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

Optimal. Leaf size=27 \[ -\left ((1-x) \log \left (-\frac {1-x}{x+1}\right )\right )-2 \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2486, 31} \[ -(1-x) \log \left (-\frac {1-x}{x+1}\right )-2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[Log[(-1 + x)/(1 + x)],x]

[Out]

-((1 - x)*Log[-((1 - x)/(1 + x))]) - 2*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 {-1+x}{1+x}\right ) \, dx &=-(1-x) \log \left (-\frac {1-x}{1+x}\right )-2 \int \frac {1}{1+x} \, dx\\ &=-(1-x) \log \left (-\frac {1-x}{1+x}\right )-2 \log (1+x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.78 \[ (x-1) \log \left (\frac {x-1}{x+1}\right )-2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(-1 + x)/(1 + x)],x]

[Out]

(-1 + x)*Log[(-1 + x)/(1 + x)] - 2*Log[1 + x]

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fricas [A]  time = 0.42, size = 21, normalized size = 0.78 \[ x \log \left (\frac {x - 1}{x + 1}\right ) - \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log((x - 1)/(x + 1)) - log(x^2 - 1)

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giac [B]  time = 0.23, size = 103, normalized size = 3.81 \[ -\frac {2 \, \log \left (\frac {\frac {\frac {x - 1}{x + 1} + 1}{\frac {x - 1}{x + 1} - 1} + 1}{\frac {\frac {x - 1}{x + 1} + 1}{\frac {x - 1}{x + 1} - 1} - 1}\right )}{\frac {x - 1}{x + 1} - 1} - 2 \, \log \left (\frac {{\left | x - 1 \right |}}{{\left | x + 1 \right |}}\right ) + 2 \, \log \left ({\left | \frac {x - 1}{x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log((((x - 1)/(x + 1) + 1)/((x - 1)/(x + 1) - 1) + 1)/(((x - 1)/(x + 1) + 1)/((x - 1)/(x + 1) - 1) - 1))/((
x - 1)/(x + 1) - 1) - 2*log(abs(x - 1)/abs(x + 1)) + 2*log(abs((x - 1)/(x + 1) - 1))

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maple [A]  time = 0.17, size = 35, normalized size = 1.30 \[ 2 \ln \left (-\frac {2}{x +1}\right )+\left (1-\frac {2}{x +1}\right ) \left (x +1\right ) \ln \left (1-\frac {2}{x +1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 25, normalized size = 0.93 \[ x \log \left (\frac {x - 1}{x + 1}\right ) - \log \left (x + 1\right ) - \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.08, size = 21, normalized size = 0.78 \[ x\,\ln \left (\frac {x-1}{x+1}\right )-\ln \left (x^2-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((x - 1)/(x + 1)),x)

[Out]

x*log((x - 1)/(x + 1)) - log(x^2 - 1)

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sympy [A]  time = 0.12, size = 15, normalized size = 0.56 \[ x \log {\left (\frac {x - 1}{x + 1} \right )} - \log {\left (x^{2} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log((x - 1)/(x + 1)) - log(x**2 - 1)

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