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

Optimal. Leaf size=46 \[ x \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (-e^{2 x}\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right )-2 n x \tanh ^{-1}\left (e^{2 x}\right ) \]

[Out]

-2*n*x*arctanh(exp(2*x))+x*ln(a*coth(x)^n)-1/2*n*polylog(2,-exp(2*x))+1/2*n*polylog(2,exp(2*x))

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Rubi [A]  time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 55, normalized size = 1.20 \[ -\frac {1}{2} \log (1-\tanh (x)) \log \left (a \coth ^n(x)\right )+\frac {1}{2} \log (\tanh (x)+1) \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Li}_2(-\tanh (x))+\frac {1}{2} n \text {Li}_2(\tanh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Log[a*Coth[x]^n]*Log[1 - Tanh[x]]) + (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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fricas [C]  time = 0.49, size = 116, normalized size = 2.52 \[ n x \log \left (\frac {\cosh \relax (x)}{\sinh \relax (x)}\right ) + n x \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - n x \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) - n x \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) + n x \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + n {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - n {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) - n {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) + n {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) + x \log \relax (a) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (a \coth \relax (x)^{n}\right )\,{d x} \]

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)

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maple [A]  time = 0.24, size = 43, normalized size = 0.93 \[ \frac {n \ln \left (\coth \relax (x )+1\right ) \ln \left (\coth \relax (x )\right )}{2}+\frac {n \dilog \left (\coth \relax (x )+1\right )}{2}+\frac {n \dilog \left (\coth \relax (x )\right )}{2}+\left (-n \ln \left (\coth \relax (x )\right )+\ln \left (a \left (\coth ^{n}\relax (x )\right )\right )\right ) x \]

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 = 1.03, size = 61, normalized size = 1.33 \[ -\frac {1}{2} \, {\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 2 \, {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_2\left (e^{x}\right )\right )} n + x \log \left (a \coth \relax (x)^{n}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \ln \left (a\,{\mathrm {coth}\relax (x)}^n\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(log(a*coth(x)^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (a \coth ^{n}{\relax (x )} \right )}\, dx \]

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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