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

Optimal. Leaf size=131 \[ -2 a c \text{PolyLog}(2,c x)-a c \text{PolyLog}(3,c x)-2 a c \text{PolyLog}(3,1-c x)+a c \log (1-c x) \text{PolyLog}(2,c x)-\frac{a \log (1-c x) \text{PolyLog}(2,c x)}{x}+2 a c \log (1-c x) \text{PolyLog}(2,1-c x)-\frac{1}{2} b \text{PolyLog}(2,c x)^2+\frac{a (1-c x) \log ^2(1-c x)}{x}+a c \log (c x) \log ^2(1-c x) \]

[Out]

(a*(1 - c*x)*Log[1 - c*x]^2)/x + a*c*Log[c*x]*Log[1 - c*x]^2 - 2*a*c*PolyLog[2, c*x] + a*c*Log[1 - c*x]*PolyLo
g[2, c*x] - (a*Log[1 - c*x]*PolyLog[2, c*x])/x - (b*PolyLog[2, c*x]^2)/2 + 2*a*c*Log[1 - c*x]*PolyLog[2, 1 - c
*x] - a*c*PolyLog[3, c*x] - 2*a*c*PolyLog[3, 1 - c*x]

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Rubi [A]  time = 0.313536, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 17, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.81, Rules used = {6742, 6591, 2395, 36, 29, 31, 6589, 6605, 6601, 12, 6603, 2397, 2391, 6596, 2396, 2433, 2374} \[ -2 a c \text{PolyLog}(2,c x)-a c \text{PolyLog}(3,c x)-2 a c \text{PolyLog}(3,1-c x)+a c \log (1-c x) \text{PolyLog}(2,c x)-\frac{a \log (1-c x) \text{PolyLog}(2,c x)}{x}+2 a c \log (1-c x) \text{PolyLog}(2,1-c x)-\frac{1}{2} b \text{PolyLog}(2,c x)^2+\frac{a (1-c x) \log ^2(1-c x)}{x}+a c \log (c x) \log ^2(1-c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(1 - c*x)*Log[1 - c*x]^2)/x + a*c*Log[c*x]*Log[1 - c*x]^2 - 2*a*c*PolyLog[2, c*x] + a*c*Log[1 - c*x]*PolyLo
g[2, c*x] - (a*Log[1 - c*x]*PolyLog[2, c*x])/x - (b*PolyLog[2, c*x]^2)/2 + 2*a*c*Log[1 - c*x]*PolyLog[2, 1 - c
*x] - a*c*PolyLog[3, c*x] - 2*a*c*PolyLog[3, 1 - c*x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 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 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 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 6605

Int[((g_.) + Log[1 + (e_.)*(x_)]*(h_.))*(Px_)*(x_)^(m_)*PolyLog[2, (c_.)*(x_)], x_Symbol] :> Dist[Coeff[Px, x,
-m - 1], Int[((g + h*Log[1 + e*x])*PolyLog[2, c*x])/x, x], x] + Int[x^m*(Px - Coeff[Px, x, -m - 1]*x^(-m - 1)
)*(g + h*Log[1 + e*x])*PolyLog[2, c*x], x] /; FreeQ[{c, e, g, h}, x] && PolyQ[Px, x] && ILtQ[m, 0] && EqQ[c +
e, 0] && NeQ[Coeff[Px, x, -m - 1], 0]

Rule 6601

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

Rule 12

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

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 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

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

Mathematica [A]  time = 0.713885, size = 135, normalized size = 1.03 \[ -a c \text{PolyLog}(3,c x)-2 a c \text{PolyLog}(3,1-c x)+\frac{a (c x-1) \log (1-c x) \text{PolyLog}(2,c x)}{x}+2 a c (\log (1-c x)+1) \text{PolyLog}(2,1-c x)-\frac{1}{2} b \text{PolyLog}(2,c x)^2-a c \log ^2(1-c x)+a c \log (c x) \log ^2(1-c x)+\frac{a \log ^2(1-c x)}{x}+2 a c \log (c x) \log (1-c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.008, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) \ln \left ( -cx+1 \right ){\it polylog} \left ( 2,cx \right ) }{{x}^{2}}}\, 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^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}\,{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^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}, 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^2,x, algorithm="fricas")

[Out]

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

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

[Out]

Integral((a + b*x)*log(-c*x + 1)*polylog(2, c*x)/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}{\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}}\,{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^2,x, algorithm="giac")

[Out]

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