\(\int x^2 \coth ^{-1}(\cosh (x)) \, dx\) [202]

Optimal result

Integrand size = 7, antiderivative size = 77 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 \operatorname {PolyLog}\left (4,-e^x\right )+2 \operatorname {PolyLog}\left (4,e^x\right ) \]

[Out]

1/3*x^3*arccoth(cosh(x))-2/3*x^3*arctanh(exp(x))-x^2*polylog(2,-exp(x))+x^2*polylog(2,exp(x))+2*x*polylog(3,-e
xp(x))-2*x*polylog(3,exp(x))-2*polylog(4,-exp(x))+2*polylog(4,exp(x))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6409, 4267, 2611, 6744, 2320, 6724} \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 \operatorname {PolyLog}\left (4,-e^x\right )+2 \operatorname {PolyLog}\left (4,e^x\right )+\frac {1}{3} x^3 \coth ^{-1}(\cosh (x)) \]

[In]

Int[x^2*ArcCoth[Cosh[x]],x]

[Out]

(x^3*ArcCoth[Cosh[x]])/3 - (2*x^3*ArcTanh[E^x])/3 - x^2*PolyLog[2, -E^x] + x^2*PolyLog[2, E^x] + 2*x*PolyLog[3
, -E^x] - 2*x*PolyLog[3, E^x] - 2*PolyLog[4, -E^x] + 2*PolyLog[4, 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} \int x^3 \text {csch}(x) \, dx \\ & = \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-\int x^2 \log \left (1-e^x\right ) \, dx+\int x^2 \log \left (1+e^x\right ) \, dx \\ & = \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 \int x \operatorname {PolyLog}\left (2,-e^x\right ) \, dx-2 \int x \operatorname {PolyLog}\left (2,e^x\right ) \, dx \\ & = \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 \int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx+2 \int \operatorname {PolyLog}\left (3,e^x\right ) \, dx \\ & = \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^x\right )+2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^x\right ) \\ & = \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 \operatorname {PolyLog}\left (4,-e^x\right )+2 \operatorname {PolyLog}\left (4,e^x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{3} \left (x^3 \coth ^{-1}(\cosh (x))+x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right ) \]

[In]

Integrate[x^2*ArcCoth[Cosh[x]],x]

[Out]

(x^3*ArcCoth[Cosh[x]] + x^3*Log[1 - E^x] - x^3*Log[1 + E^x] - 3*x^2*PolyLog[2, -E^x] + 3*x^2*PolyLog[2, E^x] +
6*x*PolyLog[3, -E^x] - 6*x*PolyLog[3, E^x] - 6*PolyLog[4, -E^x] + 6*PolyLog[4, E^x])/3

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.34 (sec) , antiderivative size = 401, normalized size of antiderivative = 5.21

method	result	size
risch	\(-\frac {x^{3} \ln \left ({\mathrm e}^{x}-1\right )}{3}-x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )+2 x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )-2 \operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )-\frac {i \pi \left ({\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )-2 \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{2}+{\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{3}+\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )-\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2}-{\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )+2 \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{2}-{\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{3}-\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2}-\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{3}-\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3}\right ) x^{3}}{12}+\frac {x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{3}+x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-2 x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+2 \operatorname {polylog}\left (4, {\mathrm e}^{x}\right )\)	\(401\)

[In]

int(x^2*arccoth(cosh(x)),x,method=_RETURNVERBOSE)

[Out]

-1/3*x^3*ln(exp(x)-1)-x^2*polylog(2,-exp(x))+2*x*polylog(3,-exp(x))-2*polylog(4,-exp(x))-1/12*I*Pi*(csgn(I*(1+
exp(x)))^2*csgn(I*(1+exp(x))^2)-2*csgn(I*(1+exp(x)))*csgn(I*(1+exp(x))^2)^2+csgn(I*(1+exp(x))^2)^3+csgn(I*(1+e
xp(x))^2)*csgn(I*exp(-x))*csgn(I*exp(-x)*(1+exp(x))^2)-csgn(I*(1+exp(x))^2)*csgn(I*exp(-x)*(1+exp(x))^2)^2-csg
n(I*(exp(x)-1))^2*csgn(I*(exp(x)-1)^2)+2*csgn(I*(exp(x)-1))*csgn(I*(exp(x)-1)^2)^2-csgn(I*(exp(x)-1)^2)^3-csgn
(I*(exp(x)-1)^2)*csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x)-1)^2)+csgn(I*(exp(x)-1)^2)*csgn(I*exp(-x)*(exp(x)-1)^2
)^2-csgn(I*exp(-x))*csgn(I*exp(-x)*(1+exp(x))^2)^2+csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x)-1)^2)^2+csgn(I*exp(-
x)*(1+exp(x))^2)^3-csgn(I*exp(-x)*(exp(x)-1)^2)^3)*x^3+1/3*x^3*ln(1-exp(x))+x^2*polylog(2,exp(x))-2*x*polylog(
3,exp(x))+2*polylog(4,exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.52 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{6} \, x^{3} \log \left (\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac {1}{3} \, x^{3} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{3} \, x^{3} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x^{2} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x^{2} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 2 \, x {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, x {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + 2 \, {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]

[In]

integrate(x^2*arccoth(cosh(x)),x, algorithm="fricas")

[Out]

1/6*x^3*log((cosh(x) + 1)/(cosh(x) - 1)) - 1/3*x^3*log(cosh(x) + sinh(x) + 1) + 1/3*x^3*log(-cosh(x) - sinh(x)
+ 1) + x^2*dilog(cosh(x) + sinh(x)) - x^2*dilog(-cosh(x) - sinh(x)) - 2*x*polylog(3, cosh(x) + sinh(x)) + 2*x
*polylog(3, -cosh(x) - sinh(x)) + 2*polylog(4, cosh(x) + sinh(x)) - 2*polylog(4, -cosh(x) - sinh(x))

Sympy [F]

\[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\int x^{2} \operatorname {acoth}{\left (\cosh {\left (x \right )} \right )}\, dx \]

[In]

integrate(x**2*acoth(cosh(x)),x)

[Out]

Integral(x**2*acoth(cosh(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (\cosh \left (x\right )\right ) - \frac {1}{3} \, x^{3} \log \left (e^{x} + 1\right ) + \frac {1}{3} \, x^{3} \log \left (-e^{x} + 1\right ) - x^{2} {\rm Li}_2\left (-e^{x}\right ) + x^{2} {\rm Li}_2\left (e^{x}\right ) + 2 \, x {\rm Li}_{3}(-e^{x}) - 2 \, x {\rm Li}_{3}(e^{x}) - 2 \, {\rm Li}_{4}(-e^{x}) + 2 \, {\rm Li}_{4}(e^{x}) \]

[In]

integrate(x^2*arccoth(cosh(x)),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\int { x^{2} \operatorname {arcoth}\left (\cosh \left (x\right )\right ) \,d x } \]

[In]

integrate(x^2*arccoth(cosh(x)),x, algorithm="giac")

[Out]

integrate(x^2*arccoth(cosh(x)), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\int x^2\,\mathrm {acoth}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \]

[In]

int(x^2*acoth(cosh(x)),x)

[Out]

int(x^2*acoth(cosh(x)), x)