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

Optimal. Leaf size=423 \[ -\frac{h x \text{PolyLog}(2,c x)}{3 c^2}+\frac{2 h \text{PolyLog}(3,1-c x)}{3 c^3}-\frac{h \log (1-c x) \text{PolyLog}(2,c x)}{3 c^3}-\frac{2 h \log (1-c x) \text{PolyLog}(2,1-c x)}{3 c^3}+\frac{1}{3} x^3 \text{PolyLog}(2,c x) (h \log (1-c x)+g)-\frac{1}{9} h x^3 \text{PolyLog}(2,c x)-\frac{h x^2 \text{PolyLog}(2,c x)}{6 c}+\frac{(1-c x)^3 (2 h \log (1-c x)+g)}{27 c^3}-\frac{(1-c x)^2 (2 h \log (1-c x)+g)}{6 c^3}+\frac{(1-c x) (2 h \log (1-c x)+g)}{3 c^3}-\frac{\log (1-c x) (2 h \log (1-c x)+g)}{9 c^3}+\frac{121 h x}{108 c^2}-\frac{2 h (1-c x)^3}{81 c^3}+\frac{h (1-c x)^2}{6 c^3}+\frac{h \log ^2(1-c x)}{9 c^3}-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{13 h \log (1-c x)}{108 c^3}+\frac{h (1-c x) \log (1-c x)}{3 c^3}+\frac{1}{9} x^3 \log (1-c x) (h \log (1-c x)+g)+\frac{13 h x^2}{216 c}-\frac{1}{27} h x^3 \log (1-c x)-\frac{h x^2 \log (1-c x)}{12 c}+\frac{h x^3}{81} \]

[Out]

(121*h*x)/(108*c^2) + (13*h*x^2)/(216*c) + (h*x^3)/81 + (h*(1 - c*x)^2)/(6*c^3) - (2*h*(1 - c*x)^3)/(81*c^3) +
(13*h*Log[1 - c*x])/(108*c^3) - (h*x^2*Log[1 - c*x])/(12*c) - (h*x^3*Log[1 - c*x])/27 + (h*(1 - c*x)*Log[1 -
c*x])/(3*c^3) + (h*Log[1 - c*x]^2)/(9*c^3) - (h*Log[c*x]*Log[1 - c*x]^2)/(3*c^3) + (x^3*Log[1 - c*x]*(g + h*Lo
g[1 - c*x]))/9 + ((1 - c*x)*(g + 2*h*Log[1 - c*x]))/(3*c^3) - ((1 - c*x)^2*(g + 2*h*Log[1 - c*x]))/(6*c^3) + (
(1 - c*x)^3*(g + 2*h*Log[1 - c*x]))/(27*c^3) - (Log[1 - c*x]*(g + 2*h*Log[1 - c*x]))/(9*c^3) - (h*x*PolyLog[2,
c*x])/(3*c^2) - (h*x^2*PolyLog[2, c*x])/(6*c) - (h*x^3*PolyLog[2, c*x])/9 - (h*Log[1 - c*x]*PolyLog[2, c*x])/
(3*c^3) + (x^3*(g + h*Log[1 - c*x])*PolyLog[2, c*x])/3 - (2*h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(3*c^3) + (2*h
*PolyLog[3, 1 - c*x])/(3*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.611175, antiderivative size = 366, normalized size of antiderivative = 0.87, number of steps used = 37, number of rules used = 20, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6603, 2439, 2410, 2389, 2295, 2395, 43, 2390, 2301, 2411, 2334, 12, 14, 6586, 6591, 6596, 2396, 2433, 2374, 6589} \[ -\frac{h x \text{PolyLog}(2,c x)}{3 c^2}+\frac{2 h \text{PolyLog}(3,1-c x)}{3 c^3}-\frac{h \log (1-c x) \text{PolyLog}(2,c x)}{3 c^3}-\frac{2 h \log (1-c x) \text{PolyLog}(2,1-c x)}{3 c^3}+\frac{1}{3} x^3 \text{PolyLog}(2,c x) (h \log (1-c x)+g)-\frac{1}{9} h x^3 \text{PolyLog}(2,c x)-\frac{h x^2 \text{PolyLog}(2,c x)}{6 c}+\frac{1}{54} \left (\frac{2 (1-c x)^3}{c^3}-\frac{9 (1-c x)^2}{c^3}+\frac{18 (1-c x)}{c^3}-\frac{6 \log (1-c x)}{c^3}\right ) (h \log (1-c x)+g)+\frac{107 h x}{108 c^2}-\frac{h (1-c x)^3}{81 c^3}+\frac{h (1-c x)^2}{12 c^3}-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{23 h \log (1-c x)}{108 c^3}+\frac{4 h (1-c x) \log (1-c x)}{9 c^3}+\frac{1}{9} x^3 \log (1-c x) (h \log (1-c x)+g)+\frac{23 h x^2}{216 c}-\frac{2}{27} h x^3 \log (1-c x)-\frac{5 h x^2 \log (1-c x)}{36 c}+\frac{2 h x^3}{81} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(107*h*x)/(108*c^2) + (23*h*x^2)/(216*c) + (2*h*x^3)/81 + (h*(1 - c*x)^2)/(12*c^3) - (h*(1 - c*x)^3)/(81*c^3)
+ (23*h*Log[1 - c*x])/(108*c^3) - (5*h*x^2*Log[1 - c*x])/(36*c) - (2*h*x^3*Log[1 - c*x])/27 + (4*h*(1 - c*x)*L
og[1 - c*x])/(9*c^3) - (h*Log[c*x]*Log[1 - c*x]^2)/(3*c^3) + (x^3*Log[1 - c*x]*(g + h*Log[1 - c*x]))/9 + (((18
*(1 - c*x))/c^3 - (9*(1 - c*x)^2)/c^3 + (2*(1 - c*x)^3)/c^3 - (6*Log[1 - c*x])/c^3)*(g + h*Log[1 - c*x]))/54 -
(h*x*PolyLog[2, c*x])/(3*c^2) - (h*x^2*PolyLog[2, c*x])/(6*c) - (h*x^3*PolyLog[2, c*x])/9 - (h*Log[1 - c*x]*P
olyLog[2, c*x])/(3*c^3) + (x^3*(g + h*Log[1 - c*x])*PolyLog[2, c*x])/3 - (2*h*Log[1 - c*x]*PolyLog[2, 1 - c*x]
)/(3*c^3) + (2*h*PolyLog[3, 1 - c*x])/(3*c^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 2389

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 2334

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6586

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

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 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]

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]

Rubi steps

\begin{align*} \int x^2 (g+h \log (1-c x)) \text{Li}_2(c x) \, dx &=\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)+\frac{1}{3} \int x^2 \log (1-c x) (g+h \log (1-c x)) \, dx+\frac{1}{3} (c h) \int \left (-\frac{\text{Li}_2(c x)}{c^3}-\frac{x \text{Li}_2(c x)}{c^2}-\frac{x^2 \text{Li}_2(c x)}{c}-\frac{\text{Li}_2(c x)}{c^3 (-1+c x)}\right ) \, dx\\ &=\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)+\frac{1}{9} c \int \frac{x^3 (g+h \log (1-c x))}{1-c x} \, dx-\frac{1}{3} h \int x^2 \text{Li}_2(c x) \, dx-\frac{h \int \text{Li}_2(c x) \, dx}{3 c^2}-\frac{h \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx}{3 c^2}-\frac{h \int x \text{Li}_2(c x) \, dx}{3 c}+\frac{1}{9} (c h) \int \frac{x^3 \log (1-c x)}{1-c x} \, dx\\ &=\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{3 c^2}-\frac{h x^2 \text{Li}_2(c x)}{6 c}-\frac{1}{9} h x^3 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{3 c^3}+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{1}{9} \operatorname{Subst}\left (\int \frac{\left (\frac{1}{c}-\frac{x}{c}\right )^3 (g+h \log (x))}{x} \, dx,x,1-c x\right )-\frac{1}{9} h \int x^2 \log (1-c x) \, dx-\frac{h \int \frac{\log ^2(1-c x)}{x} \, dx}{3 c^3}-\frac{h \int \log (1-c x) \, dx}{3 c^2}-\frac{h \int x \log (1-c x) \, dx}{6 c}+\frac{1}{9} (c h) \int \left (-\frac{\log (1-c x)}{c^3}-\frac{x \log (1-c x)}{c^2}-\frac{x^2 \log (1-c x)}{c}-\frac{\log (1-c x)}{c^3 (-1+c x)}\right ) \, dx\\ &=-\frac{h x^2 \log (1-c x)}{12 c}-\frac{1}{27} h x^3 \log (1-c x)-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))+\frac{1}{54} \left (\frac{18 (1-c x)}{c^3}-\frac{9 (1-c x)^2}{c^3}+\frac{2 (1-c x)^3}{c^3}-\frac{6 \log (1-c x)}{c^3}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{3 c^2}-\frac{h x^2 \text{Li}_2(c x)}{6 c}-\frac{1}{9} h x^3 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{3 c^3}+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{1}{12} h \int \frac{x^2}{1-c x} \, dx-\frac{1}{9} h \int x^2 \log (1-c x) \, dx+\frac{1}{9} h \operatorname{Subst}\left (\int \frac{x \left (-18+9 x-2 x^2\right )+6 \log (x)}{6 c^3 x} \, dx,x,1-c x\right )+\frac{h \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{3 c^3}-\frac{h \int \log (1-c x) \, dx}{9 c^2}-\frac{h \int \frac{\log (1-c x)}{-1+c x} \, dx}{9 c^2}-\frac{(2 h) \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx}{3 c^2}-\frac{h \int x \log (1-c x) \, dx}{9 c}-\frac{1}{27} (c h) \int \frac{x^3}{1-c x} \, dx\\ &=\frac{h x}{3 c^2}-\frac{5 h x^2 \log (1-c x)}{36 c}-\frac{2}{27} h x^3 \log (1-c x)+\frac{h (1-c x) \log (1-c x)}{3 c^3}-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))+\frac{1}{54} \left (\frac{18 (1-c x)}{c^3}-\frac{9 (1-c x)^2}{c^3}+\frac{2 (1-c x)^3}{c^3}-\frac{6 \log (1-c x)}{c^3}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{3 c^2}-\frac{h x^2 \text{Li}_2(c x)}{6 c}-\frac{1}{9} h x^3 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{3 c^3}+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{1}{18} h \int \frac{x^2}{1-c x} \, dx-\frac{1}{12} h \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx+\frac{h \operatorname{Subst}\left (\int \frac{x \left (-18+9 x-2 x^2\right )+6 \log (x)}{x} \, dx,x,1-c x\right )}{54 c^3}+\frac{h \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{9 c^3}-\frac{h \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )}{9 c^3}+\frac{(2 h) \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 )}{3 c^3}-\frac{1}{27} (c h) \int \frac{x^3}{1-c x} \, dx-\frac{1}{27} (c h) \int \left (-\frac{1}{c^3}-\frac{x}{c^2}-\frac{x^2}{c}-\frac{1}{c^3 (-1+c x)}\right ) \, dx\\ &=\frac{61 h x}{108 c^2}+\frac{13 h x^2}{216 c}+\frac{h x^3}{81}+\frac{13 h \log (1-c x)}{108 c^3}-\frac{5 h x^2 \log (1-c x)}{36 c}-\frac{2}{27} h x^3 \log (1-c x)+\frac{4 h (1-c x) \log (1-c x)}{9 c^3}-\frac{h \log ^2(1-c x)}{18 c^3}-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))+\frac{1}{54} \left (\frac{18 (1-c x)}{c^3}-\frac{9 (1-c x)^2}{c^3}+\frac{2 (1-c x)^3}{c^3}-\frac{6 \log (1-c x)}{c^3}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{3 c^2}-\frac{h x^2 \text{Li}_2(c x)}{6 c}-\frac{1}{9} h x^3 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{3 c^3}+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{2 h \log (1-c x) \text{Li}_2(1-c x)}{3 c^3}-\frac{1}{18} h \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx+\frac{h \operatorname{Subst}\left (\int \left (-18+9 x-2 x^2+\frac{6 \log (x)}{x}\right ) \, dx,x,1-c x\right )}{54 c^3}+\frac{(2 h) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )}{3 c^3}-\frac{1}{27} (c h) \int \left (-\frac{1}{c^3}-\frac{x}{c^2}-\frac{x^2}{c}-\frac{1}{c^3 (-1+c x)}\right ) \, dx\\ &=\frac{107 h x}{108 c^2}+\frac{23 h x^2}{216 c}+\frac{2 h x^3}{81}+\frac{h (1-c x)^2}{12 c^3}-\frac{h (1-c x)^3}{81 c^3}+\frac{23 h \log (1-c x)}{108 c^3}-\frac{5 h x^2 \log (1-c x)}{36 c}-\frac{2}{27} h x^3 \log (1-c x)+\frac{4 h (1-c x) \log (1-c x)}{9 c^3}-\frac{h \log ^2(1-c x)}{18 c^3}-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))+\frac{1}{54} \left (\frac{18 (1-c x)}{c^3}-\frac{9 (1-c x)^2}{c^3}+\frac{2 (1-c x)^3}{c^3}-\frac{6 \log (1-c x)}{c^3}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{3 c^2}-\frac{h x^2 \text{Li}_2(c x)}{6 c}-\frac{1}{9} h x^3 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{3 c^3}+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{2 h \log (1-c x) \text{Li}_2(1-c x)}{3 c^3}+\frac{2 h \text{Li}_3(1-c x)}{3 c^3}+\frac{h \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )}{9 c^3}\\ &=\frac{107 h x}{108 c^2}+\frac{23 h x^2}{216 c}+\frac{2 h x^3}{81}+\frac{h (1-c x)^2}{12 c^3}-\frac{h (1-c x)^3}{81 c^3}+\frac{23 h \log (1-c x)}{108 c^3}-\frac{5 h x^2 \log (1-c x)}{36 c}-\frac{2}{27} h x^3 \log (1-c x)+\frac{4 h (1-c x) \log (1-c x)}{9 c^3}-\frac{h \log (c x) \log ^2(1-c x)}{3 c^3}+\frac{1}{9} x^3 \log (1-c x) (g+h \log (1-c x))+\frac{1}{54} \left (\frac{18 (1-c x)}{c^3}-\frac{9 (1-c x)^2}{c^3}+\frac{2 (1-c x)^3}{c^3}-\frac{6 \log (1-c x)}{c^3}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{3 c^2}-\frac{h x^2 \text{Li}_2(c x)}{6 c}-\frac{1}{9} h x^3 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{3 c^3}+\frac{1}{3} x^3 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{2 h \log (1-c x) \text{Li}_2(1-c x)}{3 c^3}+\frac{2 h \text{Li}_3(1-c x)}{3 c^3}\\ \end{align*}

Mathematica [A]  time = 0.46092, size = 252, normalized size = 0.6 \[ \frac{g \left (18 c^3 x^3 \text{PolyLog}(2,c x)-c x \left (2 c^2 x^2+3 c x+6\right )+6 \left (c^3 x^3-1\right ) \log (1-c x)\right )}{54 c^3}+\frac{h \left (12 \left (6 \left (c^3 x^3-1\right ) \log (1-c x)-c x \left (2 c^2 x^2+3 c x+6\right )\right ) \text{PolyLog}(2,c x)+144 \text{PolyLog}(3,1-c x)-144 \log (1-c x) \text{PolyLog}(2,1-c x)+8 c^3 x^3+33 c^2 x^2+24 c^3 x^3 \log ^2(1-c x)-24 c^3 x^3 \log (1-c x)-42 c^2 x^2 \log (1-c x)+186 c x-72 \log (c x) \log ^2(1-c x)-24 \log ^2(1-c x)-120 c x \log (1-c x)+186 \log (1-c x)\right )}{216 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(-(c*x*(6 + 3*c*x + 2*c^2*x^2)) + 6*(-1 + c^3*x^3)*Log[1 - c*x] + 18*c^3*x^3*PolyLog[2, c*x]))/(54*c^3) + (
h*(186*c*x + 33*c^2*x^2 + 8*c^3*x^3 + 186*Log[1 - c*x] - 120*c*x*Log[1 - c*x] - 42*c^2*x^2*Log[1 - c*x] - 24*c
^3*x^3*Log[1 - c*x] - 24*Log[1 - c*x]^2 + 24*c^3*x^3*Log[1 - c*x]^2 - 72*Log[c*x]*Log[1 - c*x]^2 + 12*(-(c*x*(
6 + 3*c*x + 2*c^2*x^2)) + 6*(-1 + c^3*x^3)*Log[1 - c*x])*PolyLog[2, c*x] - 144*Log[1 - c*x]*PolyLog[2, 1 - c*x
] + 144*PolyLog[3, 1 - c*x]))/(216*c^3)

________________________________________________________________________________________

Maple [F]  time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( g+h\ln \left ( -cx+1 \right ) \right ){\it polylog} \left ( 2,cx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{18} \, h{\left (\frac{{\left (2 \, c^{3} x^{3} + 3 \, c^{2} x^{2} + 6 \, c x - 6 \,{\left (c^{3} x^{3} - 1\right )} \log \left (-c x + 1\right )\right )}{\rm Li}_2\left (c x\right )}{c^{3}} - \frac{\frac{4}{9} \, c^{3} x^{3} - \frac{1}{9} \,{\left (6 \, x^{3} \log \left (-c x + 1\right ) - c{\left (\frac{2 \, c^{2} x^{3} + 3 \, c x^{2} + 6 \, x}{c^{3}} + \frac{6 \, \log \left (c x - 1\right )}{c^{4}}\right )}\right )} c^{3} + \frac{5}{3} \, c^{2} x^{2} - \frac{3}{4} \,{\left (2 \, x^{2} \log \left (-c x + 1\right ) - c{\left (\frac{c x^{2} + 2 \, x}{c^{2}} + \frac{2 \, \log \left (c x - 1\right )}{c^{3}}\right )}\right )} c^{2} + 2 \,{\left (c^{3} x^{3} - 1\right )} \log \left (-c x + 1\right )^{2} - 6 \, \log \left (c x\right ) \log \left (-c x + 1\right )^{2} + \frac{40}{3} \, c x - \frac{2}{3} \,{\left (2 \, c^{3} x^{3} + 3 \, c^{2} x^{2} + 6 \, c x - 11\right )} \log \left (-c x + 1\right ) - 6 \,{\left (c x - 1\right )} \log \left (-c x + 1\right ) - 12 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) + 12 \,{\rm Li}_{3}(-c x + 1) - 6}{c^{3}}\right )} + \frac{{\left (18 \, c^{3} x^{3}{\rm Li}_2\left (c x\right ) - 2 \, c^{3} x^{3} - 3 \, c^{2} x^{2} - 6 \, c x + 6 \,{\left (c^{3} x^{3} - 1\right )} \log \left (-c x + 1\right )\right )} g}{54 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*h*((2*c^3*x^3 + 3*c^2*x^2 + 6*c*x - 6*(c^3*x^3 - 1)*log(-c*x + 1))*dilog(c*x)/c^3 - integrate((6*(c^3*x^
3 - 1)*log(-c*x + 1)^2 - (2*c^3*x^3 + 3*c^2*x^2 + 6*c*x)*log(-c*x + 1))/x, x)/c^3) + 1/54*(18*c^3*x^3*dilog(c*
x) - 2*c^3*x^3 - 3*c^2*x^2 - 6*c*x + 6*(c^3*x^3 - 1)*log(-c*x + 1))*g/c^3

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (h x^{2}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ) + g x^{2}{\rm Li}_2\left (c x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(h*x^2*dilog(c*x)*log(-c*x + 1) + g*x^2*dilog(c*x), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((h*log(-c*x + 1) + g)*x^2*dilog(c*x), x)