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

Optimal. Leaf size=41 \[ \frac{\text{PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac{\text{PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0164433, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 5913} \[ \frac{\text{PolyLog}\left (2,-e^{-a-b x}\right )}{2 b}-\frac{\text{PolyLog}\left (2,e^{-a-b x}\right )}{2 b} \]

Antiderivative was successfully verified.

[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 2282

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 5913

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

Rubi steps

\begin{align*} \int \coth ^{-1}\left (e^{a+b x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\text{Li}_2\left (-e^{-a-b x}\right )}{2 b}-\frac{\text{Li}_2\left (e^{-a-b x}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0725544, size = 68, normalized size = 1.66 \[ \frac{-\text{PolyLog}\left (2,-e^{a+b x}\right )+\text{PolyLog}\left (2,e^{a+b x}\right )+b x \left (\log \left (1-e^{a+b x}\right )-\log \left (e^{a+b x}+1\right )+2 \coth ^{-1}\left (e^{a+b x}\right )\right )}{2 b} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.072, size = 67, normalized size = 1.6 \begin{align*}{\frac{\ln \left ({{\rm e}^{bx+a}} \right ){\rm arccoth} \left ({{\rm e}^{bx+a}}\right )}{b}}-{\frac{{\it dilog} \left ({{\rm e}^{bx+a}} \right ) }{2\,b}}-{\frac{{\it dilog} \left ({{\rm e}^{bx+a}}+1 \right ) }{2\,b}}-{\frac{\ln \left ({{\rm e}^{bx+a}} \right ) \ln \left ({{\rm e}^{bx+a}}+1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(exp(b*x+a)),x)

[Out]

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

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Maxima [B]  time = 1.07206, size = 144, normalized size = 3.51 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.67866, size = 417, normalized size = 10.17 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acoth}{\left (e^{a + b x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(exp(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcoth}\left (e^{\left (b x + a\right )}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccoth(e^(b*x + a)), x)