3.346 \(\int \tanh ^{-1}(e^x) \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \text{PolyLog}\left (2,e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,-e^x\right ) \]

[Out]

-PolyLog[2, -E^x]/2 + PolyLog[2, E^x]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0115109, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2282, 5912} \[ \frac{1}{2} \text{PolyLog}\left (2,e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,-e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[E^x],x]

[Out]

-PolyLog[2, -E^x]/2 + PolyLog[2, E^x]/2

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 5912

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

Rubi steps

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

Mathematica [B]  time = 0.0349408, size = 51, normalized size = 2.43 \[ -\frac{1}{2} \text{PolyLog}\left (2,-e^x\right )+\frac{1}{2} \text{PolyLog}\left (2,e^x\right )+\frac{1}{2} x \log \left (1-e^x\right )-\frac{1}{2} x \log \left (e^x+1\right )+x \tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[E^x],x]

[Out]

x*ArcTanh[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]  time = 0.044, size = 31, normalized size = 1.5 \begin{align*} \ln \left ({{\rm e}^{x}} \right ){\it Artanh} \left ({{\rm e}^{x}} \right ) -{\frac{{\it dilog} \left ({{\rm e}^{x}} \right ) }{2}}-{\frac{{\it dilog} \left ({{\rm e}^{x}}+1 \right ) }{2}}-{\frac{\ln \left ({{\rm e}^{x}} \right ) \ln \left ({{\rm e}^{x}}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(exp(x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.964469, size = 78, normalized size = 3.71 \begin{align*} -\frac{1}{2} \, x{\left (\log \left (e^{x} + 1\right ) - \log \left (e^{x} - 1\right )\right )} + x \operatorname{artanh}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.58103, size = 263, normalized size = 12.52 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(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(-co
sh(x) - sinh(x) + 1) + 1/2*dilog(cosh(x) + sinh(x)) - 1/2*dilog(-cosh(x) - sinh(x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(exp(x)),x)

[Out]

Integral(atanh(exp(x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arctanh(e^x), x)