3.176 \(\int \frac{(g+h \log (1-c x)) \text{PolyLog}(2,c x)}{x^4} \, dx\)

Optimal. Leaf size=340 \[ -\frac{2}{9} c^3 h \text{PolyLog}\left (2,\frac{1}{1-c x}\right )-\frac{1}{3} c^3 h \text{PolyLog}(3,c x)-\frac{2}{3} c^3 h \text{PolyLog}(3,1-c x)+\frac{c^2 h \text{PolyLog}(2,c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{PolyLog}(2,c x)+\frac{2}{3} c^3 h \log (1-c x) \text{PolyLog}(2,1-c x)-\frac{\text{PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}+\frac{c h \text{PolyLog}(2,c x)}{6 x^2}+\frac{1}{9} c^3 \log \left (1-\frac{1}{1-c x}\right ) (2 h \log (1-c x)+g)-\frac{c^2 (1-c x) (2 h \log (1-c x)+g)}{9 x}+\frac{7 c^2 h}{36 x}+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac{3}{4} c^3 h \log (x)+\frac{19}{36} c^3 h \log (1-c x)-\frac{c^2 h \log (1-c x)}{3 x}-\frac{c (2 h \log (1-c x)+g)}{18 x^2}+\frac{\log (1-c x) (h \log (1-c x)+g)}{9 x^3}-\frac{c h \log (1-c x)}{12 x^2} \]

[Out]

(7*c^2*h)/(36*x) - (3*c^3*h*Log[x])/4 + (19*c^3*h*Log[1 - c*x])/36 - (c*h*Log[1 - c*x])/(12*x^2) - (c^2*h*Log[
1 - c*x])/(3*x) + (c^3*h*Log[c*x]*Log[1 - c*x]^2)/3 + (Log[1 - c*x]*(g + h*Log[1 - c*x]))/(9*x^3) - (c*(g + 2*
h*Log[1 - c*x]))/(18*x^2) - (c^2*(1 - c*x)*(g + 2*h*Log[1 - c*x]))/(9*x) + (c^3*(g + 2*h*Log[1 - c*x])*Log[1 -
(1 - c*x)^(-1)])/9 + (c*h*PolyLog[2, c*x])/(6*x^2) + (c^2*h*PolyLog[2, c*x])/(3*x) + (c^3*h*Log[1 - c*x]*Poly
Log[2, c*x])/3 - ((g + h*Log[1 - c*x])*PolyLog[2, c*x])/(3*x^3) - (2*c^3*h*PolyLog[2, (1 - c*x)^(-1)])/9 + (2*
c^3*h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/3 - (c^3*h*PolyLog[3, c*x])/3 - (2*c^3*h*PolyLog[3, 1 - c*x])/3

________________________________________________________________________________________

Rubi [A]  time = 0.650804, antiderivative size = 351, normalized size of antiderivative = 1.03, number of steps used = 42, number of rules used = 24, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.2, Rules used = {6603, 2439, 2410, 2395, 44, 36, 29, 31, 2391, 2390, 2301, 2411, 2347, 2344, 2316, 2315, 2314, 2319, 6591, 6589, 6596, 2396, 2433, 2374} \[ -\frac{2}{9} c^3 h \text{PolyLog}(2,c x)-\frac{1}{3} c^3 h \text{PolyLog}(3,c x)-\frac{2}{3} c^3 h \text{PolyLog}(3,1-c x)+\frac{c^2 h \text{PolyLog}(2,c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{PolyLog}(2,c x)+\frac{2}{3} c^3 h \log (1-c x) \text{PolyLog}(2,1-c x)-\frac{\text{PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}+\frac{c h \text{PolyLog}(2,c x)}{6 x^2}-\frac{c^3 (h \log (1-c x)+g)^2}{18 h}-\frac{c^2 (1-c x) (h \log (1-c x)+g)}{9 x}+\frac{1}{9} c^3 g \log (x)+\frac{7 c^2 h}{36 x}-\frac{1}{18} c^3 h \log ^2(1-c x)+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac{3}{4} c^3 h \log (x)+\frac{23}{36} c^3 h \log (1-c x)-\frac{4 c^2 h \log (1-c x)}{9 x}-\frac{c (h \log (1-c x)+g)}{18 x^2}+\frac{\log (1-c x) (h \log (1-c x)+g)}{9 x^3}-\frac{5 c h \log (1-c x)}{36 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((g + h*Log[1 - c*x])*PolyLog[2, c*x])/x^4,x]

[Out]

(7*c^2*h)/(36*x) + (c^3*g*Log[x])/9 - (3*c^3*h*Log[x])/4 + (23*c^3*h*Log[1 - c*x])/36 - (5*c*h*Log[1 - c*x])/(
36*x^2) - (4*c^2*h*Log[1 - c*x])/(9*x) - (c^3*h*Log[1 - c*x]^2)/18 + (c^3*h*Log[c*x]*Log[1 - c*x]^2)/3 - (c*(g
+ h*Log[1 - c*x]))/(18*x^2) - (c^2*(1 - c*x)*(g + h*Log[1 - c*x]))/(9*x) + (Log[1 - c*x]*(g + h*Log[1 - c*x])
)/(9*x^3) - (c^3*(g + h*Log[1 - c*x])^2)/(18*h) - (2*c^3*h*PolyLog[2, c*x])/9 + (c*h*PolyLog[2, c*x])/(6*x^2)
+ (c^2*h*PolyLog[2, c*x])/(3*x) + (c^3*h*Log[1 - c*x]*PolyLog[2, c*x])/3 - ((g + h*Log[1 - c*x])*PolyLog[2, c*
x])/(3*x^3) + (2*c^3*h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/3 - (c^3*h*PolyLog[3, c*x])/3 - (2*c^3*h*PolyLog[3, 1
- c*x])/3

Rule 6603

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(x_)^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x
_Symbol] :> Simp[(x^(m + 1)*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(m + 1), x] + (Dist[b/(m + 1),
Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], x^(m + 1)/(a + b*x), x], x], x] - Dist[(
e*h*n)/(m + 1), Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], x^(m + 1)/(d + e*x), x], x], x]) /; FreeQ[{a, b,
c, d, e, f, g, h, n}, x] && IntegerQ[m] && NeQ[m, -1]

Rule 2439

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

Rule 2410

Int[(Log[(c_.)*((d_) + (e_.)*(x_))]*(x_)^(m_.))/((f_) + (g_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[Log[c*(d
+ e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m
]

Rule 2395

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 2390

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

Rule 2301

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

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 6596

Int[PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 - a*c - b*c*x]*PolyL
og[2, c*(a + b*x)])/e, x] + Dist[b/e, Int[Log[1 - a*c - b*c*x]^2/(a + b*x), x], x] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c*(b*d - a*e) + e, 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]

Rubi steps

\begin{align*} \int \frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{x^4} \, dx &=-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}-\frac{1}{3} \int \frac{\log (1-c x) (g+h \log (1-c x))}{x^4} \, dx-\frac{1}{3} (c h) \int \left (\frac{\text{Li}_2(c x)}{x^3}+\frac{c \text{Li}_2(c x)}{x^2}+\frac{c^2 \text{Li}_2(c x)}{x}-\frac{c^3 \text{Li}_2(c x)}{-1+c x}\right ) \, dx\\ &=\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}+\frac{1}{9} c \int \frac{g+h \log (1-c x)}{x^3 (1-c x)} \, dx+\frac{1}{9} (c h) \int \frac{\log (1-c x)}{x^3 (1-c x)} \, dx-\frac{1}{3} (c h) \int \frac{\text{Li}_2(c x)}{x^3} \, dx-\frac{1}{3} \left (c^2 h\right ) \int \frac{\text{Li}_2(c x)}{x^2} \, dx-\frac{1}{3} \left (c^3 h\right ) \int \frac{\text{Li}_2(c x)}{x} \, dx+\frac{1}{3} \left (c^4 h\right ) \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx\\ &=\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}+\frac{c h \text{Li}_2(c x)}{6 x^2}+\frac{c^2 h \text{Li}_2(c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{Li}_2(c x)-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}-\frac{1}{3} c^3 h \text{Li}_3(c x)-\frac{1}{9} \operatorname{Subst}\left (\int \frac{g+h \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-c x\right )+\frac{1}{9} (c h) \int \left (\frac{\log (1-c x)}{x^3}+\frac{c \log (1-c x)}{x^2}+\frac{c^2 \log (1-c x)}{x}-\frac{c^3 \log (1-c x)}{-1+c x}\right ) \, dx+\frac{1}{6} (c h) \int \frac{\log (1-c x)}{x^3} \, dx+\frac{1}{3} \left (c^2 h\right ) \int \frac{\log (1-c x)}{x^2} \, dx+\frac{1}{3} \left (c^3 h\right ) \int \frac{\log ^2(1-c x)}{x} \, dx\\ &=-\frac{c h \log (1-c x)}{12 x^2}-\frac{c^2 h \log (1-c x)}{3 x}+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)+\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}+\frac{c h \text{Li}_2(c x)}{6 x^2}+\frac{c^2 h \text{Li}_2(c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{Li}_2(c x)-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}-\frac{1}{3} c^3 h \text{Li}_3(c x)-\frac{1}{9} \operatorname{Subst}\left (\int \frac{g+h \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-c x\right )-\frac{1}{9} c \operatorname{Subst}\left (\int \frac{g+h \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x\right )+\frac{1}{9} (c h) \int \frac{\log (1-c x)}{x^3} \, dx-\frac{1}{12} \left (c^2 h\right ) \int \frac{1}{x^2 (1-c x)} \, dx+\frac{1}{9} \left (c^2 h\right ) \int \frac{\log (1-c x)}{x^2} \, dx+\frac{1}{9} \left (c^3 h\right ) \int \frac{\log (1-c x)}{x} \, dx-\frac{1}{3} \left (c^3 h\right ) \int \frac{1}{x (1-c x)} \, dx-\frac{1}{9} \left (c^4 h\right ) \int \frac{\log (1-c x)}{-1+c x} \, dx+\frac{1}{3} \left (2 c^4 h\right ) \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx\\ &=-\frac{5 c h \log (1-c x)}{36 x^2}-\frac{4 c^2 h \log (1-c x)}{9 x}+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac{c (g+h \log (1-c x))}{18 x^2}+\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac{1}{9} c^3 h \text{Li}_2(c x)+\frac{c h \text{Li}_2(c x)}{6 x^2}+\frac{c^2 h \text{Li}_2(c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{Li}_2(c x)-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}-\frac{1}{3} c^3 h \text{Li}_3(c x)-\frac{1}{9} c \operatorname{Subst}\left (\int \frac{g+h \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x\right )-\frac{1}{9} c^2 \operatorname{Subst}\left (\int \frac{g+h \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-c x\right )+\frac{1}{18} (c h) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x\right )-\frac{1}{18} \left (c^2 h\right ) \int \frac{1}{x^2 (1-c x)} \, dx-\frac{1}{12} \left (c^2 h\right ) \int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx-\frac{1}{9} \left (c^3 h\right ) \int \frac{1}{x (1-c x)} \, dx-\frac{1}{9} \left (c^3 h\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )-\frac{1}{3} \left (c^3 h\right ) \int \frac{1}{x} \, dx-\frac{1}{3} \left (2 c^3 h\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (c \left (\frac{1}{c}-\frac{x}{c}\right )\right )}{x} \, dx,x,1-c x\right )-\frac{1}{3} \left (c^4 h\right ) \int \frac{1}{1-c x} \, dx\\ &=\frac{c^2 h}{12 x}-\frac{5}{12} c^3 h \log (x)+\frac{5}{12} c^3 h \log (1-c x)-\frac{5 c h \log (1-c x)}{36 x^2}-\frac{4 c^2 h \log (1-c x)}{9 x}-\frac{1}{18} c^3 h \log ^2(1-c x)+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac{c (g+h \log (1-c x))}{18 x^2}-\frac{c^2 (1-c x) (g+h \log (1-c x))}{9 x}+\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac{1}{9} c^3 h \text{Li}_2(c x)+\frac{c h \text{Li}_2(c x)}{6 x^2}+\frac{c^2 h \text{Li}_2(c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{Li}_2(c x)-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}+\frac{2}{3} c^3 h \log (1-c x) \text{Li}_2(1-c x)-\frac{1}{3} c^3 h \text{Li}_3(c x)-\frac{1}{9} c^2 \operatorname{Subst}\left (\int \frac{g+h \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x\right )-\frac{1}{9} c^3 \operatorname{Subst}\left (\int \frac{g+h \log (x)}{x} \, dx,x,1-c x\right )+\frac{1}{18} (c h) \operatorname{Subst}\left (\int \left (\frac{c^2}{(-1+x)^2}-\frac{c^2}{-1+x}+\frac{c^2}{x}\right ) \, dx,x,1-c x\right )-\frac{1}{18} \left (c^2 h\right ) \int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx+\frac{1}{9} \left (c^2 h\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x\right )-\frac{1}{9} \left (c^3 h\right ) \int \frac{1}{x} \, dx-\frac{1}{3} \left (2 c^3 h\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )-\frac{1}{9} \left (c^4 h\right ) \int \frac{1}{1-c x} \, dx\\ &=\frac{7 c^2 h}{36 x}+\frac{1}{9} c^3 g \log (x)-\frac{3}{4} c^3 h \log (x)+\frac{23}{36} c^3 h \log (1-c x)-\frac{5 c h \log (1-c x)}{36 x^2}-\frac{4 c^2 h \log (1-c x)}{9 x}-\frac{1}{18} c^3 h \log ^2(1-c x)+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac{c (g+h \log (1-c x))}{18 x^2}-\frac{c^2 (1-c x) (g+h \log (1-c x))}{9 x}+\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac{c^3 (g+h \log (1-c x))^2}{18 h}-\frac{1}{9} c^3 h \text{Li}_2(c x)+\frac{c h \text{Li}_2(c x)}{6 x^2}+\frac{c^2 h \text{Li}_2(c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{Li}_2(c x)-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}+\frac{2}{3} c^3 h \log (1-c x) \text{Li}_2(1-c x)-\frac{1}{3} c^3 h \text{Li}_3(c x)-\frac{2}{3} c^3 h \text{Li}_3(1-c x)-\frac{1}{9} \left (c^2 h\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x\right )\\ &=\frac{7 c^2 h}{36 x}+\frac{1}{9} c^3 g \log (x)-\frac{3}{4} c^3 h \log (x)+\frac{23}{36} c^3 h \log (1-c x)-\frac{5 c h \log (1-c x)}{36 x^2}-\frac{4 c^2 h \log (1-c x)}{9 x}-\frac{1}{18} c^3 h \log ^2(1-c x)+\frac{1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac{c (g+h \log (1-c x))}{18 x^2}-\frac{c^2 (1-c x) (g+h \log (1-c x))}{9 x}+\frac{\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac{c^3 (g+h \log (1-c x))^2}{18 h}-\frac{2}{9} c^3 h \text{Li}_2(c x)+\frac{c h \text{Li}_2(c x)}{6 x^2}+\frac{c^2 h \text{Li}_2(c x)}{3 x}+\frac{1}{3} c^3 h \log (1-c x) \text{Li}_2(c x)-\frac{(g+h \log (1-c x)) \text{Li}_2(c x)}{3 x^3}+\frac{2}{3} c^3 h \log (1-c x) \text{Li}_2(1-c x)-\frac{1}{3} c^3 h \text{Li}_3(c x)-\frac{2}{3} c^3 h \text{Li}_3(1-c x)\\ \end{align*}

Mathematica [A]  time = 0.200583, size = 301, normalized size = 0.89 \[ \frac{h \left (-12 c^3 x^3 \text{PolyLog}(3,c x)-24 c^3 x^3 \text{PolyLog}(3,1-c x)+8 c^3 x^3 (3 \log (1-c x)+1) \text{PolyLog}(2,1-c x)+6 \left (2 \left (c^3 x^3-1\right ) \log (1-c x)+c x (2 c x+1)\right ) \text{PolyLog}(2,c x)-4 c^3 x^3+7 c^2 x^2-4 c^3 x^3 \log ^2(1-c x)+12 c^3 x^3 \log (c x) \log ^2(1-c x)-15 c^3 x^3 \log (x)-12 c^3 x^3 \log (c x)+27 c^3 x^3 \log (1-c x)+8 c^3 x^3 \log (c x) \log (1-c x)-20 c^2 x^2 \log (1-c x)+4 \log ^2(1-c x)-7 c x \log (1-c x)\right )}{36 x^3}-\frac{g \left (6 \text{PolyLog}(2,c x)-2 c^3 x^3 \log (x)+2 \left (c^3 x^3-1\right ) \log (1-c x)+c x (2 c x+1)\right )}{18 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((g + h*Log[1 - c*x])*PolyLog[2, c*x])/x^4,x]

[Out]

-(g*(c*x*(1 + 2*c*x) - 2*c^3*x^3*Log[x] + 2*(-1 + c^3*x^3)*Log[1 - c*x] + 6*PolyLog[2, c*x]))/(18*x^3) + (h*(7
*c^2*x^2 - 4*c^3*x^3 - 15*c^3*x^3*Log[x] - 12*c^3*x^3*Log[c*x] - 7*c*x*Log[1 - c*x] - 20*c^2*x^2*Log[1 - c*x]
+ 27*c^3*x^3*Log[1 - c*x] + 8*c^3*x^3*Log[c*x]*Log[1 - c*x] + 4*Log[1 - c*x]^2 - 4*c^3*x^3*Log[1 - c*x]^2 + 12
*c^3*x^3*Log[c*x]*Log[1 - c*x]^2 + 6*(c*x*(1 + 2*c*x) + 2*(-1 + c^3*x^3)*Log[1 - c*x])*PolyLog[2, c*x] + 8*c^3
*x^3*(1 + 3*Log[1 - c*x])*PolyLog[2, 1 - c*x] - 12*c^3*x^3*PolyLog[3, c*x] - 24*c^3*x^3*PolyLog[3, 1 - c*x]))/
(36*x^3)

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Maple [F]  time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( g+h\ln \left ( -cx+1 \right ) \right ){\it polylog} \left ( 2,cx \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g+h*ln(-c*x+1))*polylog(2,c*x)/x^4,x)

[Out]

int((g+h*ln(-c*x+1))*polylog(2,c*x)/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{18} \,{\left (2 \, c^{3} \log \left (x\right ) - \frac{2 \, c^{2} x^{2} + c x + 2 \,{\left (c^{3} x^{3} - 1\right )} \log \left (-c x + 1\right ) + 6 \,{\rm Li}_2\left (c x\right )}{x^{3}}\right )} g + h \int \frac{{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g+h*log(-c*x+1))*polylog(2,c*x)/x^4,x, algorithm="maxima")

[Out]

1/18*(2*c^3*log(x) - (2*c^2*x^2 + c*x + 2*(c^3*x^3 - 1)*log(-c*x + 1) + 6*dilog(c*x))/x^3)*g + h*integrate(dil
og(c*x)*log(-c*x + 1)/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{h{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ) + g{\rm Li}_2\left (c x\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g+h*log(-c*x+1))*polylog(2,c*x)/x^4,x, algorithm="fricas")

[Out]

integral((h*dilog(c*x)*log(-c*x + 1) + g*dilog(c*x))/x^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g+h*ln(-c*x+1))*polylog(2,c*x)/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (h \log \left (-c x + 1\right ) + g\right )}{\rm Li}_2\left (c x\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g+h*log(-c*x+1))*polylog(2,c*x)/x^4,x, algorithm="giac")

[Out]

integrate((h*log(-c*x + 1) + g)*dilog(c*x)/x^4, x)