Integrand size = 8, antiderivative size = 70 \[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,-e^{-x}\right )-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{-x}\right )+x \operatorname {PolyLog}\left (3,-e^{-x}\right )-x \operatorname {PolyLog}\left (3,e^{-x}\right )+\operatorname {PolyLog}\left (4,-e^{-x}\right )-\operatorname {PolyLog}\left (4,e^{-x}\right ) \]
1/2*x^2*polylog(2,-1/exp(x))-1/2*x^2*polylog(2,exp(-x))+x*polylog(3,-1/exp (x))-x*polylog(3,exp(-x))+polylog(4,-1/exp(x))-polylog(4,exp(-x))
Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.33 \[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\frac {1}{6} \left (2 x^3 \coth ^{-1}\left (e^x\right )+x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right ) \]
(2*x^3*ArcCoth[E^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])/6
Time = 0.48 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6768, 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.
\(\displaystyle \int x^2 \coth ^{-1}\left (e^x\right ) \, dx\) |
\(\Big \downarrow \) 6768 |
\(\displaystyle \frac {1}{2} \int x^2 \log \left (1+e^{-x}\right )dx-\frac {1}{2} \int x^2 \log \left (1-e^{-x}\right )dx\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} \left (x^2 \operatorname {PolyLog}\left (2,-e^{-x}\right )-2 \int x \operatorname {PolyLog}\left (2,-e^{-x}\right )dx\right )+\frac {1}{2} \left (2 \int x \operatorname {PolyLog}\left (2,e^{-x}\right )dx-x^2 \operatorname {PolyLog}\left (2,e^{-x}\right )\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {1}{2} \left (x^2 \operatorname {PolyLog}\left (2,-e^{-x}\right )-2 \left (\int \operatorname {PolyLog}\left (3,-e^{-x}\right )dx-x \operatorname {PolyLog}\left (3,-e^{-x}\right )\right )\right )+\frac {1}{2} \left (2 \left (\int \operatorname {PolyLog}\left (3,e^{-x}\right )dx-x \operatorname {PolyLog}\left (3,e^{-x}\right )\right )-x^2 \operatorname {PolyLog}\left (2,e^{-x}\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} \left (x^2 \operatorname {PolyLog}\left (2,-e^{-x}\right )-2 \left (-\int e^x \operatorname {PolyLog}\left (3,-e^{-x}\right )de^{-x}-x \operatorname {PolyLog}\left (3,-e^{-x}\right )\right )\right )+\frac {1}{2} \left (2 \left (-\int e^x \operatorname {PolyLog}\left (3,e^{-x}\right )de^{-x}-x \operatorname {PolyLog}\left (3,e^{-x}\right )\right )-x^2 \operatorname {PolyLog}\left (2,e^{-x}\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} \left (x^2 \operatorname {PolyLog}\left (2,-e^{-x}\right )-2 \left (-x \operatorname {PolyLog}\left (3,-e^{-x}\right )-\operatorname {PolyLog}\left (4,-e^{-x}\right )\right )\right )+\frac {1}{2} \left (2 \left (-x \operatorname {PolyLog}\left (3,e^{-x}\right )-\operatorname {PolyLog}\left (4,e^{-x}\right )\right )-x^2 \operatorname {PolyLog}\left (2,e^{-x}\right )\right )\) |
(x^2*PolyLog[2, -E^(-x)] - 2*(-(x*PolyLog[3, -E^(-x)]) - PolyLog[4, -E^(-x )]))/2 + (-(x^2*PolyLog[2, E^(-x)]) + 2*(-(x*PolyLog[3, E^(-x)]) - PolyLog [4, E^(-x)]))/2
3.3.85.3.1 Defintions of rubi rules used
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[ArcCoth[(a_.) + (b_.)*(f_)^((c_.) + (d_.)*(x_))]*(x_)^(m_.), x_Symbol] :> Simp[1/2 Int[x^m*Log[1 + 1/(a + b*f^(c + d*x))], x], x] - Simp[1/2 I nt[x^m*Log[1 - 1/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]
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]
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01
method | result | size |
risch | \(-\frac {x^{3} \ln \left ({\mathrm e}^{x}-1\right )}{6}-\frac {x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{2}+x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )-\operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )+\frac {x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{6}+\frac {x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{2}-x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+\operatorname {polylog}\left (4, {\mathrm e}^{x}\right )\) | \(71\) |
default | \(\frac {x^{3} \operatorname {arccoth}\left ({\mathrm e}^{x}\right )}{3}+\frac {x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{6}+\frac {x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{2}-x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+\operatorname {polylog}\left (4, {\mathrm e}^{x}\right )-\frac {x^{3} \ln \left (1+{\mathrm e}^{x}\right )}{6}-\frac {x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{2}+x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )-\operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )\) | \(79\) |
parts | \(\frac {x^{3} \operatorname {arccoth}\left ({\mathrm e}^{x}\right )}{3}+\frac {x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{6}+\frac {x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{2}-x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+\operatorname {polylog}\left (4, {\mathrm e}^{x}\right )-\frac {x^{3} \ln \left (1+{\mathrm e}^{x}\right )}{6}-\frac {x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{2}+x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )-\operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )\) | \(79\) |
-1/6*x^3*ln(exp(x)-1)-1/2*x^2*polylog(2,-exp(x))+x*polylog(3,-exp(x))-poly log(4,-exp(x))+1/6*x^3*ln(1-exp(x))+1/2*x^2*polylog(2,exp(x))-x*polylog(3, exp(x))+polylog(4,exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (58) = 116\).
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.70 \[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\frac {1}{6} \, x^{3} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac {1}{6} \, x^{3} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{6} \, x^{3} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x^{2} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{2} \, x^{2} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - x {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + x {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
1/6*x^3*log((cosh(x) + sinh(x) + 1)/(cosh(x) + sinh(x) - 1)) - 1/6*x^3*log (cosh(x) + sinh(x) + 1) + 1/6*x^3*log(-cosh(x) - sinh(x) + 1) + 1/2*x^2*di log(cosh(x) + sinh(x)) - 1/2*x^2*dilog(-cosh(x) - sinh(x)) - x*polylog(3, cosh(x) + sinh(x)) + x*polylog(3, -cosh(x) - sinh(x)) + polylog(4, cosh(x) + sinh(x)) - polylog(4, -cosh(x) - sinh(x))
\[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\int x^{2} \operatorname {acoth}{\left (e^{x} \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.09 \[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (e^{x}\right ) - \frac {1}{6} \, x^{3} \log \left (e^{x} + 1\right ) + \frac {1}{6} \, x^{3} \log \left (-e^{x} + 1\right ) - \frac {1}{2} \, x^{2} {\rm Li}_2\left (-e^{x}\right ) + \frac {1}{2} \, x^{2} {\rm Li}_2\left (e^{x}\right ) + x {\rm Li}_{3}(-e^{x}) - x {\rm Li}_{3}(e^{x}) - {\rm Li}_{4}(-e^{x}) + {\rm Li}_{4}(e^{x}) \]
1/3*x^3*arccoth(e^x) - 1/6*x^3*log(e^x + 1) + 1/6*x^3*log(-e^x + 1) - 1/2* x^2*dilog(-e^x) + 1/2*x^2*dilog(e^x) + x*polylog(3, -e^x) - x*polylog(3, e ^x) - polylog(4, -e^x) + polylog(4, e^x)
\[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\int { x^{2} \operatorname {arcoth}\left (e^{x}\right ) \,d x } \]
Timed out. \[ \int x^2 \coth ^{-1}\left (e^x\right ) \, dx=\int x^2\,\mathrm {acoth}\left ({\mathrm {e}}^x\right ) \,d x \]