\(\int \coth ^{-1}(e^x) \, dx\) [283]

Optimal result

Integrand size = 4, antiderivative size = 25 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{-x}\right )}{2}-\frac {\operatorname {PolyLog}\left (2,e^{-x}\right )}{2} \]

[Out]

1/2*polylog(2,-1/exp(x))-1/2*polylog(2,exp(-x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 6032} \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{-x}\right )}{2}-\frac {\operatorname {PolyLog}\left (2,e^{-x}\right )}{2} \]

[In]

Int[ArcCoth[E^x],x]

[Out]

PolyLog[2, -E^(-x)]/2 - PolyLog[2, E^(-x)]/2

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6032

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b/2)*PolyLog[2, -(c*x)^(
-1)], x] - Simp[(b/2)*PolyLog[2, 1/(c*x)], x]) /; FreeQ[{a, b, c}, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(25)=50\).

Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=x \coth ^{-1}\left (e^x\right )+\frac {1}{2} x \log \left (1-e^x\right )-\frac {1}{2} x \log \left (1+e^x\right )-\frac {\operatorname {PolyLog}\left (2,-e^x\right )}{2}+\frac {\operatorname {PolyLog}\left (2,e^x\right )}{2} \]

[In]

Integrate[ArcCoth[E^x],x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
risch	\(-\frac {x \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\operatorname {dilog}\left ({\mathrm e}^{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )}{2}\)	\(22\)
derivativedivides	\(\ln \left ({\mathrm e}^{x}\right ) \operatorname {arccoth}\left ({\mathrm e}^{x}\right )-\frac {\operatorname {dilog}\left ({\mathrm e}^{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )}{2}\)	\(31\)
default	\(\ln \left ({\mathrm e}^{x}\right ) \operatorname {arccoth}\left ({\mathrm e}^{x}\right )-\frac {\operatorname {dilog}\left ({\mathrm e}^{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )}{2}\)	\(31\)
parts	\(x \,\operatorname {arccoth}\left ({\mathrm e}^{x}\right )+\frac {x \ln \left (1-{\mathrm e}^{x}\right )}{2}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{2}-\frac {x \ln \left (1+{\mathrm e}^{x}\right )}{2}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{2}\)	\(39\)

[In]

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

[Out]

-1/2*x*ln(exp(x)-1)-1/2*dilog(exp(x))-1/2*dilog(1+exp(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\frac {1}{2} \, x \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac {1}{2} \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{2} \, {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(acoth(exp(x)),x)

[Out]

Integral(acoth(exp(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=-\frac {1}{2} \, x {\left (\log \left (e^{x} + 1\right ) - \log \left (e^{x} - 1\right )\right )} + x \operatorname {arcoth}\left (e^{x}\right ) + \frac {1}{2} \, \log \left (-e^{x}\right ) \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x \log \left (e^{x} - 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (e^{x} + 1\right ) - \frac {1}{2} \, {\rm Li}_2\left (-e^{x} + 1\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(arccoth(e^x), x)

Mupad [F(-1)]

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

[In]

int(acoth(exp(x)),x)

[Out]

int(acoth(exp(x)), x)