\(\int \coth ^{-1}(e^{a+b x}) \, dx\) [286]

Optimal result

Integrand size = 8, antiderivative size = 41 \[ \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 6032} \[ \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]

[In]

Int[ArcCoth[E^(a + b*x)],x]

[Out]

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

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}& = \frac {\text {Subst}\left (\int \frac {\coth ^{-1}(x)}{x} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.66 \[ \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {b x \left (2 \coth ^{-1}\left (e^{a+b x}\right )+\log \left (1-e^{a+b x}\right )-\log \left (1+e^{a+b x}\right )\right )-\operatorname {PolyLog}\left (2,-e^{a+b x}\right )+\operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b} \]

[In]

Integrate[ArcCoth[E^(a + b*x)],x]

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20

method	result	size
risch	\(-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right )}{2 b}-\frac {\ln \left (-1+{\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}\right )}{2 b}-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2 b}\)	\(49\)
derivativedivides	\(\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \operatorname {arccoth}\left ({\mathrm e}^{b x +a}\right )-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2}-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}}{b}\)	\(59\)
default	\(\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \operatorname {arccoth}\left ({\mathrm e}^{b x +a}\right )-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}\right )}{2}-\frac {\operatorname {dilog}\left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\ln \left ({\mathrm e}^{b x +a}\right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}}{b}\)	\(59\)
parts	\(x \,\operatorname {arccoth}\left ({\mathrm e}^{b x +a}\right )+\frac {\frac {\left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2}-\frac {\left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2}+a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\)	\(81\)

[In]

int(arccoth(exp(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (33) = 66\).

Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 3.34 \[ \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {b x \log \left (\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b} \]

[In]

integrate(arccoth(exp(b*x+a)),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(acoth(exp(b*x+a)),x)

[Out]

Integral(acoth(exp(a + b*x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (33) = 66\).

Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.61 \[ \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {{\left (b x + a\right )} \operatorname {arcoth}\left (e^{\left (b x + a\right )}\right )}{b} - \frac {{\left (b x + a\right )} {\left (\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left (e^{\left (b x + a\right )} - 1\right )\right )} - \log \left (-e^{\left (b x + a\right )}\right ) \log \left (e^{\left (b x + a\right )} + 1\right ) + {\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - {\rm Li}_2\left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )} + 1\right )}{2 \, b} \]

[In]

integrate(arccoth(exp(b*x+a)),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(arccoth(exp(b*x+a)),x, algorithm="giac")

[Out]

integrate(arccoth(e^(b*x + a)), x)

Mupad [F(-1)]

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

[In]

int(acoth(exp(a + b*x)),x)

[Out]

int(acoth(exp(a + b*x)), x)