

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ x \log \left (-\frac {x}{x+1}\right )-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[-(x/(1 + x))],x]

[Out]

x*Log[-(x/(1 + x))] - Log[1 + x]

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fricas [A]  time = 0.45, size = 18, normalized size = 1.00 \[ x \log \left (-\frac {x}{x + 1}\right ) - \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-x/(x + 1)) - log(x + 1)

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giac [B]  time = 0.16, size = 80, normalized size = 4.44 \[ -\frac {\log \left (-\frac {x}{{\left (x + 1\right )} {\left (\frac {x}{x + 1} - 1\right )} {\left (\frac {x}{{\left (x + 1\right )} {\left (\frac {x}{x + 1} - 1\right )}} - 1\right )}}\right )}{\frac {x}{x + 1} - 1} - \log \left (\frac {{\left | x \right |}}{{\left | x + 1 \right |}}\right ) + \log \left ({\left | -\frac {x}{x + 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x/((x + 1)*(x/(x + 1) - 1)*(x/((x + 1)*(x/(x + 1) - 1)) - 1)))/(x/(x + 1) - 1) - log(abs(x)/abs(x + 1))
+ log(abs(-x/(x + 1) + 1))

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maple [A]  time = 0.20, size = 28, normalized size = 1.56 \[ \ln \left (\frac {1}{x +1}\right )-\left (-1+\frac {1}{x +1}\right ) \left (x +1\right ) \ln \left (-1+\frac {1}{x +1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(-1/(x+1)*x),x)

[Out]

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

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maxima [A]  time = 0.64, size = 18, normalized size = 1.00 \[ x \log \left (-\frac {x}{x + 1}\right ) - \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-x/(x + 1)) - log(x + 1)

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mupad [B]  time = 0.36, size = 18, normalized size = 1.00 \[ x\,\ln \left (-\frac {x}{x+1}\right )-\ln \left (x+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(-x/(x + 1)),x)

[Out]

x*log(-x/(x + 1)) - log(x + 1)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.78 \[ x \log {\left (- \frac {x}{x + 1} \right )} - \log {\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-x/(x + 1)) - log(x + 1)

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