\(\int x^2 \text {arctanh}(\cosh (x)) \, dx\) [283]

Optimal result

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

-2/3*x^3*arctanh(exp(x))+1/3*x^3*arctanh(cosh(x))-x^2*polylog(2,-exp(x))+x 
^2*polylog(2,exp(x))+2*x*polylog(3,-exp(x))-2*x*polylog(3,exp(x))-2*polylo 
g(4,-exp(x))+2*polylog(4,exp(x))
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\frac {1}{3} \left (x^3 \text {arctanh}(\cosh (x))+x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right ) \] Input:

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

Output:

(x^3*ArcTanh[Cosh[x]] + x^3*Log[1 - E^x] - x^3*Log[1 + E^x] - 3*x^2*PolyLo 
g[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
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {6827, 25, 3042, 26, 4670, 3011, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \text {arctanh}(\cosh (x)) \, dx\)

\(\Big \downarrow \) 6827

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))-\frac {1}{3} \int -x^3 \text {csch}(x)dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \int x^3 \text {csch}(x)dx+\frac {1}{3} x^3 \text {arctanh}(\cosh (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} \int i x^3 \csc (i x)dx\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} i \int x^3 \csc (i x)dx\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} i \left (3 i \int x^2 \log \left (1-e^x\right )dx-3 i \int x^2 \log \left (1+e^x\right )dx+2 i x^3 \text {arctanh}\left (e^x\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} i \left (-3 i \left (2 \int x \operatorname {PolyLog}\left (2,-e^x\right )dx-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \int x \operatorname {PolyLog}\left (2,e^x\right )dx-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} i \left (-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\int \operatorname {PolyLog}\left (3,-e^x\right )dx\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\int \operatorname {PolyLog}\left (3,e^x\right )dx\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} i \left (-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\int e^{-x} \operatorname {PolyLog}\left (3,-e^x\right )de^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\int e^{-x} \operatorname {PolyLog}\left (3,e^x\right )de^x\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{3} x^3 \text {arctanh}(\cosh (x))+\frac {1}{3} i \left (2 i x^3 \text {arctanh}\left (e^x\right )-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (4,-e^x\right )\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\operatorname {PolyLog}\left (4,e^x\right )\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[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 6827

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*((a + b*ArcTanh[u])/(d*(m + 1))), x] - Simp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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 7163

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] - Simp[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]
 

Maple [C] (warning: unable to verify)

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

Time = 0.42 (sec) , antiderivative size = 422, normalized size of antiderivative = 5.48

method	result	size
risch	\(-\frac {x^{3} \ln \left ({\mathrm e}^{x}-1\right )}{3}+\frac {i \pi \left (2 \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2}-{\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}-2\right ) x^{3}}{12}-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 {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 )\)	\(422\)

Input:

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

Output:

-1/3*x^3*ln(exp(x)-1)+1/12*I*Pi*(2*csgn(I*exp(-x)*(exp(x)-1)^2)^2-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+exp(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*ex 
p(-x)*(exp(x)-1)^2)^2-csgn(I*exp(-x)*(1+exp(x))^2)^3-csgn(I*exp(-x)*(exp(x 
)-1)^2)^3-2)*x^3-x^2*polylog(2,-exp(x))+2*x*polylog(3,-exp(x))-2*polylog(4 
,-exp(x))+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)

Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.53 \[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\frac {1}{6} \, x^{3} \log \left (-\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac {1}{3} \, x^{3} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{3} \, x^{3} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x^{2} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x^{2} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 2 \, x {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, x {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + 2 \, {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \] Input:

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

Output:

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*polyl 
og(4, -cosh(x) - sinh(x))
 

Sympy [F]

\[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\int x^{2} \operatorname {atanh}{\left (\cosh {\left (x \right )} \right )}\, dx \] Input:

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

Output:

Integral(x**2*atanh(cosh(x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\frac {1}{3} \, x^{3} \operatorname {artanh}\left (\cosh \left (x\right )\right ) - \frac {1}{3} \, x^{3} \log \left (e^{x} + 1\right ) + \frac {1}{3} \, x^{3} \log \left (-e^{x} + 1\right ) - x^{2} {\rm Li}_2\left (-e^{x}\right ) + x^{2} {\rm Li}_2\left (e^{x}\right ) + 2 \, x {\rm Li}_{3}(-e^{x}) - 2 \, x {\rm Li}_{3}(e^{x}) - 2 \, {\rm Li}_{4}(-e^{x}) + 2 \, {\rm Li}_{4}(e^{x}) \] Input:

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

Output:

1/3*x^3*arctanh(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]

\[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\int { x^{2} \operatorname {artanh}\left (\cosh \left (x\right )\right ) \,d x } \] Input:

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

Output:

integrate(x^2*arctanh(cosh(x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\int x^2\,\mathrm {atanh}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^2 \text {arctanh}(\cosh (x)) \, dx=\int \mathit {atanh} \left (\cosh \left (x \right )\right ) x^{2}d x \] Input:

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

Output:

int(atanh(cosh(x))*x**2,x)