3.3.1 \(\int x \coth ^{-1}(\cosh (x)) \, dx\) [201]

Optimal. Leaf size=51 \[ \frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {PolyLog}\left (2,-e^x\right )+x \text {PolyLog}\left (2,e^x\right )+\text {PolyLog}\left (3,-e^x\right )-\text {PolyLog}\left (3,e^x\right ) \]

[Out]

1/2*x^2*arccoth(cosh(x))-x^2*arctanh(exp(x))-x*polylog(2,-exp(x))+x*polylog(2,exp(x))+polylog(3,-exp(x))-polyl
og(3,exp(x))

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Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6409, 4267, 2611, 2320, 6724} \begin {gather*} -x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Li}_3\left (-e^x\right )-\text {Li}_3\left (e^x\right )-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \coth ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*ArcCoth[Cosh[x]],x]

[Out]

(x^2*ArcCoth[Cosh[x]])/2 - x^2*ArcTanh[E^x] - x*PolyLog[2, -E^x] + x*PolyLog[2, E^x] + PolyLog[3, -E^x] - Poly
Log[3, E^x]

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 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4267

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 6409

Int[((a_.) + ArcCoth[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcCot
h[u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 - u^2)), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int x \coth ^{-1}(\cosh (x)) \, dx &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))+\frac {1}{2} \int x^2 \text {csch}(x) \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-\int x \log \left (1-e^x\right ) \, dx+\int x \log \left (1+e^x\right ) \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\int \text {Li}_2\left (-e^x\right ) \, dx-\int \text {Li}_2\left (e^x\right ) \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Li}_3\left (-e^x\right )-\text {Li}_3\left (e^x\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 81, normalized size = 1.59 \begin {gather*} \frac {1}{2} \left (x^2 \coth ^{-1}(\cosh (x))+x^2 \log \left (1-e^{-x}\right )-x^2 \log \left (1+e^{-x}\right )+2 x \text {PolyLog}\left (2,-e^{-x}\right )-2 x \text {PolyLog}\left (2,e^{-x}\right )+2 \text {PolyLog}\left (3,-e^{-x}\right )-2 \text {PolyLog}\left (3,e^{-x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*ArcCoth[Cosh[x]],x]

[Out]

(x^2*ArcCoth[Cosh[x]] + x^2*Log[1 - E^(-x)] - x^2*Log[1 + E^(-x)] + 2*x*PolyLog[2, -E^(-x)] - 2*x*PolyLog[2, E
^(-x)] + 2*PolyLog[3, -E^(-x)] - 2*PolyLog[3, E^(-x)])/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 449, normalized size = 8.80

method	result	size
risch	\(-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{8}+\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{4}+\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) x^{2}}{8}+x \polylog \left (2, {\mathrm e}^{x}\right )-x \polylog \left (2, -{\mathrm e}^{x}\right )+\polylog \left (3, -{\mathrm e}^{x}\right )-\polylog \left (3, {\mathrm e}^{x}\right )+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{2}-\frac {x^{2} \ln \left ({\mathrm e}^{x}-1\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{4}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right )^{3} x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right ) x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{3} x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right ) x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{8}+\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x^{2}}{8}\)	\(449\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccoth(cosh(x)),x,method=_RETURNVERBOSE)

[Out]

-1/8*I*Pi*csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x)-1)^2)^2*x^2+1/8*I*Pi*csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x)+1
)^2)^2*x^2+1/8*I*Pi*csgn(I*exp(-x)*(exp(x)-1)^2)^3*x^2-1/4*I*Pi*csgn(I*(exp(x)-1))*csgn(I*(exp(x)-1)^2)^2*x^2+
1/8*I*Pi*csgn(I*(exp(x)-1))^2*csgn(I*(exp(x)-1)^2)*x^2+x*polylog(2,exp(x))-x*polylog(2,-exp(x))+polylog(3,-exp
(x))-polylog(3,exp(x))+1/2*x^2*ln(1-exp(x))-1/2*x^2*ln(exp(x)-1)+1/4*I*Pi*csgn(I*(exp(x)+1))*csgn(I*(exp(x)+1)
^2)^2*x^2-1/8*I*Pi*csgn(I*(exp(x)-1)^2)*csgn(I*exp(-x)*(exp(x)-1)^2)^2*x^2-1/8*I*Pi*csgn(I*(exp(x)+1)^2)^3*x^2
-1/8*I*Pi*csgn(I*(exp(x)+1)^2)*csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x)+1)^2)*x^2-1/8*I*Pi*csgn(I*(exp(x)+1))^2*
csgn(I*(exp(x)+1)^2)*x^2-1/8*I*Pi*csgn(I*exp(-x)*(exp(x)+1)^2)^3*x^2+1/8*I*Pi*csgn(I*(exp(x)-1)^2)*csgn(I*exp(
-x))*csgn(I*exp(-x)*(exp(x)-1)^2)*x^2+1/8*I*Pi*csgn(I*(exp(x)+1)^2)*csgn(I*exp(-x)*(exp(x)+1)^2)^2*x^2+1/8*I*P
i*csgn(I*(exp(x)-1)^2)^3*x^2

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Maxima [A]
time = 0.30, size = 56, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\cosh \left (x\right )\right ) - \frac {1}{2} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-e^{x} + 1\right ) - x {\rm Li}_2\left (-e^{x}\right ) + x {\rm Li}_2\left (e^{x}\right ) + {\rm Li}_{3}(-e^{x}) - {\rm Li}_{3}(e^{x}) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccoth(cosh(x)),x, algorithm="maxima")

[Out]

1/2*x^2*arccoth(cosh(x)) - 1/2*x^2*log(e^x + 1) + 1/2*x^2*log(-e^x + 1) - x*dilog(-e^x) + x*dilog(e^x) + polyl
og(3, -e^x) - polylog(3, e^x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (42) = 84\).
time = 0.37, size = 87, normalized size = 1.71 \begin {gather*} \frac {1}{4} \, x^{2} \log \left (\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac {1}{2} \, x^{2} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccoth(cosh(x)),x, algorithm="fricas")

[Out]

1/4*x^2*log((cosh(x) + 1)/(cosh(x) - 1)) - 1/2*x^2*log(cosh(x) + sinh(x) + 1) + 1/2*x^2*log(-cosh(x) - sinh(x)
+ 1) + x*dilog(cosh(x) + sinh(x)) - x*dilog(-cosh(x) - sinh(x)) - polylog(3, cosh(x) + sinh(x)) + polylog(3,
-cosh(x) - sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \operatorname {acoth}{\left (\cosh {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acoth(cosh(x)),x)

[Out]

Integral(x*acoth(cosh(x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccoth(cosh(x)),x, algorithm="giac")

[Out]

integrate(x*arccoth(cosh(x)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\mathrm {acoth}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*acoth(cosh(x)),x)

[Out]

int(x*acoth(cosh(x)), x)

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