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*Log[-(x/(1 + x))] - Log[1 + x]

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Rubi [A]  time = 0.0035066, antiderivative size = 18, normalized size of antiderivative = 1., 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 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]

Rule 31

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

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*}

Mathematica [A]  time = 0.0016447, size = 18, normalized size = 1. \[ x \log \left (-\frac{x}{x+1}\right )-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[-(x/(1 + x))],x]

[Out]

x*Log[-(x/(1 + x))] - Log[1 + x]

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Maple [A]  time = 0.011, size = 28, normalized size = 1.6 \begin{align*} \ln \left ( \left ( 1+x \right ) ^{-1} \right ) -\ln \left ( -1+ \left ( 1+x \right ) ^{-1} \right ) \left ( -1+ \left ( 1+x \right ) ^{-1} \right ) \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07712, size = 24, normalized size = 1.33 \begin{align*} x \log \left (-\frac{x}{x + 1}\right ) - \log \left (x + 1\right ) \end{align*}

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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Fricas [A]  time = 1.94007, size = 43, normalized size = 2.39 \begin{align*} x \log \left (-\frac{x}{x + 1}\right ) - \log \left (x + 1\right ) \end{align*}

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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Sympy [A]  time = 0.105937, size = 14, normalized size = 0.78 \begin{align*} x \log{\left (- \frac{x}{x + 1} \right )} - \log{\left (x + 1 \right )} \end{align*}

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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Giac [A]  time = 1.30919, size = 26, normalized size = 1.44 \begin{align*} x \log \left (-\frac{x}{x + 1}\right ) - \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-x/(x + 1)) - log(abs(x + 1))