3.2.74 \(\int \frac {(g+h \log (1-c x)) \text {PolyLog}(2,c x)}{x^2} \, dx\) [174]

Optimal. Leaf size=156 \[ c h \log (c x) \log ^2(1-c x)+\frac {\log (1-c x) (g+h \log (1-c x))}{x}+c (g+2 h \log (1-c x)) \log \left (1-\frac {1}{1-c x}\right )+c h \log (1-c x) \text {PolyLog}(2,c x)-\frac {(g+h \log (1-c x)) \text {PolyLog}(2,c x)}{x}-2 c h \text {PolyLog}\left (2,\frac {1}{1-c x}\right )+2 c h \log (1-c x) \text {PolyLog}(2,1-c x)-c h \text {PolyLog}(3,c x)-2 c h \text {PolyLog}(3,1-c x) \]

[Out]

c*h*ln(c*x)*ln(-c*x+1)^2+ln(-c*x+1)*(g+h*ln(-c*x+1))/x+c*(g+2*h*ln(-c*x+1))*ln(1-1/(-c*x+1))+c*h*ln(-c*x+1)*po
lylog(2,c*x)-(g+h*ln(-c*x+1))*polylog(2,c*x)/x-2*c*h*polylog(2,1/(-c*x+1))+2*c*h*ln(-c*x+1)*polylog(2,-c*x+1)-
c*h*polylog(3,c*x)-2*c*h*polylog(3,-c*x+1)

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Rubi [A]
time = 0.18, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6738, 2483, 2458, 2379, 2438, 6724, 6731, 2443, 2481, 2421} \begin {gather*} -\frac {\text {Li}_2(c x) (h \log (1-c x)+g)}{x}+\frac {\log (1-c x) (h \log (1-c x)+g)}{x}+c \log \left (1-\frac {1}{1-c x}\right ) (2 h \log (1-c x)+g)-2 c h \text {Li}_2\left (\frac {1}{1-c x}\right )-c h \text {Li}_3(c x)-2 c h \text {Li}_3(1-c x)+c h \text {Li}_2(c x) \log (1-c x)+2 c h \text {Li}_2(1-c x) \log (1-c x)+c h \log (c x) \log ^2(1-c x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2379

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

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 2438

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 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 2458

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 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 2483

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

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 6738

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]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 150, normalized size = 0.96 \begin {gather*} \frac {g (c x \log (x)+(1-c x) \log (1-c x)-\text {PolyLog}(2,c x))}{x}+h \left (2 c \log (c x) \log (1-c x)-c \log ^2(1-c x)+\frac {\log ^2(1-c x)}{x}+c \log (c x) \log ^2(1-c x)+\frac {(-1+c x) \log (1-c x) \text {PolyLog}(2,c x)}{x}+2 c (1+\log (1-c x)) \text {PolyLog}(2,1-c x)-c \text {PolyLog}(3,c x)-2 c \text {PolyLog}(3,1-c x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (g +h \ln \left (-c x +1\right )\right ) \polylog \left (2, c x \right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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