\(\int x \coth ^{-1}(c+d \coth (a+b x)) \, dx\) [218]

Optimal result

Integrand size = 13, antiderivative size = 229 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {1}{2} x^2 \coth ^{-1}(c+d \coth (a+b x))+\frac {1}{4} x^2 \log \left (1-\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{4} x^2 \log \left (1-\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac {x \operatorname {PolyLog}\left (2,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (2,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (3,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{8 b^2}+\frac {\operatorname {PolyLog}\left (3,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{8 b^2} \]

[Out]

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)*exp(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)*exp(2*b*x+2*a)/(1+c-d))/b^2

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6381, 2221, 2611, 2320, 6724} \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=-\frac {\operatorname {PolyLog}\left (3,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{8 b^2}+\frac {\operatorname {PolyLog}\left (3,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{8 b^2}+\frac {x \operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac {x \operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac {1}{4} x^2 \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )-\frac {1}{4} x^2 \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )+\frac {1}{2} x^2 \coth ^{-1}(d \coth (a+b x)+c) \]

[In]

Int[x*ArcCoth[c + d*Coth[a + b*x]],x]

[Out]

(x^2*ArcCoth[c + d*Coth[a + b*x]])/2 + (x^2*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/4 - (x^2*Log[1
- ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/4 + (x*PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(
4*b) - (x*PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b) - PolyLog[3, ((1 - c - d)*E^(2*a + 2*b*
x))/(1 - c + d)]/(8*b^2) + PolyLog[3, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(8*b^2)

Rule 2221

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]
- Dist[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]

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 6381

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] + (-Dist[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] + Dist[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]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \coth ^{-1}(c+d \coth (a+b x))-\frac {1}{2} (b (1-c-d)) \int \frac {e^{2 a+2 b x} x^2}{1-c+d+(-1+c+d) e^{2 a+2 b x}} \, dx+\frac {1}{2} (b (1+c+d)) \int \frac {e^{2 a+2 b x} x^2}{1+c-d+(-1-c-d) e^{2 a+2 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 {1}{2} \int x \log \left (1+\frac {(-1-c-d) e^{2 a+2 b x}}{1+c-d}\right ) \, dx-\frac {1}{2} \int x \log \left (1+\frac {(-1+c+d) e^{2 a+2 b x}}{1-c+d}\right ) \, 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 {\int \operatorname {PolyLog}\left (2,-\frac {(-1-c-d) e^{2 a+2 b x}}{1+c-d}\right ) \, dx}{4 b}-\frac {\int \operatorname {PolyLog}\left (2,-\frac {(-1+c+d) e^{2 a+2 b x}}{1-c+d}\right ) \, dx}{4 b} \\ & = \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 {\text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {(-1+c+d) x}{-1+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^2}+\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {(1+c+d) x}{1+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.87 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {4 b^2 x^2 \coth ^{-1}(c+d \coth (a+b x))+2 b^2 x^2 \log \left (1+\frac {(1-c+d) e^{-2 (a+b x)}}{-1+c+d}\right )-2 b^2 x^2 \log \left (1+\frac {(-1-c+d) e^{-2 (a+b x)}}{1+c+d}\right )-2 b x \operatorname {PolyLog}\left (2,\frac {(-1+c-d) e^{-2 (a+b x)}}{-1+c+d}\right )+2 b x \operatorname {PolyLog}\left (2,\frac {(1+c-d) e^{-2 (a+b x)}}{1+c+d}\right )-\operatorname {PolyLog}\left (3,\frac {(-1+c-d) e^{-2 (a+b x)}}{-1+c+d}\right )+\operatorname {PolyLog}\left (3,\frac {(1+c-d) e^{-2 (a+b x)}}{1+c+d}\right )}{8 b^2} \]

[In]

Integrate[x*ArcCoth[c + d*Coth[a + b*x]],x]

[Out]

(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)

Maple [C] (warning: unable to verify)

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

method	result	size
risch	\(\text {Expression too large to display}\)	\(4881\)

[In]

int(x*arccoth(c+d*coth(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

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)*polylog(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)*poly
log(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)*exp(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*(cs
gn(I/(exp(2*b*x+2*a)-1))*csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d-exp(2*b*x+2*a)+1))*csgn(I*((exp(2*b
*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d-exp(2*b*x+2*a)+1)/(exp(2*b*x+2*a)-1))-csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*((ex
p(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d+exp(2*b*x+2*a)-1))*csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d+ex
p(2*b*x+2*a)-1)/(exp(2*b*x+2*a)-1))-csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)
*d-exp(2*b*x+2*a)+1)/(exp(2*b*x+2*a)-1))^2+csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+
2*a)+1)*d+exp(2*b*x+2*a)-1)/(exp(2*b*x+2*a)-1))^2-csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d-exp(2*b*x+
2*a)+1))*csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d-exp(2*b*x+2*a)+1)/(exp(2*b*x+2*a)-1))^2+csgn(I*((ex
p(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d+exp(2*b*x+2*a)-1))*csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d+ex
p(2*b*x+2*a)-1)/(exp(2*b*x+2*a)-1))^2+csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d-exp(2*b*x+2*a)+1)/(exp
(2*b*x+2*a)-1))^3-csgn(I*((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d+exp(2*b*x+2*a)-1)/(exp(2*b*x+2*a)-1))^3)*x
^2-1/2/b^2*a*d/(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*c/(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*d/(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/(c+d-1)*ln(1-(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*x^2+1/8/b^2/(c+d-1)*polylog(3,(c+d-1)*exp(2*b
*x+2*a)/(c-d-1))-1/2/b*a*d/(c+d-1)*x*ln((-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*a*d/(c+d-1)*x*ln((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*c/(c+d-1)*ln(1-(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a*x+1/2/b*d/(c+d-1)*ln(1-(c+d-1)*exp(
2*b*x+2*a)/(c-d-1))*a*x+1/2/b^2*a*d/(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*d/(1+c+d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a^2-1/4/b*d/(1+c+d)*polylog
(2,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*x-1/4/b^2*d/(1+c+d)*polylog(2,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a-1/2/b/(1+c+
d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a*x-1/4/b^2*a^2*c/(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*a^2*d/(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*c/(1+c
+d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a^2-1/4/b*c/(1+c+d)*polylog(2,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*x-1/4/b
^2*c/(1+c+d)*polylog(2,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a+1/2/b*a/(1+c+d)*x*ln((-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*a/(1+c+d)*x*ln((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/(c+d-1)*ln(1-(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a*x
+1/2/b*a/(c+d-1)*x*ln((-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*a/(c+d-1)*x*ln((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*c/(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/(c+d-1)*ln(1-(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a^2+1/4/b*c/(c+d-1)*polylog(2,(c+d-1)*exp(2*b*x+2*a
)/(c-d-1))*x+1/4/b^2*c/(c+d-1)*polylog(2,(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a+1/4/b^2*a^2*c/(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/b^2*a^2*d/(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/b^2*d/(c+d-1)*ln(1-(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a^2+1/4/b*d/(c+d-1)*polylog(2,(c+d-1)*
exp(2*b*x+2*a)/(c-d-1))*x+1/4/b^2*d/(c+d-1)*polylog(2,(c+d-1)*exp(2*b*x+2*a)/(c-d-1))*a-1/2/b^2*a*c/(c+d-1)*di
log((-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*d/(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*c/
(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*a
*c/(c+d-1)*x*ln((-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
*a*c/(c+d-1)*x*ln((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/
(1+c+d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*x^2+1/8/b^2/(1+c+d)*polylog(3,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))-1/2
/b*c/(1+c+d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a*x-1/2/b*d/(1+c+d)*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))*a*x
+1/2/b*a*c/(1+c+d)*x*ln((-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*a*c/(1+c+d)*x*ln((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*a*d/(1+c+d)*x*ln((-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*a*d/(1+c+d)*x*ln((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^2/(1+c+d)*ln((-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^2/(1+c+d)*ln((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^2/(c+d-1)*ln((-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^2/(c+d-1)*ln((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*x^2*ln((exp(2*b*x+2*a)-1)*c+(exp(2*b*x+2*a)+1)*d+exp(2*b*x+2*a)-1)-1/4*x^2*ln((exp(2*b*x+2*a)
-1)*c+(exp(2*b*x+2*a)+1)*d-exp(2*b*x+2*a)+1)+1/2/b^2*a^2*c/(1+c+d)*ln((-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^2*c/(1+c+d)*ln((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^2*d/(1+c+d)*ln((-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^2*d/(1+c+d)*ln((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^2*c/(c+d-1)*ln((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^2*d/(c+d-1)*ln((-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^2*d/(c+d-1)*ln((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^2*c/(c+d-1)*ln((-exp(b*x+a)*c-e
xp(b*x+a)*d+((c-d-1)*(c+d-1))^(1/2)+exp(b*x+a))/((c-d-1)*(c+d-1))^(1/2))

Fricas [B] (verification not implemented)

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}} \]

[In]

integrate(x*arccoth(c+d*coth(b*x+a)),x, algorithm="fricas")

[Out]

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))*(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
))) + 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)*sinh(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(3, -sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a
))))/b^2

Sympy [F]

\[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\int x \operatorname {acoth}{\left (c + d \coth {\left (a + b x \right )} \right )}\, dx \]

[In]

integrate(x*acoth(c+d*coth(b*x+a)),x)

[Out]

Integral(x*acoth(c + d*coth(a + b*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.93 \[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=-\frac {1}{8} \, b d {\left (\frac {2 \, b^{2} x^{2} \log \left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + 2 \, b x {\rm Li}_2\left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right ) - {\rm Li}_{3}(\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1})}{b^{3} d} - \frac {2 \, b^{2} x^{2} \log \left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + 2 \, b x {\rm Li}_2\left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right ) - {\rm Li}_{3}(\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1})}{b^{3} d}\right )} + \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + c\right ) \]

[In]

integrate(x*arccoth(c+d*coth(b*x+a)),x, algorithm="maxima")

[Out]

-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)

Giac [F]

\[ \int x \coth ^{-1}(c+d \coth (a+b x)) \, dx=\int { x \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + c\right ) \,d x } \]

[In]

integrate(x*arccoth(c+d*coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(x*arccoth(d*coth(b*x + a) + c), x)

Mupad [F(-1)]

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

[In]

int(x*acoth(c + d*coth(a + b*x)),x)

[Out]

int(x*acoth(c + d*coth(a + b*x)), x)