3.3.17 \(\int \log (\frac {1+x}{1-x}) \, dx\) [217]

Optimal. Leaf size=25 \[ 2 \log (1-x)+(1+x) \log \left (\frac {1+x}{1-x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2535, 31} \begin {gather*} 2 \log (1-x)+(x+1) \log \left (\frac {x+1}{1-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[(1 + x)/(1 - x)],x]

[Out]

2*Log[1 - x] + (1 + x)*Log[(1 + x)/(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 2535

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.), x_Symbol] :> Simp[(a +
b*x)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/b), x] - Dist[B*n*p*((b*c - a*d)/b), Int[(A + B*Log[e*((a + b*
x)/(c + d*x))^n])^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[b*c - a*d, 0] && IGtQ
[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=(1+x) \log \left (\frac {1+x}{1-x}\right )-2 \int \frac {1}{1-x} \, dx\\ &=2 \log (1-x)+(1+x) \log \left (\frac {1+x}{1-x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} 2 \log (1-x)+(1+x) \log \left (\frac {1+x}{1-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[(1 + x)/(1 - x)],x]

[Out]

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

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Maple [A]
time = 0.06, size = 36, normalized size = 1.44

method	result	size
risch	\(x \ln \left (\frac {x +1}{1-x}\right )+\ln \left (x^{2}-1\right )\)	\(22\)
parts	\(x \ln \left (\frac {x +1}{1-x}\right )+\ln \left (\left (x -1\right ) \left (x +1\right )\right )\)	\(24\)
meijerg	\(\frac {\left (2 x +2\right ) \ln \left (x +1\right )}{2}+\frac {\left (-2 x +2\right ) \ln \left (1-x \right )}{2}\)	\(26\)
parallelrisch	\(\ln \left (-\frac {x +1}{x -1}\right ) x +2 \ln \left (x -1\right )+\ln \left (-\frac {x +1}{x -1}\right )\)	\(32\)
derivativedivides	\(-2 \ln \left (-\frac {2}{x -1}\right )-\ln \left (-1-\frac {2}{x -1}\right ) \left (-1-\frac {2}{x -1}\right ) \left (x -1\right )\)	\(36\)
default	\(-2 \ln \left (-\frac {2}{x -1}\right )-\ln \left (-1-\frac {2}{x -1}\right ) \left (-1-\frac {2}{x -1}\right ) \left (x -1\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 22, normalized size = 0.88 \begin {gather*} x \log \left (-\frac {x + 1}{x - 1}\right ) + \log \left (x + 1\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.57, size = 20, normalized size = 0.80 \begin {gather*} x \log \left (-\frac {x + 1}{x - 1}\right ) + \log \left (x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.04, size = 15, normalized size = 0.60 \begin {gather*} x \log {\left (\frac {x + 1}{1 - x} \right )} + \log {\left (x^{2} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (24) = 48\).
time = 0.43, size = 107, normalized size = 4.28 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((x+1)/(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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Mupad [B]
time = 0.10, size = 20, normalized size = 0.80 \begin {gather*} \ln \left (x^2-1\right )+x\,\ln \left (-\frac {x+1}{x-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(-(x + 1)/(x - 1)),x)

[Out]

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

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Chatgpt [F] Failed to verify
time = 1.00, size = 16, normalized size = 0.64 \begin {gather*} \frac {\ln \left (\frac {x +1}{1-x}\right )^{2}}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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