\(\int \coth ^{-1}(\cosh (x)) \, dx\) [200]

Optimal result

Integrand size = 3, antiderivative size = 27 \[ \int \coth ^{-1}(\cosh (x)) \, dx=x \coth ^{-1}(\cosh (x))-2 x \text {arctanh}\left (e^x\right )-\operatorname {PolyLog}\left (2,-e^x\right )+\operatorname {PolyLog}\left (2,e^x\right ) \]

[Out]

x*arccoth(cosh(x))-2*x*arctanh(exp(x))-polylog(2,-exp(x))+polylog(2,exp(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6407, 4267, 2317, 2438} \[ \int \coth ^{-1}(\cosh (x)) \, dx=-2 x \text {arctanh}\left (e^x\right )-\operatorname {PolyLog}\left (2,-e^x\right )+\operatorname {PolyLog}\left (2,e^x\right )+x \coth ^{-1}(\cosh (x)) \]

[In]

Int[ArcCoth[Cosh[x]],x]

[Out]

x*ArcCoth[Cosh[x]] - 2*x*ArcTanh[E^x] - PolyLog[2, -E^x] + PolyLog[2, E^x]

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 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 6407

Int[ArcCoth[u_], x_Symbol] :> Simp[x*ArcCoth[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 - u^2)), x], x] /; I
nverseFunctionFreeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = x \coth ^{-1}(\cosh (x))+\int x \text {csch}(x) \, dx \\ & = x \coth ^{-1}(\cosh (x))-2 x \text {arctanh}\left (e^x\right )-\int \log \left (1-e^x\right ) \, dx+\int \log \left (1+e^x\right ) \, dx \\ & = x \coth ^{-1}(\cosh (x))-2 x \text {arctanh}\left (e^x\right )-\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right ) \\ & = x \coth ^{-1}(\cosh (x))-2 x \text {arctanh}\left (e^x\right )-\operatorname {PolyLog}\left (2,-e^x\right )+\operatorname {PolyLog}\left (2,e^x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \coth ^{-1}(\cosh (x)) \, dx=x \coth ^{-1}(\cosh (x))+x \left (\log \left (1-e^x\right )-\log \left (1+e^x\right )\right )-\operatorname {PolyLog}\left (2,-e^x\right )+\operatorname {PolyLog}\left (2,e^x\right ) \]

[In]

Integrate[ArcCoth[Cosh[x]],x]

[Out]

x*ArcCoth[Cosh[x]] + x*(Log[1 - E^x] - Log[1 + E^x]) - PolyLog[2, -E^x] + PolyLog[2, E^x]

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33

method	result	size
default	\(x \,\operatorname {arccoth}\left (\cosh \left (x \right )\right )+x \ln \left (1-{\mathrm e}^{x}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-x \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )\)	\(36\)
parts	\(x \,\operatorname {arccoth}\left (\cosh \left (x \right )\right )+x \ln \left (1-{\mathrm e}^{x}\right )+\operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-x \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )\)	\(36\)
risch	\(-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x}{4}+\frac {i \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{3} x}{4}-x \ln \left ({\mathrm e}^{x}-1\right )-\operatorname {dilog}\left ({\mathrm e}^{x}\right )-\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{3} x}{4}+\frac {i \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2} x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2} x}{4}-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right ) x}{4}-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{3} x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right ) x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{2} x}{2}\)	\(392\)

[In]

int(arccoth(cosh(x)),x,method=_RETURNVERBOSE)

[Out]

x*arccoth(cosh(x))+x*ln(1-exp(x))+polylog(2,exp(x))-x*ln(1+exp(x))-polylog(2,-exp(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).

Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{2} \, x \log \left (\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]

[In]

integrate(arccoth(cosh(x)),x, algorithm="fricas")

[Out]

1/2*x*log((cosh(x) + 1)/(cosh(x) - 1)) - x*log(cosh(x) + sinh(x) + 1) + x*log(-cosh(x) - sinh(x) + 1) + dilog(
cosh(x) + sinh(x)) - dilog(-cosh(x) - sinh(x))

Sympy [F]

\[ \int \coth ^{-1}(\cosh (x)) \, dx=\int \operatorname {acoth}{\left (\cosh {\left (x \right )} \right )}\, dx \]

[In]

integrate(acoth(cosh(x)),x)

[Out]

Integral(acoth(cosh(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \coth ^{-1}(\cosh (x)) \, dx=x \operatorname {arcoth}\left (\cosh \left (x\right )\right ) - x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{x}\right ) + {\rm Li}_2\left (e^{x}\right ) \]

[In]

integrate(arccoth(cosh(x)),x, algorithm="maxima")

[Out]

x*arccoth(cosh(x)) - x*log(e^x + 1) + x*log(-e^x + 1) - dilog(-e^x) + dilog(e^x)

Giac [F]

\[ \int \coth ^{-1}(\cosh (x)) \, dx=\int { \operatorname {arcoth}\left (\cosh \left (x\right )\right ) \,d x } \]

[In]

integrate(arccoth(cosh(x)),x, algorithm="giac")

[Out]

integrate(arccoth(cosh(x)), x)

Mupad [F(-1)]

Timed out. \[ \int \coth ^{-1}(\cosh (x)) \, dx=\int \mathrm {acoth}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \]

[In]

int(acoth(cosh(x)),x)

[Out]

int(acoth(cosh(x)), x)