3.249 \(\int \frac{\log (\frac{1+x}{-1+x})}{x^2} \, dx\)

Optimal. Leaf size=35 \[ 2 \log \left (-\frac{x}{1-x}\right )-\frac{(x+1) \log \left (-\frac{x+1}{1-x}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.0142757, antiderivative size = 34, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2490, 36, 29, 31} \[ 2 \log (x)-2 \log (x+1)-\frac{(1-x) \log \left (-\frac{x+1}{1-x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Log[(1 + x)/(-1 + x)]/x^2,x]

[Out]

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

Rule 2490

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A]  time = 0.0053786, size = 30, normalized size = 0.86 \[ -\log \left (1-x^2\right )+2 \log (x)-\frac{\log \left (\frac{x+1}{x-1}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(1 + x)/(-1 + x)]/x^2,x]

[Out]

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

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Maple [A]  time = 0.131, size = 46, normalized size = 1.3 \begin{align*} 2\,\ln \left ( 2\, \left ( x-1 \right ) ^{-1}+2 \right ) -2\,{\ln \left ( 1+2\, \left ( x-1 \right ) ^{-1} \right ) \left ( 1+2\, \left ( x-1 \right ) ^{-1} \right ) \left ( 2\, \left ( x-1 \right ) ^{-1}+2 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((1+x)/(x-1))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.12857, size = 43, normalized size = 1.23 \begin{align*} -\frac{\log \left (\frac{x + 1}{x - 1}\right )}{x} - \log \left (x + 1\right ) - \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((1+x)/(-1+x))/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.947874, size = 77, normalized size = 2.2 \begin{align*} -\frac{x \log \left (x^{2} - 1\right ) - 2 \, x \log \left (x\right ) + \log \left (\frac{x + 1}{x - 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((1+x)/(-1+x))/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.140983, size = 20, normalized size = 0.57 \begin{align*} 2 \log{\left (x \right )} - \log{\left (x^{2} - 1 \right )} - \frac{\log{\left (\frac{x + 1}{x - 1} \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26509, size = 39, normalized size = 1.11 \begin{align*} -\frac{\log \left (\frac{x + 1}{x - 1}\right )}{x} + \log \left (x^{2}\right ) - \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((1+x)/(-1+x))/x^2,x, algorithm="giac")

[Out]

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