Integrand size = 13, antiderivative size = 229 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {1}{2} x^2 \coth ^{-1}(c+d \coth (a+b x))+\frac {1}{4} x^2 \log \left (1-\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{4} x^2 \log \left (1-\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac {x \operatorname {PolyLog}\left (2,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (2,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (3,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{8 b^2}+\frac {\operatorname {PolyLog}\left (3,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{8 b^2} \]
1/2*x^2*arccoth(c+d*coth(b*x+a))+1/4*x^2*ln(1-(1-c-d)*exp(2*b*x+2*a)/(1-c+ d))-1/4*x^2*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))+1/4*x*polylog(2,(1-c-d)*e xp(2*b*x+2*a)/(1-c+d))/b-1/4*x*polylog(2,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))/b -1/8*polylog(3,(1-c-d)*exp(2*b*x+2*a)/(1-c+d))/b^2+1/8*polylog(3,(1+c+d)*e xp(2*b*x+2*a)/(1+c-d))/b^2
Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.87 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {4 b^2 x^2 \coth ^{-1}(c+d \coth (a+b x))+2 b^2 x^2 \log \left (1+\frac {(1-c+d) e^{-2 (a+b x)}}{-1+c+d}\right )-2 b^2 x^2 \log \left (1+\frac {(-1-c+d) e^{-2 (a+b x)}}{1+c+d}\right )-2 b x \operatorname {PolyLog}\left (2,\frac {(-1+c-d) e^{-2 (a+b x)}}{-1+c+d}\right )+2 b x \operatorname {PolyLog}\left (2,\frac {(1+c-d) e^{-2 (a+b x)}}{1+c+d}\right )-\operatorname {PolyLog}\left (3,\frac {(-1+c-d) e^{-2 (a+b x)}}{-1+c+d}\right )+\operatorname {PolyLog}\left (3,\frac {(1+c-d) e^{-2 (a+b x)}}{1+c+d}\right )}{8 b^2} \]
(4*b^2*x^2*ArcCoth[c + d*Coth[a + b*x]] + 2*b^2*x^2*Log[1 + (1 - c + d)/(( -1 + c + d)*E^(2*(a + b*x)))] - 2*b^2*x^2*Log[1 + (-1 - c + d)/((1 + c + d )*E^(2*(a + b*x)))] - 2*b*x*PolyLog[2, (-1 + c - d)/((-1 + c + d)*E^(2*(a + b*x)))] + 2*b*x*PolyLog[2, (1 + c - d)/((1 + c + d)*E^(2*(a + b*x)))] - PolyLog[3, (-1 + c - d)/((-1 + c + d)*E^(2*(a + b*x)))] + PolyLog[3, (1 + c - d)/((1 + c + d)*E^(2*(a + b*x)))])/(8*b^2)
Time = 1.02 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6800, 2620, 3011, 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.
\(\displaystyle \int x \coth ^{-1}(d \coth (a+b x)+c) \, dx\) |
\(\Big \downarrow \) 6800 |
\(\displaystyle -\frac {1}{2} b (-c-d+1) \int \frac {e^{2 a+2 b x} x^2}{-c-(-c-d+1) e^{2 a+2 b x}+d+1}dx+\frac {1}{2} b (c+d+1) \int \frac {e^{2 a+2 b x} x^2}{c-(c+d+1) e^{2 a+2 b x}-d+1}dx+\frac {1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {1}{2} b (-c-d+1) \left (\frac {\int x \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )dx}{b (-c-d+1)}-\frac {x^2 \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b (-c-d+1)}\right )+\frac {1}{2} b (c+d+1) \left (\frac {\int x \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )dx}{b (c+d+1)}-\frac {x^2 \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b (c+d+1)}\right )+\frac {1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {1}{2} b (-c-d+1) \left (\frac {\frac {\int \operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )dx}{2 b}-\frac {x \operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b}}{b (-c-d+1)}-\frac {x^2 \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b (-c-d+1)}\right )+\frac {1}{2} b (c+d+1) \left (\frac {\frac {\int \operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )dx}{2 b}-\frac {x \operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b}}{b (c+d+1)}-\frac {x^2 \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b (c+d+1)}\right )+\frac {1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {1}{2} b (-c-d+1) \left (\frac {\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )de^{2 a+2 b x}}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b}}{b (-c-d+1)}-\frac {x^2 \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b (-c-d+1)}\right )+\frac {1}{2} b (c+d+1) \left (\frac {\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )de^{2 a+2 b x}}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b}}{b (c+d+1)}-\frac {x^2 \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b (c+d+1)}\right )+\frac {1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {1}{2} b (-c-d+1) \left (\frac {\frac {\operatorname {PolyLog}\left (3,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b}}{b (-c-d+1)}-\frac {x^2 \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{2 b (-c-d+1)}\right )+\frac {1}{2} b (c+d+1) \left (\frac {\frac {\operatorname {PolyLog}\left (3,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b}}{b (c+d+1)}-\frac {x^2 \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{2 b (c+d+1)}\right )+\frac {1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c)\) |
(x^2*ArcCoth[c + d*Coth[a + b*x]])/2 - (b*(1 - c - d)*(-1/2*(x^2*Log[1 - ( (1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(b*(1 - c - d)) + (-1/2*(x*Poly Log[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/b + PolyLog[3, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(4*b^2))/(b*(1 - c - d))))/2 + (b*(1 + c + d)*(-1/2*(x^2*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(b*( 1 + c + d)) + (-1/2*(x*PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d )])/b + PolyLog[3, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(4*b^2))/(b* (1 + c + d))))/2
3.3.18.3.1 Defintions of rubi rules used
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCoth[c + d*Coth[a + b*x]]/(f*( m + 1))), x] + (-Simp[b*((1 - c - d)/(f*(m + 1))) Int[(e + f*x)^(m + 1)*( E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x))), x], x] + Simp[b *((1 + c + d)/(f*(m + 1))) Int[(e + f*x)^(m + 1)*(E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x ] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.95 (sec) , antiderivative size = 4881, normalized size of antiderivative = 21.31
1/2/b^2*a/(c+d-1)*dilog((exp(b*x+a)*c+exp(b*x+a)*d+((c-d-1)*(c+d-1))^(1/2) -exp(b*x+a))/((c-d-1)*(c+d-1))^(1/2))-1/4/b^2/(c+d-1)*ln(1-(c+d-1)*exp(2*b *x+2*a)/(c-d-1))*a^2-1/4/b/(c+d-1)*polylog(2,(c+d-1)*exp(2*b*x+2*a)/(c-d-1 ))*x-1/4/b^2/(c+d-1)*polylog(2,(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a-1/8/b^2*d /(c+d-1)*polylog(3,(c+d-1)*exp(2*b*x+2*a)/(c-d-1))-1/8/b^2*c/(c+d-1)*polyl og(3,(c+d-1)*exp(2*b*x+2*a)/(c-d-1))-1/4/b^2*a^2/(c+d-1)*ln(c*exp(2*b*x+2* a)+d*exp(2*b*x+2*a)-exp(2*b*x+2*a)-c+d+1)+1/4*c/(c+d-1)*ln(1-(c+d-1)*exp(2 *b*x+2*a)/(c-d-1))*x^2+1/4*d/(c+d-1)*ln(1-(c+d-1)*exp(2*b*x+2*a)/(c-d-1))* x^2+1/2/b^2*a/(c+d-1)*dilog((-exp(b*x+a)*c-exp(b*x+a)*d+((c-d-1)*(c+d-1))^ (1/2)+exp(b*x+a))/((c-d-1)*(c+d-1))^(1/2))+1/2/b^2*a/(1+c+d)*dilog((-exp(b *x+a)*c-exp(b*x+a)*d+((1+c-d)*(1+c+d))^(1/2)-exp(b*x+a))/((1+c-d)*(1+c+d)) ^(1/2))+1/2/b^2*a/(1+c+d)*dilog((exp(b*x+a)*c+exp(b*x+a)*d+((1+c-d)*(1+c+d ))^(1/2)+exp(b*x+a))/((1+c-d)*(1+c+d))^(1/2))-1/4/b^2*a^2/(1+c+d)*ln(c*exp (2*b*x+2*a)+d*exp(2*b*x+2*a)+exp(2*b*x+2*a)-c+d-1)-1/4/b^2/(1+c+d)*ln(1-(1 +c+d)*exp(2*b*x+2*a)/(1+c-d))*a^2-1/4/b/(1+c+d)*polylog(2,(1+c+d)*exp(2*b* x+2*a)/(1+c-d))*x+1/8/b^2*c/(1+c+d)*polylog(3,(1+c+d)*exp(2*b*x+2*a)/(1+c- d))+1/8/b^2*d/(1+c+d)*polylog(3,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))-1/4*d/(1+c +d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*x^2-1/4*c/(1+c+d)*ln(1-(1+c+d)*ex p(2*b*x+2*a)/(1+c-d))*x^2-1/4/b^2/(1+c+d)*polylog(2,(1+c+d)*exp(2*b*x+2*a) /(1+c-d))*a+1/8*I*Pi*(csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*((exp(2*b*x+2*a...
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (195) = 390\).
Time = 0.30 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.18 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {b^{2} x^{2} \log \left (\frac {d \cosh \left (b x + a\right ) + {\left (c + 1\right )} \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + {\left (c - 1\right )} \sinh \left (b x + a\right )}\right ) - 2 \, b x {\rm Li}_2\left (\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, b x {\rm Li}_2\left (-\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 \, b x {\rm Li}_2\left (\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 \, b x {\rm Li}_2\left (-\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - a^{2} \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d + 1\right )} \sqrt {\frac {c + d + 1}{c - d + 1}}\right ) - a^{2} \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d + 1\right )} \sqrt {\frac {c + d + 1}{c - d + 1}}\right ) + a^{2} \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d - 1\right )} \sqrt {\frac {c + d - 1}{c - d - 1}}\right ) + a^{2} \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d - 1\right )} \sqrt {\frac {c + d - 1}{c - d - 1}}\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \log \left (\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \log \left (\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + 2 \, {\rm polylog}\left (3, \sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 \, {\rm polylog}\left (3, -\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\rm polylog}\left (3, \sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\rm polylog}\left (3, -\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{4 \, b^{2}} \]
1/4*(b^2*x^2*log((d*cosh(b*x + a) + (c + 1)*sinh(b*x + a))/(d*cosh(b*x + a ) + (c - 1)*sinh(b*x + a))) - 2*b*x*dilog(sqrt((c + d + 1)/(c - d + 1))*(c osh(b*x + a) + sinh(b*x + a))) - 2*b*x*dilog(-sqrt((c + d + 1)/(c - d + 1) )*(cosh(b*x + a) + sinh(b*x + a))) + 2*b*x*dilog(sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))) + 2*b*x*dilog(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))) - a^2*log(2*(c + d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) + 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) - a^2*log(2*(c + d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) - 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) + a^2*log(2*(c + d - 1)* cosh(b*x + a) + 2*(c + d - 1)*sinh(b*x + a) + 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1))) + a^2*log(2*(c + d - 1)*cosh(b*x + a) + 2*(c + d - 1)*sin h(b*x + a) - 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1))) - (b^2*x^2 - a^2 )*log(sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) - (b^2*x^2 - a^2)*log(-sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh( b*x + a)) + 1) + (b^2*x^2 - a^2)*log(sqrt((c + d - 1)/(c - d - 1))*(cosh(b *x + a) + sinh(b*x + a)) + 1) + (b^2*x^2 - a^2)*log(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) + 2*polylog(3, sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a))) + 2*polylog(3, -sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a))) - 2*polylog(3, sqrt ((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))) - 2*polylog(...
\[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\int x \operatorname {acoth}{\left (c + d \coth {\left (a + b x \right )} \right )}\, dx \]
Time = 0.46 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.93 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=-\frac {1}{8} \, b d {\left (\frac {2 \, b^{2} x^{2} \log \left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + 2 \, b x {\rm Li}_2\left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right ) - {\rm Li}_{3}(\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1})}{b^{3} d} - \frac {2 \, b^{2} x^{2} \log \left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + 2 \, b x {\rm Li}_2\left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right ) - {\rm Li}_{3}(\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1})}{b^{3} d}\right )} + \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + c\right ) \]
-1/8*b*d*((2*b^2*x^2*log(-(c + d + 1)*e^(2*b*x + 2*a)/(c - d + 1) + 1) + 2 *b*x*dilog((c + d + 1)*e^(2*b*x + 2*a)/(c - d + 1)) - polylog(3, (c + d + 1)*e^(2*b*x + 2*a)/(c - d + 1)))/(b^3*d) - (2*b^2*x^2*log(-(c + d - 1)*e^( 2*b*x + 2*a)/(c - d - 1) + 1) + 2*b*x*dilog((c + d - 1)*e^(2*b*x + 2*a)/(c - d - 1)) - polylog(3, (c + d - 1)*e^(2*b*x + 2*a)/(c - d - 1)))/(b^3*d)) + 1/2*x^2*arccoth(d*coth(b*x + a) + c)
\[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\int { x \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + c\right ) \,d x } \]
Timed out. \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\int x\,\mathrm {acoth}\left (c+d\,\mathrm {coth}\left (a+b\,x\right )\right ) \,d x \]