Integrand size = 7, antiderivative size = 77 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 \operatorname {PolyLog}\left (4,-e^x\right )+2 \operatorname {PolyLog}\left (4,e^x\right ) \]
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,-exp(x))-2*x*polylog(3,exp(x))-2*polylog (4,-exp(x))+2*polylog(4,exp(x))
Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{3} \left (x^3 \coth ^{-1}(\cosh (x))+x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right ) \]
(x^3*ArcCoth[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
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 = {6828, 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.
\(\displaystyle \int x^2 \coth ^{-1}(\cosh (x)) \, dx\) |
\(\Big \downarrow \) 6828 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac {1}{3} \int -x^3 \text {csch}(x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \int x^3 \text {csch}(x)dx+\frac {1}{3} x^3 \coth ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} \int i x^3 \csc (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} i \int x^3 \csc (i x)dx\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} i \left (3 i \int x^2 \log \left (1-e^x\right )dx-3 i \int x^2 \log \left (1+e^x\right )dx+2 i x^3 \text {arctanh}\left (e^x\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} i \left (-3 i \left (2 \int x \operatorname {PolyLog}\left (2,-e^x\right )dx-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \int x \operatorname {PolyLog}\left (2,e^x\right )dx-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} i \left (-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\int \operatorname {PolyLog}\left (3,-e^x\right )dx\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\int \operatorname {PolyLog}\left (3,e^x\right )dx\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} i \left (-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\int e^{-x} \operatorname {PolyLog}\left (3,-e^x\right )de^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\int e^{-x} \operatorname {PolyLog}\left (3,e^x\right )de^x\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac {1}{3} i \left (2 i x^3 \text {arctanh}\left (e^x\right )-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (4,-e^x\right )\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\operatorname {PolyLog}\left (4,e^x\right )\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )\right )\) |
(x^3*ArcCoth[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])))
3.3.2.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]]]
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]
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]
Int[((a_.) + ArcCoth[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*((a + b*ArcCoth[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]]
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]
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]
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 )\) | \(401\) |
-1/3*x^3*ln(exp(x)-1)-x^2*polylog(2,-exp(x))+2*x*polylog(3,-exp(x))-2*poly log(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+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-csgn(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))
Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.52 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{6} \, x^{3} \log \left (\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac {1}{3} \, x^{3} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{3} \, x^{3} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x^{2} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x^{2} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 2 \, x {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, x {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + 2 \, {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
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*p olylog(3, -cosh(x) - sinh(x)) + 2*polylog(4, cosh(x) + sinh(x)) - 2*polylo g(4, -cosh(x) - sinh(x))
\[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\int x^{2} \operatorname {acoth}{\left (\cosh {\left (x \right )} \right )}\, dx \]
Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (\cosh \left (x\right )\right ) - \frac {1}{3} \, x^{3} \log \left (e^{x} + 1\right ) + \frac {1}{3} \, x^{3} \log \left (-e^{x} + 1\right ) - x^{2} {\rm Li}_2\left (-e^{x}\right ) + x^{2} {\rm Li}_2\left (e^{x}\right ) + 2 \, x {\rm Li}_{3}(-e^{x}) - 2 \, x {\rm Li}_{3}(e^{x}) - 2 \, {\rm Li}_{4}(-e^{x}) + 2 \, {\rm Li}_{4}(e^{x}) \]
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)
\[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\int { x^{2} \operatorname {arcoth}\left (\cosh \left (x\right )\right ) \,d x } \]
Timed out. \[ \int x^2 \coth ^{-1}(\cosh (x)) \, dx=\int x^2\,\mathrm {acoth}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \]