\(\int \log (a \coth (x)) \, dx\) [212]

Optimal result

Integrand size = 5, antiderivative size = 41 \[ \int \log (a \coth (x)) \, dx=-2 x \text {arctanh}\left (e^{2 x}\right )+x \log (a \coth (x))-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2628, 5569, 4267, 2317, 2438} \[ \int \log (a \coth (x)) \, dx=x \log (a \coth (x))-2 x \text {arctanh}\left (e^{2 x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2} \]

[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 2317

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 2438

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 2628

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

Rule 4267

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 5569

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*} \text {integral}& = 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} \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac {1}{2} \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20 \[ \int \log (a \coth (x)) \, dx=-\frac {1}{2} \log (a \coth (x)) \log (1-\tanh (x))+\frac {1}{2} \log (a \coth (x)) \log (1+\tanh (x))-\frac {1}{2} \operatorname {PolyLog}(2,-\tanh (x))+\frac {\operatorname {PolyLog}(2,\tanh (x))}{2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(34)=68\).

Time = 1.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.85

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\operatorname {dilog}\left (-\frac {a \coth \left (x \right )-a}{a}\right )+\ln \left (a \coth \left (x \right )\right ) \ln \left (-\frac {a \coth \left (x \right )-a}{a}\right )\right )}{2}+\frac {a \left (\operatorname {dilog}\left (\frac {a \coth \left (x \right )+a}{a}\right )+\ln \left (a \coth \left (x \right )\right ) \ln \left (\frac {a \coth \left (x \right )+a}{a}\right )\right )}{2}}{a}\)	\(76\)
default	\(\frac {-\frac {a \left (\operatorname {dilog}\left (-\frac {a \coth \left (x \right )-a}{a}\right )+\ln \left (a \coth \left (x \right )\right ) \ln \left (-\frac {a \coth \left (x \right )-a}{a}\right )\right )}{2}+\frac {a \left (\operatorname {dilog}\left (\frac {a \coth \left (x \right )+a}{a}\right )+\ln \left (a \coth \left (x \right )\right ) \ln \left (\frac {a \coth \left (x \right )+a}{a}\right )\right )}{2}}{a}\)	\(76\)
risch	\(-x \ln \left (-1+{\mathrm e}^{2 x}\right )-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right )}^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right )}^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (\frac {i}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-1+{\mathrm e}^{2 x}}\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right )}^{2} x}{2}+\ln \left (a \right ) x +\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )+x \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {dilog}\left ({\mathrm e}^{x}\right )-\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right )}^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right ) {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )}{-1+{\mathrm e}^{2 x}}\right )}^{2} x}{2}+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )-\operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )-\operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\)	\(380\)

[In]

int(ln(a*coth(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*(-1/2*a*(dilog(-(a*coth(x)-a)/a)+ln(a*coth(x))*ln(-(a*coth(x)-a)/a))+1/2*a*(dilog((a*coth(x)+a)/a)+ln(a*co
th(x))*ln((a*coth(x)+a)/a)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.49 \[ \int \log (a \coth (x)) \, dx=x \log \left (\frac {a \cosh \left (x\right )}{\sinh \left (x\right )}\right ) + x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]

[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))

Sympy [F]

\[ \int \log (a \coth (x)) \, dx=\int \log {\left (a \coth {\left (x \right )} \right )}\, dx \]

[In]

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

[Out]

Integral(log(a*coth(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \log (a \coth (x)) \, dx=x \log \left (a \coth \left (x\right )\right ) - x \log \left (e^{\left (2 \, x\right )} + 1\right ) + x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) - \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) + {\rm Li}_2\left (-e^{x}\right ) + {\rm Li}_2\left (e^{x}\right ) \]

[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)

Giac [F]

\[ \int \log (a \coth (x)) \, dx=\int { \log \left (a \coth \left (x\right )\right ) \,d x } \]

[In]

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

[Out]

integrate(log(a*coth(x)), x)

Mupad [F(-1)]

Timed out. \[ \int \log (a \coth (x)) \, dx=\int \ln \left (a\,\mathrm {coth}\left (x\right )\right ) \,d x \]

[In]

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

[Out]

int(log(a*coth(x)), x)