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

Optimal. Leaf size=330 \[ \frac{h \text{PolyLog}(3,1-c x)}{c^2}-\frac{h \log (1-c x) \text{PolyLog}(2,c x)}{2 c^2}-\frac{h \log (1-c x) \text{PolyLog}(2,1-c x)}{c^2}+\frac{1}{2} x^2 \text{PolyLog}(2,c x) (h \log (1-c x)+g)-\frac{1}{4} h x^2 \text{PolyLog}(2,c x)-\frac{h x \text{PolyLog}(2,c x)}{2 c}-\frac{(1-c x)^2 (2 h \log (1-c x)+g)}{8 c^2}+\frac{(1-c x) (2 h \log (1-c x)+g)}{2 c^2}-\frac{\log (1-c x) (2 h \log (1-c x)+g)}{4 c^2}+\frac{h (1-c x)^2}{8 c^2}+\frac{h \log ^2(1-c x)}{4 c^2}-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{h \log (1-c x)}{8 c^2}+\frac{h (1-c x) \log (1-c x)}{2 c^2}+\frac{1}{4} x^2 \log (1-c x) (h \log (1-c x)+g)-\frac{1}{8} h x^2 \log (1-c x)+\frac{13 h x}{8 c}+\frac{h x^2}{16} \]

[Out]

(13*h*x)/(8*c) + (h*x^2)/16 + (h*(1 - c*x)^2)/(8*c^2) + (h*Log[1 - c*x])/(8*c^2) - (h*x^2*Log[1 - c*x])/8 + (h
*(1 - c*x)*Log[1 - c*x])/(2*c^2) + (h*Log[1 - c*x]^2)/(4*c^2) - (h*Log[c*x]*Log[1 - c*x]^2)/(2*c^2) + (x^2*Log
[1 - c*x]*(g + h*Log[1 - c*x]))/4 + ((1 - c*x)*(g + 2*h*Log[1 - c*x]))/(2*c^2) - ((1 - c*x)^2*(g + 2*h*Log[1 -
c*x]))/(8*c^2) - (Log[1 - c*x]*(g + 2*h*Log[1 - c*x]))/(4*c^2) - (h*x*PolyLog[2, c*x])/(2*c) - (h*x^2*PolyLog
[2, c*x])/4 - (h*Log[1 - c*x]*PolyLog[2, c*x])/(2*c^2) + (x^2*(g + h*Log[1 - c*x])*PolyLog[2, c*x])/2 - (h*Log
[1 - c*x]*PolyLog[2, 1 - c*x])/c^2 + (h*PolyLog[3, 1 - c*x])/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.44859, antiderivative size = 287, normalized size of antiderivative = 0.87, number of steps used = 30, number of rules used = 20, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.111, Rules used = {6603, 2439, 2410, 2389, 2295, 2395, 43, 2390, 2301, 2411, 2334, 12, 14, 6586, 6591, 6596, 2396, 2433, 2374, 6589} \[ \frac{h \text{PolyLog}(3,1-c x)}{c^2}-\frac{h \log (1-c x) \text{PolyLog}(2,c x)}{2 c^2}-\frac{h \log (1-c x) \text{PolyLog}(2,1-c x)}{c^2}+\frac{1}{2} x^2 \text{PolyLog}(2,c x) (h \log (1-c x)+g)-\frac{1}{4} h x^2 \text{PolyLog}(2,c x)-\frac{h x \text{PolyLog}(2,c x)}{2 c}+\frac{1}{8} \left (-\frac{(1-c x)^2}{c^2}+\frac{4 (1-c x)}{c^2}-\frac{2 \log (1-c x)}{c^2}\right ) (h \log (1-c x)+g)+\frac{h (1-c x)^2}{16 c^2}-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{h \log (1-c x)}{4 c^2}+\frac{3 h (1-c x) \log (1-c x)}{4 c^2}+\frac{1}{4} x^2 \log (1-c x) (h \log (1-c x)+g)-\frac{1}{4} h x^2 \log (1-c x)+\frac{3 h x}{2 c}+\frac{h x^2}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*h*x)/(2*c) + (h*x^2)/8 + (h*(1 - c*x)^2)/(16*c^2) + (h*Log[1 - c*x])/(4*c^2) - (h*x^2*Log[1 - c*x])/4 + (3*
h*(1 - c*x)*Log[1 - c*x])/(4*c^2) - (h*Log[c*x]*Log[1 - c*x]^2)/(2*c^2) + (x^2*Log[1 - c*x]*(g + h*Log[1 - c*x
]))/4 + (((4*(1 - c*x))/c^2 - (1 - c*x)^2/c^2 - (2*Log[1 - c*x])/c^2)*(g + h*Log[1 - c*x]))/8 - (h*x*PolyLog[2
, c*x])/(2*c) - (h*x^2*PolyLog[2, c*x])/4 - (h*Log[1 - c*x]*PolyLog[2, c*x])/(2*c^2) + (x^2*(g + h*Log[1 - c*x
])*PolyLog[2, c*x])/2 - (h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/c^2 + (h*PolyLog[3, 1 - c*x])/c^2

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 (g+h \log (1-c x)) \text{Li}_2(c x) \, dx &=\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)+\frac{1}{2} \int x \log (1-c x) (g+h \log (1-c x)) \, dx+\frac{1}{2} (c h) \int \left (-\frac{\text{Li}_2(c x)}{c^2}-\frac{x \text{Li}_2(c x)}{c}-\frac{\text{Li}_2(c x)}{c^2 (-1+c x)}\right ) \, dx\\ &=\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)+\frac{1}{4} c \int \frac{x^2 (g+h \log (1-c x))}{1-c x} \, dx-\frac{1}{2} h \int x \text{Li}_2(c x) \, dx-\frac{h \int \text{Li}_2(c x) \, dx}{2 c}-\frac{h \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx}{2 c}+\frac{1}{4} (c h) \int \frac{x^2 \log (1-c x)}{1-c x} \, dx\\ &=\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{2 c}-\frac{1}{4} h x^2 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (\frac{1}{c}-\frac{x}{c}\right )^2 (g+h \log (x))}{x} \, dx,x,1-c x\right )-\frac{1}{4} h \int x \log (1-c x) \, dx-\frac{h \int \frac{\log ^2(1-c x)}{x} \, dx}{2 c^2}-\frac{h \int \log (1-c x) \, dx}{2 c}+\frac{1}{4} (c h) \int \left (-\frac{\log (1-c x)}{c^2}-\frac{x \log (1-c x)}{c}-\frac{\log (1-c x)}{c^2 (-1+c x)}\right ) \, dx\\ &=-\frac{1}{8} h x^2 \log (1-c x)-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))+\frac{1}{8} \left (\frac{4 (1-c x)}{c^2}-\frac{(1-c x)^2}{c^2}-\frac{2 \log (1-c x)}{c^2}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{2 c}-\frac{1}{4} h x^2 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{1}{4} h \int x \log (1-c x) \, dx+\frac{1}{4} h \operatorname{Subst}\left (\int \frac{(-4+x) x+2 \log (x)}{2 c^2 x} \, dx,x,1-c x\right )+\frac{h \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{2 c^2}-\frac{h \int \log (1-c x) \, dx}{4 c}-\frac{h \int \frac{\log (1-c x)}{-1+c x} \, dx}{4 c}-\frac{h \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx}{c}-\frac{1}{8} (c h) \int \frac{x^2}{1-c x} \, dx\\ &=\frac{h x}{2 c}-\frac{1}{4} h x^2 \log (1-c x)+\frac{h (1-c x) \log (1-c x)}{2 c^2}-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))+\frac{1}{8} \left (\frac{4 (1-c x)}{c^2}-\frac{(1-c x)^2}{c^2}-\frac{2 \log (1-c x)}{c^2}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{2 c}-\frac{1}{4} h x^2 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)+\frac{h \operatorname{Subst}\left (\int \frac{(-4+x) x+2 \log (x)}{x} \, dx,x,1-c x\right )}{8 c^2}+\frac{h \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{4 c^2}-\frac{h \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )}{4 c^2}+\frac{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 )}{c^2}-\frac{1}{8} (c h) \int \frac{x^2}{1-c x} \, dx-\frac{1}{8} (c h) \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx\\ &=\frac{7 h x}{8 c}+\frac{h x^2}{16}+\frac{h \log (1-c x)}{8 c^2}-\frac{1}{4} h x^2 \log (1-c x)+\frac{3 h (1-c x) \log (1-c x)}{4 c^2}-\frac{h \log ^2(1-c x)}{8 c^2}-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))+\frac{1}{8} \left (\frac{4 (1-c x)}{c^2}-\frac{(1-c x)^2}{c^2}-\frac{2 \log (1-c x)}{c^2}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{2 c}-\frac{1}{4} h x^2 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(1-c x)}{c^2}+\frac{h \operatorname{Subst}\left (\int \left (-4+x+\frac{2 \log (x)}{x}\right ) \, dx,x,1-c x\right )}{8 c^2}+\frac{h \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )}{c^2}-\frac{1}{8} (c h) \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx\\ &=\frac{3 h x}{2 c}+\frac{h x^2}{8}+\frac{h (1-c x)^2}{16 c^2}+\frac{h \log (1-c x)}{4 c^2}-\frac{1}{4} h x^2 \log (1-c x)+\frac{3 h (1-c x) \log (1-c x)}{4 c^2}-\frac{h \log ^2(1-c x)}{8 c^2}-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))+\frac{1}{8} \left (\frac{4 (1-c x)}{c^2}-\frac{(1-c x)^2}{c^2}-\frac{2 \log (1-c x)}{c^2}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{2 c}-\frac{1}{4} h x^2 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(1-c x)}{c^2}+\frac{h \text{Li}_3(1-c x)}{c^2}+\frac{h \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-c x\right )}{4 c^2}\\ &=\frac{3 h x}{2 c}+\frac{h x^2}{8}+\frac{h (1-c x)^2}{16 c^2}+\frac{h \log (1-c x)}{4 c^2}-\frac{1}{4} h x^2 \log (1-c x)+\frac{3 h (1-c x) \log (1-c x)}{4 c^2}-\frac{h \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{1}{4} x^2 \log (1-c x) (g+h \log (1-c x))+\frac{1}{8} \left (\frac{4 (1-c x)}{c^2}-\frac{(1-c x)^2}{c^2}-\frac{2 \log (1-c x)}{c^2}\right ) (g+h \log (1-c x))-\frac{h x \text{Li}_2(c x)}{2 c}-\frac{1}{4} h x^2 \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} x^2 (g+h \log (1-c x)) \text{Li}_2(c x)-\frac{h \log (1-c x) \text{Li}_2(1-c x)}{c^2}+\frac{h \text{Li}_3(1-c x)}{c^2}\\ \end{align*}

Mathematica [A]  time = 0.339235, size = 211, normalized size = 0.64 \[ \frac{g \left (4 c^2 x^2 \text{PolyLog}(2,c x)+2 \left (c^2 x^2-1\right ) \log (1-c x)-c x (c x+2)\right )}{8 c^2}+\frac{h \left (\left (8 \left (c^2 x^2-1\right ) \log (1-c x)-4 c x (c x+2)\right ) \text{PolyLog}(2,c x)+16 \text{PolyLog}(3,1-c x)-16 \log (1-c x) \text{PolyLog}(2,1-c x)+3 c^2 x^2+4 c^2 x^2 \log ^2(1-c x)-6 c^2 x^2 \log (1-c x)+22 c x-8 \log (c x) \log ^2(1-c x)-4 \log ^2(1-c x)-16 c x \log (1-c x)+22 \log (1-c x)-14\right )}{16 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(-(c*x*(2 + c*x)) + 2*(-1 + c^2*x^2)*Log[1 - c*x] + 4*c^2*x^2*PolyLog[2, c*x]))/(8*c^2) + (h*(-14 + 22*c*x
+ 3*c^2*x^2 + 22*Log[1 - c*x] - 16*c*x*Log[1 - c*x] - 6*c^2*x^2*Log[1 - c*x] - 4*Log[1 - c*x]^2 + 4*c^2*x^2*Lo
g[1 - c*x]^2 - 8*Log[c*x]*Log[1 - c*x]^2 + (-4*c*x*(2 + c*x) + 8*(-1 + c^2*x^2)*Log[1 - c*x])*PolyLog[2, c*x]
- 16*Log[1 - c*x]*PolyLog[2, 1 - c*x] + 16*PolyLog[3, 1 - c*x]))/(16*c^2)

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int x \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*(g+h*ln(-c*x+1))*polylog(2,c*x),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, h{\left (\frac{{\left (c^{2} x^{2} + 2 \, c x - 2 \,{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )}{\rm Li}_2\left (c x\right )}{c^{2}} - \frac{\frac{1}{2} \, c^{2} x^{2} - \frac{1}{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} +{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )^{2} - 2 \, \log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 5 \, c x -{\left (c^{2} x^{2} + 2 \, c x - 3\right )} \log \left (-c x + 1\right ) - 2 \,{\left (c x - 1\right )} \log \left (-c x + 1\right ) - 4 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) + 4 \,{\rm Li}_{3}(-c x + 1) - 2}{c^{2}}\right )} + \frac{{\left (4 \, c^{2} x^{2}{\rm Li}_2\left (c x\right ) - c^{2} x^{2} - 2 \, c x + 2 \,{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} g}{8 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*h*((c^2*x^2 + 2*c*x - 2*(c^2*x^2 - 1)*log(-c*x + 1))*dilog(c*x)/c^2 - integrate((2*(c^2*x^2 - 1)*log(-c*x
+ 1)^2 - (c^2*x^2 + 2*c*x)*log(-c*x + 1))/x, x)/c^2) + 1/8*(4*c^2*x^2*dilog(c*x) - c^2*x^2 - 2*c*x + 2*(c^2*x
^2 - 1)*log(-c*x + 1))*g/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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