3.2.64 \(\int \log (1-c x) \text {PolyLog}(2,c x) \, dx\) [164]

Optimal. Leaf size=132 \[ 3 x+\frac {3 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-\frac {\log (c x) \log ^2(1-c x)}{c}-x \text {PolyLog}(2,c x)-\frac {\log (1-c x) \text {PolyLog}(2,c x)}{c}+x \log (1-c x) \text {PolyLog}(2,c x)-\frac {2 \log (1-c x) \text {PolyLog}(2,1-c x)}{c}+\frac {2 \text {PolyLog}(3,1-c x)}{c} \]

[Out]

3*x+3*(-c*x+1)*ln(-c*x+1)/c-(-c*x+1)*ln(-c*x+1)^2/c-ln(c*x)*ln(-c*x+1)^2/c-x*polylog(2,c*x)-ln(-c*x+1)*polylog
(2,c*x)/c+x*ln(-c*x+1)*polylog(2,c*x)-2*ln(-c*x+1)*polylog(2,-c*x+1)/c+2*polylog(3,-c*x+1)/c

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Rubi [A]
time = 0.14, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {6721, 2436, 2332, 6735, 2333, 6820, 6874, 6731, 2443, 2481, 2421, 6724} \begin {gather*} -x \text {Li}_2(c x)+\frac {2 \text {Li}_3(1-c x)}{c}+x \text {Li}_2(c x) \log (1-c x)-\frac {\text {Li}_2(c x) \log (1-c x)}{c}-\frac {2 \text {Li}_2(1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-\frac {\log (c x) \log ^2(1-c x)}{c}+\frac {3 (1-c x) \log (1-c x)}{c}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[1 - c*x]*PolyLog[2, c*x],x]

[Out]

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

Rule 2332

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

Rule 2333

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2436

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 2443

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 2481

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 6721

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 6724

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 6731

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 6735

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

Rule 6820

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

Rule 6874

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

Rubi steps

\begin {align*} \int \log (1-c x) \text {Li}_2(c x) \, dx &=x \log (1-c x) \text {Li}_2(c x)+c \int \left (-\frac {1}{c}-\frac {1}{c (-1+c x)}\right ) \text {Li}_2(c x) \, dx+\int \log ^2(1-c x) \, dx\\ &=x \log (1-c x) \text {Li}_2(c x)-\frac {\text {Subst}\left (\int \log ^2(x) \, dx,x,1-c x\right )}{c}+c \int \frac {x \text {Li}_2(c x)}{1-c x} \, dx\\ &=-\frac {(1-c x) \log ^2(1-c x)}{c}+x \log (1-c x) \text {Li}_2(c x)+\frac {2 \text {Subst}(\int \log (x) \, dx,x,1-c x)}{c}+c \int \left (-\frac {\text {Li}_2(c x)}{c}-\frac {\text {Li}_2(c x)}{c (-1+c x)}\right ) \, dx\\ &=2 x+\frac {2 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}+x \log (1-c x) \text {Li}_2(c x)-\int \text {Li}_2(c x) \, dx-\int \frac {\text {Li}_2(c x)}{-1+c x} \, dx\\ &=2 x+\frac {2 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-x \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{c}+x \log (1-c x) \text {Li}_2(c x)-\frac {\int \frac {\log ^2(1-c x)}{x} \, dx}{c}-\int \log (1-c x) \, dx\\ &=2 x+\frac {2 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-\frac {\log (c x) \log ^2(1-c x)}{c}-x \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{c}+x \log (1-c x) \text {Li}_2(c x)-2 \int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx+\frac {\text {Subst}(\int \log (x) \, dx,x,1-c x)}{c}\\ &=3 x+\frac {3 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-\frac {\log (c x) \log ^2(1-c x)}{c}-x \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{c}+x \log (1-c x) \text {Li}_2(c x)+\frac {2 \text {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}\\ &=3 x+\frac {3 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-\frac {\log (c x) \log ^2(1-c x)}{c}-x \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{c}+x \log (1-c x) \text {Li}_2(c x)-\frac {2 \log (1-c x) \text {Li}_2(1-c x)}{c}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x\right )}{c}\\ &=3 x+\frac {3 (1-c x) \log (1-c x)}{c}-\frac {(1-c x) \log ^2(1-c x)}{c}-\frac {\log (c x) \log ^2(1-c x)}{c}-x \text {Li}_2(c x)-\frac {\log (1-c x) \text {Li}_2(c x)}{c}+x \log (1-c x) \text {Li}_2(c x)-\frac {2 \log (1-c x) \text {Li}_2(1-c x)}{c}+\frac {2 \text {Li}_3(1-c x)}{c}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 119, normalized size = 0.90 \begin {gather*} \frac {-2+3 c x+3 \log (1-c x)-3 c x \log (1-c x)-\log ^2(1-c x)+c x \log ^2(1-c x)-\log (c x) \log ^2(1-c x)+(-c x+(-1+c x) \log (1-c x)) \text {PolyLog}(2,c x)-2 \log (1-c x) \text {PolyLog}(2,1-c x)+2 \text {PolyLog}(3,1-c x)}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[1 - c*x]*PolyLog[2, c*x],x]

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \ln \left (-c x +1\right ) \polylog \left (2, c x \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 141, normalized size = 1.07 \begin {gather*} c {\left (\frac {x}{c} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} + \frac {{\left (c x {\rm Li}_2\left (c x\right ) - c x + {\left (c x - 1\right )} \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{c} - \frac {\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)}{c} + \frac {2 \, c x - {\left (c x + \log \left (-c x + 1\right )\right )} {\rm Li}_2\left (c x\right ) - 2 \, {\left (c x - 1\right )} \log \left (-c x + 1\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(x/c + log(c*x - 1)/c^2) + (c*x*dilog(c*x) - c*x + (c*x - 1)*log(-c*x + 1))*log(-c*x + 1)/c - (log(c*x)*log(
-c*x + 1)^2 + 2*dilog(-c*x + 1)*log(-c*x + 1) - 2*polylog(3, -c*x + 1))/c + (2*c*x - (c*x + log(-c*x + 1))*dil
og(c*x) - 2*(c*x - 1)*log(-c*x + 1))/c

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(dilog(c*x)*log(-c*x + 1), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \log {\left (- c x + 1 \right )} \operatorname {Li}_{2}\left (c x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(1 - c*x)*polylog(2, c*x),x)

[Out]

int(log(1 - c*x)*polylog(2, c*x), x)

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