3.281 \(\int \tanh ^{-1}(\cosh (x)) \, dx\)

Optimal. Leaf size=27 \[ -\text{PolyLog}\left (2,-e^x\right )+\text{PolyLog}\left (2,e^x\right )-2 x \tanh ^{-1}\left (e^x\right )+x \tanh ^{-1}(\cosh (x)) \]

[Out]

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

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Rubi [A]  time = 0.0319938, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 3, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.333, Rules used = {6271, 4182, 2279, 2391} \[ -\text{PolyLog}\left (2,-e^x\right )+\text{PolyLog}\left (2,e^x\right )-2 x \tanh ^{-1}\left (e^x\right )+x \tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Cosh[x]],x]

[Out]

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

Rule 6271

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

Rule 4182

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 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.0205727, size = 47, normalized size = 1.74 \[ \text{PolyLog}\left (2,-e^{-x}\right )-\text{PolyLog}\left (2,e^{-x}\right )+x \left (\log \left (1-e^{-x}\right )-\log \left (e^{-x}+1\right )\right )+x \tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Cosh[x]],x]

[Out]

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

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Maple [A]  time = 0.04, size = 21, normalized size = 0.8 \begin{align*} x{\it Artanh} \left ( \cosh \left ( x \right ) \right ) +2\,{\it dilog} \left ({{\rm e}^{-x}} \right ) -{\frac{{\it dilog} \left ({{\rm e}^{-2\,x}} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(cosh(x)),x)

[Out]

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

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Maxima [A]  time = 1.12202, size = 45, normalized size = 1.67 \begin{align*} x \operatorname{artanh}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.23709, size = 215, normalized size = 7.96 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(cosh(x)),x)

[Out]

Integral(atanh(cosh(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arctanh(cosh(x)), x)