### 3.212 $$\int \log (a \coth (x)) \, dx$$

Optimal. Leaf size=41 $-\frac{1}{2} \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )+x \log (a \coth (x))-2 x \tanh ^{-1}\left (e^{2 x}\right )$

[Out]

-2*x*ArcTanh[E^(2*x)] + x*Log[a*Coth[x]] - PolyLog[2, -E^(2*x)]/2 + PolyLog[2, E^(2*x)]/2

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Rubi [A]  time = 0.0442162, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 5, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 1., Rules used = {2548, 5461, 4182, 2279, 2391} $-\frac{1}{2} \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )+x \log (a \coth (x))-2 x \tanh ^{-1}\left (e^{2 x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[a*Coth[x]],x]

[Out]

-2*x*ArcTanh[E^(2*x)] + x*Log[a*Coth[x]] - PolyLog[2, -E^(2*x)]/2 + PolyLog[2, E^(2*x)]/2

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \log (a \coth (x)) \, dx &=x \log (a \coth (x))+\int x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log (a \coth (x))+2 \int x \text{csch}(2 x) \, dx\\ &=-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (a \coth (x))-\int \log \left (1-e^{2 x}\right ) \, dx+\int \log \left (1+e^{2 x}\right ) \, dx\\ &=-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (a \coth (x))-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (a \coth (x))-\frac{1}{2} \text{Li}_2\left (-e^{2 x}\right )+\frac{\text{Li}_2\left (e^{2 x}\right )}{2}\\ \end{align*}

Mathematica [A]  time = 0.0076459, size = 49, normalized size = 1.2 $\frac{1}{2} \text{PolyLog}(2,-\coth (x))-\frac{1}{2} \text{PolyLog}(2,\coth (x))-\frac{1}{2} \log (1-\coth (x)) \log (a \coth (x))+\frac{1}{2} \log (\coth (x)+1) \log (a \coth (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[a*Coth[x]],x]

[Out]

-(Log[1 - Coth[x]]*Log[a*Coth[x]])/2 + (Log[a*Coth[x]]*Log[1 + Coth[x]])/2 + PolyLog[2, -Coth[x]]/2 - PolyLog[
2, Coth[x]]/2

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Maple [B]  time = 0.014, size = 70, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( a{\rm coth} \left (x\right ) \right ) }{2}\ln \left ({\frac{a{\rm coth} \left (x\right )+a}{a}} \right ) }+{\frac{1}{2}{\it dilog} \left ({\frac{a{\rm coth} \left (x\right )+a}{a}} \right ) }-{\frac{\ln \left ( a{\rm coth} \left (x\right ) \right ) }{2}\ln \left ( -{\frac{a{\rm coth} \left (x\right )-a}{a}} \right ) }-{\frac{1}{2}{\it dilog} \left ( -{\frac{a{\rm coth} \left (x\right )-a}{a}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*coth(x)),x)

[Out]

1/2*ln(a*coth(x))*ln((a*coth(x)+a)/a)+1/2*dilog((a*coth(x)+a)/a)-1/2*ln(a*coth(x))*ln(-(a*coth(x)-a)/a)-1/2*di
log(-(a*coth(x)-a)/a)

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Maxima [A]  time = 1.55698, size = 69, normalized size = 1.68 \begin{align*} x \log \left (a \coth \left (x\right )\right ) - x \log \left (e^{\left (2 \, x\right )} + 1\right ) + x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) - \frac{1}{2} \,{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) +{\rm Li}_2\left (-e^{x}\right ) +{\rm Li}_2\left (e^{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*coth(x)),x, algorithm="maxima")

[Out]

x*log(a*coth(x)) - x*log(e^(2*x) + 1) + x*log(e^x + 1) + x*log(-e^x + 1) - 1/2*dilog(-e^(2*x)) + dilog(-e^x) +
dilog(e^x)

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Fricas [C]  time = 1.93172, size = 375, normalized size = 9.15 \begin{align*} x \log \left (\frac{a \cosh \left (x\right )}{\sinh \left (x\right )}\right ) + x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) +{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) -{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) -{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) +{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*coth(x)),x, algorithm="fricas")

[Out]

x*log(a*cosh(x)/sinh(x)) + x*log(cosh(x) + sinh(x) + 1) - x*log(I*cosh(x) + I*sinh(x) + 1) - x*log(-I*cosh(x)
- I*sinh(x) + 1) + x*log(-cosh(x) - sinh(x) + 1) + dilog(cosh(x) + sinh(x)) - dilog(I*cosh(x) + I*sinh(x)) - d
ilog(-I*cosh(x) - I*sinh(x)) + dilog(-cosh(x) - sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (a \coth{\left (x \right )} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*coth(x)),x)

[Out]

Integral(log(a*coth(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (a \coth \left (x\right )\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*coth(x)),x, algorithm="giac")

[Out]

integrate(log(a*coth(x)), x)