### 3.214 $$\int \log (a \coth ^n(x)) \, dx$$

Optimal. Leaf size=46 $-\frac{1}{2} n \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} n \text{PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 x}\right )$

[Out]

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

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Rubi [A]  time = 0.0494245, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.857, Rules used = {2548, 12, 5461, 4182, 2279, 2391} $-\frac{1}{2} n \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} n \text{PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 x}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-2*n*x*ArcTanh[E^(2*x)] + x*Log[a*Coth[x]^n] - (n*PolyLog[2, -E^(2*x)])/2 + (n*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 \left (a \coth ^n(x)\right ) \, dx &=x \log \left (a \coth ^n(x)\right )+\int n x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log \left (a \coth ^n(x)\right )+n \int x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log \left (a \coth ^n(x)\right )+(2 n) \int x \text{csch}(2 x) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-n \int \log \left (1-e^{2 x}\right ) \, dx+n \int \log \left (1+e^{2 x}\right ) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac{1}{2} n \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac{1}{2} n \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac{1}{2} n \text{Li}_2\left (-e^{2 x}\right )+\frac{1}{2} n \text{Li}_2\left (e^{2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0116304, size = 55, normalized size = 1.2 $-\frac{1}{2} n \text{PolyLog}(2,-\tanh (x))+\frac{1}{2} n \text{PolyLog}(2,\tanh (x))-\frac{1}{2} \log (1-\tanh (x)) \log \left (a \coth ^n(x)\right )+\frac{1}{2} \log (\tanh (x)+1) \log \left (a \coth ^n(x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.021, size = 43, normalized size = 0.9 \begin{align*} \left ( \ln \left ( a \left ({\rm coth} \left (x\right ) \right ) ^{n} \right ) -n\ln \left ({\rm coth} \left (x\right ) \right ) \right ) x+{\frac{n{\it dilog} \left ({\rm coth} \left (x\right ) \right ) }{2}}+{\frac{n{\it dilog} \left ({\rm coth} \left (x\right )+1 \right ) }{2}}+{\frac{n\ln \left ({\rm coth} \left (x\right ) \right ) \ln \left ({\rm coth} \left (x\right )+1 \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln(a*coth(x)^n)-n*ln(coth(x)))*x+1/2*n*dilog(coth(x))+1/2*n*dilog(coth(x)+1)+1/2*n*ln(coth(x))*ln(coth(x)+1)

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Maxima [A]  time = 1.74351, size = 82, normalized size = 1.78 \begin{align*} -\frac{1}{2} \,{\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 2 \,{\rm Li}_2\left (-e^{x}\right ) - 2 \,{\rm Li}_2\left (e^{x}\right )\right )} n + x \log \left (a \coth \left (x\right )^{n}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.92849, size = 412, normalized size = 8.96 \begin{align*} n x \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right ) + n x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - n x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - n x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + n x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + n{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - n{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - n{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + n{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) + x \log \left (a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

n*x*log(cosh(x)/sinh(x)) + n*x*log(cosh(x) + sinh(x) + 1) - n*x*log(I*cosh(x) + I*sinh(x) + 1) - n*x*log(-I*co
sh(x) - I*sinh(x) + 1) + n*x*log(-cosh(x) - sinh(x) + 1) + n*dilog(cosh(x) + sinh(x)) - n*dilog(I*cosh(x) + I*
sinh(x)) - n*dilog(-I*cosh(x) - I*sinh(x)) + n*dilog(-cosh(x) - sinh(x)) + x*log(a)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a*coth(x)**n), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a*coth(x)^n), x)