3.186 \(\int (a+b x) \log (1-c x) \text{PolyLog}(2,c x) \, dx\)

Optimal. Leaf size=390 \[ \frac{(2 a c+b) \text{PolyLog}(3,1-c x)}{c^2}-\frac{(2 a c+b) \log (1-c x) \text{PolyLog}(2,c x)}{2 c^2}-\frac{(2 a c+b) \log (1-c x) \text{PolyLog}(2,1-c x)}{c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{PolyLog}(2,c x)-\frac{x (2 a c+b) \text{PolyLog}(2,c x)}{2 c}-\frac{1}{4} b x^2 \text{PolyLog}(2,c x)-\frac{(2 a c+b) \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{(1-c x) (2 a c+b) \log (1-c x)}{2 c^2}+\frac{x (2 a c+b)}{2 c}-\frac{a (1-c x) \log ^2(1-c x)}{c}+\frac{2 a (1-c x) \log (1-c x)}{c}+2 a x+\frac{b (1-c x)^2}{8 c^2}+\frac{b (1-c x)^2 \log ^2(1-c x)}{4 c^2}-\frac{b (1-c x) \log ^2(1-c x)}{2 c^2}-\frac{b (1-c x)^2 \log (1-c x)}{4 c^2}+\frac{b (1-c x) \log (1-c x)}{c^2}+\frac{b \log (1-c x)}{8 c^2}-\frac{1}{8} b x^2 \log (1-c x)+\frac{9 b x}{8 c}+\frac{b x^2}{16} \]

[Out]

2*a*x + (9*b*x)/(8*c) + ((b + 2*a*c)*x)/(2*c) + (b*x^2)/16 + (b*(1 - c*x)^2)/(8*c^2) + (b*Log[1 - c*x])/(8*c^2
) - (b*x^2*Log[1 - c*x])/8 + (b*(1 - c*x)*Log[1 - c*x])/c^2 + (2*a*(1 - c*x)*Log[1 - c*x])/c + ((b + 2*a*c)*(1
- c*x)*Log[1 - c*x])/(2*c^2) - (b*(1 - c*x)^2*Log[1 - c*x])/(4*c^2) - (b*(1 - c*x)*Log[1 - c*x]^2)/(2*c^2) -
(a*(1 - c*x)*Log[1 - c*x]^2)/c + (b*(1 - c*x)^2*Log[1 - c*x]^2)/(4*c^2) - ((b + 2*a*c)*Log[c*x]*Log[1 - c*x]^2
)/(2*c^2) - ((b + 2*a*c)*x*PolyLog[2, c*x])/(2*c) - (b*x^2*PolyLog[2, c*x])/4 - ((b + 2*a*c)*Log[1 - c*x]*Poly
Log[2, c*x])/(2*c^2) + ((2*a*x + b*x^2)*Log[1 - c*x]*PolyLog[2, c*x])/2 - ((b + 2*a*c)*Log[1 - c*x]*PolyLog[2,
1 - c*x])/c^2 + ((b + 2*a*c)*PolyLog[3, 1 - c*x])/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.44939, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 20, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.111, Rules used = {6598, 43, 2416, 2389, 2295, 2391, 2395, 6604, 2296, 2401, 2390, 2305, 2304, 6586, 6591, 6596, 2396, 2433, 2374, 6589} \[ \frac{(2 a c+b) \text{PolyLog}(3,1-c x)}{c^2}-\frac{(2 a c+b) \log (1-c x) \text{PolyLog}(2,c x)}{2 c^2}-\frac{(2 a c+b) \log (1-c x) \text{PolyLog}(2,1-c x)}{c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{PolyLog}(2,c x)-\frac{x (2 a c+b) \text{PolyLog}(2,c x)}{2 c}-\frac{1}{4} b x^2 \text{PolyLog}(2,c x)-\frac{(2 a c+b) \log (c x) \log ^2(1-c x)}{2 c^2}+\frac{(1-c x) (2 a c+b) \log (1-c x)}{2 c^2}+\frac{x (2 a c+b)}{2 c}-\frac{a (1-c x) \log ^2(1-c x)}{c}+\frac{2 a (1-c x) \log (1-c x)}{c}+2 a x+\frac{b (1-c x)^2}{8 c^2}+\frac{b (1-c x)^2 \log ^2(1-c x)}{4 c^2}-\frac{b (1-c x) \log ^2(1-c x)}{2 c^2}-\frac{b (1-c x)^2 \log (1-c x)}{4 c^2}+\frac{b (1-c x) \log (1-c x)}{c^2}+\frac{b \log (1-c x)}{8 c^2}-\frac{1}{8} b x^2 \log (1-c x)+\frac{9 b x}{8 c}+\frac{b x^2}{16} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x],x]

[Out]

2*a*x + (9*b*x)/(8*c) + ((b + 2*a*c)*x)/(2*c) + (b*x^2)/16 + (b*(1 - c*x)^2)/(8*c^2) + (b*Log[1 - c*x])/(8*c^2
) - (b*x^2*Log[1 - c*x])/8 + (b*(1 - c*x)*Log[1 - c*x])/c^2 + (2*a*(1 - c*x)*Log[1 - c*x])/c + ((b + 2*a*c)*(1
- c*x)*Log[1 - c*x])/(2*c^2) - (b*(1 - c*x)^2*Log[1 - c*x])/(4*c^2) - (b*(1 - c*x)*Log[1 - c*x]^2)/(2*c^2) -
(a*(1 - c*x)*Log[1 - c*x]^2)/c + (b*(1 - c*x)^2*Log[1 - c*x]^2)/(4*c^2) - ((b + 2*a*c)*Log[c*x]*Log[1 - c*x]^2
)/(2*c^2) - ((b + 2*a*c)*x*PolyLog[2, c*x])/(2*c) - (b*x^2*PolyLog[2, c*x])/4 - ((b + 2*a*c)*Log[1 - c*x]*Poly
Log[2, c*x])/(2*c^2) + ((2*a*x + b*x^2)*Log[1 - c*x]*PolyLog[2, c*x])/2 - ((b + 2*a*c)*Log[1 - c*x]*PolyLog[2,
1 - c*x])/c^2 + ((b + 2*a*c)*PolyLog[3, 1 - c*x])/c^2

Rule 6598

Int[((d_.) + (e_.)*(x_))^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> Simp[((d + e*x)^(m + 1)*Po
lyLog[2, c*(a + b*x)])/(e*(m + 1)), x] + Dist[b/(e*(m + 1)), Int[((d + e*x)^(m + 1)*Log[1 - a*c - b*c*x])/(a +
b*x), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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 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 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 6604

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(Px_)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symb
ol] :> With[{u = IntHide[Px, x]}, Simp[u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] + (Dist[b, Int
[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x], x] - Dist[e*h*n, Int[Ex
pandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && P
olyQ[Px, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 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 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 (a+b x) \log (1-c x) \text{Li}_2(c x) \, dx &=\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)+c \int \left (\frac{(-b-2 a c) \text{Li}_2(c x)}{2 c^2}-\frac{b x \text{Li}_2(c x)}{2 c}+\frac{(-b-2 a c) \text{Li}_2(c x)}{2 c^2 (-1+c x)}\right ) \, dx+\int \left (a \log ^2(1-c x)+\frac{1}{2} b x \log ^2(1-c x)\right ) \, dx\\ &=\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)+a \int \log ^2(1-c x) \, dx+\frac{1}{2} b \int x \log ^2(1-c x) \, dx-\frac{1}{2} b \int x \text{Li}_2(c x) \, dx-\frac{(b+2 a c) \int \text{Li}_2(c x) \, dx}{2 c}-\frac{(b+2 a c) \int \frac{\text{Li}_2(c x)}{-1+c x} \, dx}{2 c}\\ &=-\frac{(b+2 a c) x \text{Li}_2(c x)}{2 c}-\frac{1}{4} b x^2 \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)-\frac{1}{4} b \int x \log (1-c x) \, dx+\frac{1}{2} b \int \left (\frac{\log ^2(1-c x)}{c}-\frac{(1-c x) \log ^2(1-c x)}{c}\right ) \, dx-\frac{a \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1-c x\right )}{c}-\frac{(b+2 a c) \int \frac{\log ^2(1-c x)}{x} \, dx}{2 c^2}-\frac{(b+2 a c) \int \log (1-c x) \, dx}{2 c}\\ &=-\frac{1}{8} b x^2 \log (1-c x)-\frac{a (1-c x) \log ^2(1-c x)}{c}-\frac{(b+2 a c) \log (c x) \log ^2(1-c x)}{2 c^2}-\frac{(b+2 a c) x \text{Li}_2(c x)}{2 c}-\frac{1}{4} b x^2 \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)+\frac{(2 a) \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{c}+\frac{b \int \log ^2(1-c x) \, dx}{2 c}-\frac{b \int (1-c x) \log ^2(1-c x) \, dx}{2 c}-\frac{1}{8} (b c) \int \frac{x^2}{1-c x} \, dx+\frac{(b+2 a c) \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{2 c^2}-\frac{(b+2 a c) \int \frac{\log (c x) \log (1-c x)}{1-c x} \, dx}{c}\\ &=2 a x+\frac{(b+2 a c) x}{2 c}-\frac{1}{8} b x^2 \log (1-c x)+\frac{2 a (1-c x) \log (1-c x)}{c}+\frac{(b+2 a c) (1-c x) \log (1-c x)}{2 c^2}-\frac{a (1-c x) \log ^2(1-c x)}{c}-\frac{(b+2 a c) \log (c x) \log ^2(1-c x)}{2 c^2}-\frac{(b+2 a c) x \text{Li}_2(c x)}{2 c}-\frac{1}{4} b x^2 \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)-\frac{b \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1-c x\right )}{2 c^2}+\frac{b \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1-c x\right )}{2 c^2}-\frac{1}{8} (b c) \int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx+\frac{(b+2 a c) \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}\\ &=2 a x+\frac{b x}{8 c}+\frac{(b+2 a c) x}{2 c}+\frac{b x^2}{16}+\frac{b \log (1-c x)}{8 c^2}-\frac{1}{8} b x^2 \log (1-c x)+\frac{2 a (1-c x) \log (1-c x)}{c}+\frac{(b+2 a c) (1-c x) \log (1-c x)}{2 c^2}-\frac{b (1-c x) \log ^2(1-c x)}{2 c^2}-\frac{a (1-c x) \log ^2(1-c x)}{c}+\frac{b (1-c x)^2 \log ^2(1-c x)}{4 c^2}-\frac{(b+2 a c) \log (c x) \log ^2(1-c x)}{2 c^2}-\frac{(b+2 a c) x \text{Li}_2(c x)}{2 c}-\frac{1}{4} b x^2 \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(1-c x)}{c^2}-\frac{b \operatorname{Subst}(\int x \log (x) \, dx,x,1-c x)}{2 c^2}+\frac{b \operatorname{Subst}(\int \log (x) \, dx,x,1-c x)}{c^2}+\frac{(b+2 a c) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-c x\right )}{c^2}\\ &=2 a x+\frac{9 b x}{8 c}+\frac{(b+2 a c) x}{2 c}+\frac{b x^2}{16}+\frac{b (1-c x)^2}{8 c^2}+\frac{b \log (1-c x)}{8 c^2}-\frac{1}{8} b x^2 \log (1-c x)+\frac{b (1-c x) \log (1-c x)}{c^2}+\frac{2 a (1-c x) \log (1-c x)}{c}+\frac{(b+2 a c) (1-c x) \log (1-c x)}{2 c^2}-\frac{b (1-c x)^2 \log (1-c x)}{4 c^2}-\frac{b (1-c x) \log ^2(1-c x)}{2 c^2}-\frac{a (1-c x) \log ^2(1-c x)}{c}+\frac{b (1-c x)^2 \log ^2(1-c x)}{4 c^2}-\frac{(b+2 a c) \log (c x) \log ^2(1-c x)}{2 c^2}-\frac{(b+2 a c) x \text{Li}_2(c x)}{2 c}-\frac{1}{4} b x^2 \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(c x)}{2 c^2}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (1-c x) \text{Li}_2(c x)-\frac{(b+2 a c) \log (1-c x) \text{Li}_2(1-c x)}{c^2}+\frac{(b+2 a c) \text{Li}_3(1-c x)}{c^2}\\ \end{align*}

Mathematica [A]  time = 0.490164, size = 285, normalized size = 0.73 \[ \frac{4 \text{PolyLog}(2,c x) (2 (c x-1) \log (1-c x) (2 a c+b c x+b)-c x (4 a c+b c x+2 b))-16 (2 a c+b) \log (1-c x) \text{PolyLog}(2,1-c x)+32 a c \text{PolyLog}(3,1-c x)+16 b \text{PolyLog}(3,1-c x)+48 a c^2 x+16 a c^2 x \log ^2(1-c x)-48 a c^2 x \log (1-c x)-16 a c \log ^2(1-c x)-16 a c \log (c x) \log ^2(1-c x)+48 a c \log (1-c x)-32 a c+3 b c^2 x^2+4 b c^2 x^2 \log ^2(1-c x)-6 b c^2 x^2 \log (1-c x)+22 b c x-4 b \log ^2(1-c x)-8 b \log (c x) \log ^2(1-c x)-16 b c x \log (1-c x)+22 b \log (1-c x)-14 b}{16 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x],x]

[Out]

(-14*b - 32*a*c + 22*b*c*x + 48*a*c^2*x + 3*b*c^2*x^2 + 22*b*Log[1 - c*x] + 48*a*c*Log[1 - c*x] - 16*b*c*x*Log
[1 - c*x] - 48*a*c^2*x*Log[1 - c*x] - 6*b*c^2*x^2*Log[1 - c*x] - 4*b*Log[1 - c*x]^2 - 16*a*c*Log[1 - c*x]^2 +
16*a*c^2*x*Log[1 - c*x]^2 + 4*b*c^2*x^2*Log[1 - c*x]^2 - 8*b*Log[c*x]*Log[1 - c*x]^2 - 16*a*c*Log[c*x]*Log[1 -
c*x]^2 + 4*(-(c*x*(2*b + 4*a*c + b*c*x)) + 2*(-1 + c*x)*(b + 2*a*c + b*c*x)*Log[1 - c*x])*PolyLog[2, c*x] - 1
6*(b + 2*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x] + 16*b*PolyLog[3, 1 - c*x] + 32*a*c*PolyLog[3, 1 - c*x])/(16*c^
2)

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Maple [F]  time = 0.005, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) \ln \left ( -cx+1 \right ){\it polylog} \left ( 2,cx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*ln(-c*x+1)*polylog(2,c*x),x)

[Out]

int((b*x+a)*ln(-c*x+1)*polylog(2,c*x),x)

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Maxima [A]  time = 1.00725, size = 348, normalized size = 0.89 \begin{align*} -\frac{1}{16} \, c{\left (\frac{8 \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \,{\rm Li}_{3}(-c x + 1)\right )}{\left (2 \, a c + b\right )}}{c^{3}} - \frac{3 \, b c^{2} x^{2} + 2 \,{\left (24 \, a c^{2} + 11 \, b c\right )} x - 4 \,{\left (b c^{2} x^{2} + 2 \,{\left (2 \, a c^{2} + b c\right )} x + 2 \,{\left (2 \, a c + b\right )} \log \left (-c x + 1\right )\right )}{\rm Li}_2\left (c x\right ) - 2 \,{\left (2 \, b c^{2} x^{2} - 24 \, a c + 2 \,{\left (8 \, a c^{2} + 3 \, b c\right )} x - 11 \, b\right )} \log \left (-c x + 1\right )}{c^{3}}\right )} + \frac{1}{8} \,{\left (\frac{8 \,{\left (c x{\rm Li}_2\left (c x\right ) - c x +{\left (c x - 1\right )} \log \left (-c x + 1\right )\right )} a}{c} + \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 )} b}{c^{2}}\right )} \log \left (-c x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*c*(8*(log(c*x)*log(-c*x + 1)^2 + 2*dilog(-c*x + 1)*log(-c*x + 1) - 2*polylog(3, -c*x + 1))*(2*a*c + b)/c
^3 - (3*b*c^2*x^2 + 2*(24*a*c^2 + 11*b*c)*x - 4*(b*c^2*x^2 + 2*(2*a*c^2 + b*c)*x + 2*(2*a*c + b)*log(-c*x + 1)
)*dilog(c*x) - 2*(2*b*c^2*x^2 - 24*a*c + 2*(8*a*c^2 + 3*b*c)*x - 11*b)*log(-c*x + 1))/c^3) + 1/8*(8*(c*x*dilog
(c*x) - c*x + (c*x - 1)*log(-c*x + 1))*a/c + (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))*b/c^2)*log(-c*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)*dilog(c*x)*log(-c*x + 1), 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((b*x+a)*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 (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)*dilog(c*x)*log(-c*x + 1), x)