3.269 \(\int \frac{(a+b \coth ^{-1}(c x)) (d+e \log (1-c^2 x^2))}{x} \, dx\)

Optimal. Leaf size=381 \[ -\frac{1}{2} a e \text{PolyLog}\left (2,c^2 x^2\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b d \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b d \text{PolyLog}\left (2,\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,\frac{c+\frac{1}{x}}{c}\right )-b e \text{PolyLog}\left (3,1-\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,-\frac{1}{c x}\right )-b e \text{PolyLog}\left (3,\frac{1}{c x}\right )+b e \log \left (1-\frac{1}{c x}\right ) \text{PolyLog}\left (2,1-\frac{1}{c x}\right )-b e \log \left (\frac{c+\frac{1}{x}}{c}\right ) \text{PolyLog}\left (2,\frac{c+\frac{1}{x}}{c}\right )+a d \log (x)+\frac{1}{2} b e \log \left (\frac{1}{c x}\right ) \log ^2\left (1-\frac{1}{c x}\right )-\frac{1}{2} b e \log ^2\left (\frac{1}{c x}+1\right ) \log \left (-\frac{1}{c x}\right ) \]

[Out]

-(b*e*Log[1 + 1/(c*x)]^2*Log[-(1/(c*x))])/2 + (b*e*Log[1 - 1/(c*x)]^2*Log[1/(c*x)])/2 + a*d*Log[x] - b*e*Log[(
c + x^(-1))/c]*PolyLog[2, (c + x^(-1))/c] + b*e*Log[1 - 1/(c*x)]*PolyLog[2, 1 - 1/(c*x)] + (b*d*PolyLog[2, -(1
/(c*x))])/2 + (b*e*Log[-(c^2*x^2)]*PolyLog[2, -(1/(c*x))])/2 - (b*e*(Log[1 - 1/(c*x)] + Log[1 + 1/(c*x)] + Log
[-(c^2*x^2)] - Log[1 - c^2*x^2])*PolyLog[2, -(1/(c*x))])/2 - (b*d*PolyLog[2, 1/(c*x)])/2 - (b*e*Log[-(c^2*x^2)
]*PolyLog[2, 1/(c*x)])/2 + (b*e*(Log[1 - 1/(c*x)] + Log[1 + 1/(c*x)] + Log[-(c^2*x^2)] - Log[1 - c^2*x^2])*Pol
yLog[2, 1/(c*x)])/2 - (a*e*PolyLog[2, c^2*x^2])/2 + b*e*PolyLog[3, (c + x^(-1))/c] - b*e*PolyLog[3, 1 - 1/(c*x
)] + b*e*PolyLog[3, -(1/(c*x))] - b*e*PolyLog[3, 1/(c*x)]

________________________________________________________________________________________

Rubi [A]  time = 0.439495, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {6080, 5913, 6078, 2391, 6076, 2454, 2396, 2433, 2374, 6589, 6070} \[ -\frac{1}{2} a e \text{PolyLog}\left (2,c^2 x^2\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b d \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b d \text{PolyLog}\left (2,\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,\frac{c+\frac{1}{x}}{c}\right )-b e \text{PolyLog}\left (3,1-\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,-\frac{1}{c x}\right )-b e \text{PolyLog}\left (3,\frac{1}{c x}\right )+b e \log \left (1-\frac{1}{c x}\right ) \text{PolyLog}\left (2,1-\frac{1}{c x}\right )-b e \log \left (\frac{c+\frac{1}{x}}{c}\right ) \text{PolyLog}\left (2,\frac{c+\frac{1}{x}}{c}\right )+a d \log (x)+\frac{1}{2} b e \log \left (\frac{1}{c x}\right ) \log ^2\left (1-\frac{1}{c x}\right )-\frac{1}{2} b e \log ^2\left (\frac{1}{c x}+1\right ) \log \left (-\frac{1}{c x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x,x]

[Out]

-(b*e*Log[1 + 1/(c*x)]^2*Log[-(1/(c*x))])/2 + (b*e*Log[1 - 1/(c*x)]^2*Log[1/(c*x)])/2 + a*d*Log[x] - b*e*Log[(
c + x^(-1))/c]*PolyLog[2, (c + x^(-1))/c] + b*e*Log[1 - 1/(c*x)]*PolyLog[2, 1 - 1/(c*x)] + (b*d*PolyLog[2, -(1
/(c*x))])/2 + (b*e*Log[-(c^2*x^2)]*PolyLog[2, -(1/(c*x))])/2 - (b*e*(Log[1 - 1/(c*x)] + Log[1 + 1/(c*x)] + Log
[-(c^2*x^2)] - Log[1 - c^2*x^2])*PolyLog[2, -(1/(c*x))])/2 - (b*d*PolyLog[2, 1/(c*x)])/2 - (b*e*Log[-(c^2*x^2)
]*PolyLog[2, 1/(c*x)])/2 + (b*e*(Log[1 - 1/(c*x)] + Log[1 + 1/(c*x)] + Log[-(c^2*x^2)] - Log[1 - c^2*x^2])*Pol
yLog[2, 1/(c*x)])/2 - (a*e*PolyLog[2, c^2*x^2])/2 + b*e*PolyLog[3, (c + x^(-1))/c] - b*e*PolyLog[3, 1 - 1/(c*x
)] + b*e*PolyLog[3, -(1/(c*x))] - b*e*PolyLog[3, 1/(c*x)]

Rule 6080

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*(Log[(f_.) + (g_.)*(x_)^2]*(e_.) + (d_)))/(x_), x_Symbol] :> Dist[d,
Int[(a + b*ArcCoth[c*x])/x, x], x] + Dist[e, Int[(Log[f + g*x^2]*(a + b*ArcCoth[c*x]))/x, x], x] /; FreeQ[{a,
b, c, d, e, f, g}, x]

Rule 5913

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b*PolyLog[2, -(c*x)^(-1)
])/2, x] - Simp[(b*PolyLog[2, 1/(c*x)])/2, x]) /; FreeQ[{a, b, c}, x]

Rule 6078

Int[(Log[(f_.) + (g_.)*(x_)^2]*(ArcCoth[(c_.)*(x_)]*(b_.) + (a_)))/(x_), x_Symbol] :> Dist[a, Int[Log[f + g*x^
2]/x, x], x] + Dist[b, Int[(Log[f + g*x^2]*ArcCoth[c*x])/x, x], x] /; FreeQ[{a, b, c, f, g}, x]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6076

Int[(ArcCoth[(c_.)*(x_)]*Log[(f_.) + (g_.)*(x_)^2])/(x_), x_Symbol] :> Dist[Log[f + g*x^2] - Log[-(c^2*x^2)] -
Log[1 - 1/(c*x)] - Log[1 + 1/(c*x)], Int[ArcCoth[c*x]/x, x], x] + (-Dist[1/2, Int[Log[1 - 1/(c*x)]^2/x, x], x
] + Dist[1/2, Int[Log[1 + 1/(c*x)]^2/x, x], x] + Int[(Log[-(c^2*x^2)]*ArcCoth[c*x])/x, x]) /; FreeQ[{c, f, g},
x] && EqQ[c^2*f + g, 0]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2396

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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]

Rule 6070

Int[(ArcCoth[(c_.)*(x_)^(n_.)]*Log[(d_.)*(x_)^(m_.)])/(x_), x_Symbol] :> Dist[1/2, Int[(Log[d*x^m]*Log[1 + 1/(
c*x^n)])/x, x], x] - Dist[1/2, Int[(Log[d*x^m]*Log[1 - 1/(c*x^n)])/x, x], x] /; FreeQ[{c, d, m, n}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx &=d \int \frac{a+b \coth ^{-1}(c x)}{x} \, dx+e \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )+(a e) \int \frac{\log \left (1-c^2 x^2\right )}{x} \, dx+(b e) \int \frac{\coth ^{-1}(c x) \log \left (1-c^2 x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-\frac{1}{2} (b e) \int \frac{\log ^2\left (1-\frac{1}{c x}\right )}{x} \, dx+\frac{1}{2} (b e) \int \frac{\log ^2\left (1+\frac{1}{c x}\right )}{x} \, dx+(b e) \int \frac{\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x} \, dx+\left (b e \left (-\log \left (1-\frac{1}{c x}\right )-\log \left (1+\frac{1}{c x}\right )-\log \left (-c^2 x^2\right )+\log \left (1-c^2 x^2\right )\right )\right ) \int \frac{\coth ^{-1}(c x)}{x} \, dx\\ &=a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-\frac{1}{2} (b e) \int \frac{\log \left (1-\frac{1}{c x}\right ) \log \left (-c^2 x^2\right )}{x} \, dx+\frac{1}{2} (b e) \int \frac{\log \left (1+\frac{1}{c x}\right ) \log \left (-c^2 x^2\right )}{x} \, dx+\frac{1}{2} (b e) \operatorname{Subst}\left (\int \frac{\log ^2\left (1-\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} (b e) \operatorname{Subst}\left (\int \frac{\log ^2\left (1+\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-(b e) \int \frac{\text{Li}_2\left (-\frac{1}{c x}\right )}{x} \, dx+(b e) \int \frac{\text{Li}_2\left (\frac{1}{c x}\right )}{x} \, dx+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (\frac{x}{c}\right ) \log \left (1-\frac{x}{c}\right )}{1-\frac{x}{c}} \, dx,x,\frac{1}{x}\right )}{c}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{x}{c}\right ) \log \left (1+\frac{x}{c}\right )}{1+\frac{x}{c}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )+b e \text{Li}_3\left (-\frac{1}{c x}\right )-b e \text{Li}_3\left (\frac{1}{c x}\right )-(b e) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{c-c x}{c}\right )}{x} \, dx,x,1-\frac{1}{c x}\right )+(b e) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-c+c x}{c}\right )}{x} \, dx,x,1+\frac{1}{c x}\right )\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+b e \log \left (1-\frac{1}{c x}\right ) \text{Li}_2\left (1-\frac{1}{c x}\right )-b e \log \left (1+\frac{1}{c x}\right ) \text{Li}_2\left (1+\frac{1}{c x}\right )+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )+b e \text{Li}_3\left (-\frac{1}{c x}\right )-b e \text{Li}_3\left (\frac{1}{c x}\right )-(b e) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-\frac{1}{c x}\right )+(b e) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+\frac{1}{c x}\right )\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+b e \log \left (1-\frac{1}{c x}\right ) \text{Li}_2\left (1-\frac{1}{c x}\right )-b e \log \left (1+\frac{1}{c x}\right ) \text{Li}_2\left (1+\frac{1}{c x}\right )+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-b e \text{Li}_3\left (1-\frac{1}{c x}\right )+b e \text{Li}_3\left (1+\frac{1}{c x}\right )+b e \text{Li}_3\left (-\frac{1}{c x}\right )-b e \text{Li}_3\left (\frac{1}{c x}\right )\\ \end{align*}

Mathematica [F]  time = 0.212522, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x,x]

[Out]

Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x, x]

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Maple [C]  time = 1.104, size = 864, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1))/x,x)

[Out]

dilog(c*x)*a*e-1/2*dilog(c*x)*b*d+ln(c*x)*a*d+1/2*b*e*ln(-c*x)*ln(c*x+1)^2+b*e*ln(c*x+1)*polylog(2,c*x+1)-1/4*
I*Pi*ln(c*x)*ln(c*x-1)*b*e*csgn(I*(c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2-1/4*I*Pi*ln(c*x)*ln(c*x-1)*b*e*csgn(I*(c*
x+1))*csgn(I*(c*x-1)*(c*x+1))^2+1/4*I*Pi*dilog(c*x)*b*e*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1)
)+b*e*polylog(3,-c*x+1)-b*e*polylog(3,c*x+1)-(-1/2*I*Pi*b*e*csgn(I*(c*x-1)*(c*x+1))^2-1/4*I*Pi*b*e*csgn(I*(c*x
-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+1/4*I*Pi*b*e*csgn(I*(c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2+1/4*I*Pi*b
*e*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2+1/4*I*Pi*b*e*csgn(I*(c*x-1)*(c*x+1))^3+1/2*I*e*Pi*b+a*e+1/2*b*d)*
dilog(c*x+1)+ln(c*x)*ln(c*x-1)*a*e-1/2*ln(c*x)*ln(c*x-1)*b*d-1/2*ln(c*x)*ln(c*x-1)^2*b*e-ln(c*x-1)*polylog(2,-
c*x+1)*b*e-1/2*I*Pi*ln(c*x)*a*e*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))-1/4*I*Pi*ln(c*x)*ln(c*
x-1)*b*e*csgn(I*(c*x-1)*(c*x+1))^3+1/2*I*Pi*ln(c*x)*a*e*csgn(I*(c*x-1)*(c*x+1))^3-1/4*I*Pi*dilog(c*x)*b*e*csgn
(I*(c*x-1)*(c*x+1))^3+1/4*I*Pi*ln(c*x)*ln(c*x-1)*b*e*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+1
/2*I*Pi*ln(c*x)*ln(c*x-1)*b*e*csgn(I*(c*x-1)*(c*x+1))^2-1/4*I*Pi*dilog(c*x)*b*e*csgn(I*(c*x-1))*csgn(I*(c*x-1)
*(c*x+1))^2-1/4*I*Pi*dilog(c*x)*b*e*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2+1/2*I*Pi*ln(c*x)*a*e*csgn(I*(c*x
-1))*csgn(I*(c*x-1)*(c*x+1))^2+1/2*I*Pi*ln(c*x)*a*e*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2+I*Pi*ln(c*x)*a*e
-I*Pi*ln(c*x)*a*e*csgn(I*(c*x-1)*(c*x+1))^2+1/2*I*Pi*dilog(c*x)*b*e*csgn(I*(c*x-1)*(c*x+1))^2-1/2*I*dilog(c*x)
*Pi*b*e-1/2*I*ln(c*x)*Pi*ln(c*x-1)*b*e

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Maxima [C]  time = 1.77925, size = 225, normalized size = 0.59 \begin{align*} i \, \pi a e \log \left (x\right ) - \frac{1}{2} \,{\left (\log \left (c x - 1\right )^{2} \log \left (c x\right ) + 2 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (c x - 1\right ) - 2 \,{\rm Li}_{3}(-c x + 1)\right )} b e + \frac{1}{2} \,{\left (\log \left (c x + 1\right )^{2} \log \left (-c x\right ) + 2 \,{\rm Li}_2\left (c x + 1\right ) \log \left (c x + 1\right ) - 2 \,{\rm Li}_{3}(c x + 1)\right )} b e + a d \log \left (x\right ) - \frac{1}{2} \,{\left (i \, \pi b e + b d - 2 \, a e\right )}{\left (\log \left (c x - 1\right ) \log \left (c x\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} - \frac{1}{2} \,{\left (-i \, \pi b e - b d - 2 \, a e\right )}{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1))/x,x, algorithm="maxima")

[Out]

I*pi*a*e*log(x) - 1/2*(log(c*x - 1)^2*log(c*x) + 2*dilog(-c*x + 1)*log(c*x - 1) - 2*polylog(3, -c*x + 1))*b*e
+ 1/2*(log(c*x + 1)^2*log(-c*x) + 2*dilog(c*x + 1)*log(c*x + 1) - 2*polylog(3, c*x + 1))*b*e + a*d*log(x) - 1/
2*(I*pi*b*e + b*d - 2*a*e)*(log(c*x - 1)*log(c*x) + dilog(-c*x + 1)) - 1/2*(-I*pi*b*e - b*d - 2*a*e)*(log(c*x
+ 1)*log(-c*x) + dilog(c*x + 1))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b d \operatorname{arcoth}\left (c x\right ) + a d +{\left (b e \operatorname{arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1))/x,x, algorithm="fricas")

[Out]

integral((b*d*arccoth(c*x) + a*d + (b*e*arccoth(c*x) + a*e)*log(-c^2*x^2 + 1))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acoth}{\left (c x \right )}\right ) \left (d + e \log{\left (- c^{2} x^{2} + 1 \right )}\right )}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acoth(c*x))*(d+e*ln(-c**2*x**2+1))/x,x)

[Out]

Integral((a + b*acoth(c*x))*(d + e*log(-c**2*x**2 + 1))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1))/x,x, algorithm="giac")

[Out]

integrate((b*arccoth(c*x) + a)*(e*log(-c^2*x^2 + 1) + d)/x, x)