3.2.76 \(\int \frac {(g+h \log (1-c x)) \text {PolyLog}(2,c x)}{x^4} \, dx\) [176]

Optimal. Leaf size=340 \[ \frac {7 c^2 h}{36 x}-\frac {3}{4} c^3 h \log (x)+\frac {19}{36} c^3 h \log (1-c x)-\frac {c h \log (1-c x)}{12 x^2}-\frac {c^2 h \log (1-c x)}{3 x}+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {c (g+2 h \log (1-c x))}{18 x^2}-\frac {c^2 (1-c x) (g+2 h \log (1-c x))}{9 x}+\frac {1}{9} c^3 (g+2 h \log (1-c x)) \log \left (1-\frac {1}{1-c x}\right )+\frac {c h \text {PolyLog}(2,c x)}{6 x^2}+\frac {c^2 h \text {PolyLog}(2,c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {PolyLog}(2,c x)-\frac {(g+h \log (1-c x)) \text {PolyLog}(2,c x)}{3 x^3}-\frac {2}{9} c^3 h \text {PolyLog}\left (2,\frac {1}{1-c x}\right )+\frac {2}{3} c^3 h \log (1-c x) \text {PolyLog}(2,1-c x)-\frac {1}{3} c^3 h \text {PolyLog}(3,c x)-\frac {2}{3} c^3 h \text {PolyLog}(3,1-c x) \]

[Out]

7/36*c^2*h/x-3/4*c^3*h*ln(x)+19/36*c^3*h*ln(-c*x+1)-1/12*c*h*ln(-c*x+1)/x^2-1/3*c^2*h*ln(-c*x+1)/x+1/3*c^3*h*l
n(c*x)*ln(-c*x+1)^2+1/9*ln(-c*x+1)*(g+h*ln(-c*x+1))/x^3-1/18*c*(g+2*h*ln(-c*x+1))/x^2-1/9*c^2*(-c*x+1)*(g+2*h*
ln(-c*x+1))/x+1/9*c^3*(g+2*h*ln(-c*x+1))*ln(1-1/(-c*x+1))+1/6*c*h*polylog(2,c*x)/x^2+1/3*c^2*h*polylog(2,c*x)/
x+1/3*c^3*h*ln(-c*x+1)*polylog(2,c*x)-1/3*(g+h*ln(-c*x+1))*polylog(2,c*x)/x^3-2/9*c^3*h*polylog(2,1/(-c*x+1))+
2/3*c^3*h*ln(-c*x+1)*polylog(2,-c*x+1)-1/3*c^3*h*polylog(3,c*x)-2/3*c^3*h*polylog(3,-c*x+1)

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 19, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.950, Rules used = {6738, 2483, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46, 6726, 2442, 36, 29, 6724, 6731, 2443, 2481, 2421} \begin {gather*} \frac {1}{9} c^3 \log \left (1-\frac {1}{1-c x}\right ) (2 h \log (1-c x)+g)-\frac {2}{9} c^3 h \text {Li}_2\left (\frac {1}{1-c x}\right )-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {2}{3} c^3 h \text {Li}_3(1-c x)+\frac {1}{3} c^3 h \text {Li}_2(c x) \log (1-c x)+\frac {2}{3} c^3 h \text {Li}_2(1-c x) \log (1-c x)+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac {3}{4} c^3 h \log (x)+\frac {19}{36} c^3 h \log (1-c x)-\frac {c^2 (1-c x) (2 h \log (1-c x)+g)}{9 x}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {7 c^2 h}{36 x}-\frac {c^2 h \log (1-c x)}{3 x}-\frac {\text {Li}_2(c x) (h \log (1-c x)+g)}{3 x^3}+\frac {\log (1-c x) (h \log (1-c x)+g)}{9 x^3}-\frac {c (2 h \log (1-c x)+g)}{18 x^2}+\frac {c h \text {Li}_2(c x)}{6 x^2}-\frac {c h \log (1-c x)}{12 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*c^2*h)/(36*x) - (3*c^3*h*Log[x])/4 + (19*c^3*h*Log[1 - c*x])/36 - (c*h*Log[1 - c*x])/(12*x^2) - (c^2*h*Log[
1 - c*x])/(3*x) + (c^3*h*Log[c*x]*Log[1 - c*x]^2)/3 + (Log[1 - c*x]*(g + h*Log[1 - c*x]))/(9*x^3) - (c*(g + 2*
h*Log[1 - c*x]))/(18*x^2) - (c^2*(1 - c*x)*(g + 2*h*Log[1 - c*x]))/(9*x) + (c^3*(g + 2*h*Log[1 - c*x])*Log[1 -
(1 - c*x)^(-1)])/9 + (c*h*PolyLog[2, c*x])/(6*x^2) + (c^2*h*PolyLog[2, c*x])/(3*x) + (c^3*h*Log[1 - c*x]*Poly
Log[2, c*x])/3 - ((g + h*Log[1 - c*x])*PolyLog[2, c*x])/(3*x^3) - (2*c^3*h*PolyLog[2, (1 - c*x)^(-1)])/9 + (2*
c^3*h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/3 - (c^3*h*PolyLog[3, c*x])/3 - (2*c^3*h*PolyLog[3, 1 - c*x])/3

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 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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 2442

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 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 6726

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 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^4} \, dx &=-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}-\frac {1}{3} \int \frac {\log (1-c x) (g+h \log (1-c x))}{x^4} \, dx-\frac {1}{3} (c h) \int \left (\frac {\text {Li}_2(c x)}{x^3}+\frac {c \text {Li}_2(c x)}{x^2}+\frac {c^2 \text {Li}_2(c x)}{x}-\frac {c^3 \text {Li}_2(c x)}{-1+c x}\right ) \, dx\\ &=\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}+\frac {1}{9} c \int \frac {g+h \log (1-c x)}{x^3 (1-c x)} \, dx+\frac {1}{9} (c h) \int \frac {\log (1-c x)}{x^3 (1-c x)} \, dx-\frac {1}{3} (c h) \int \frac {\text {Li}_2(c x)}{x^3} \, dx-\frac {1}{3} \left (c^2 h\right ) \int \frac {\text {Li}_2(c x)}{x^2} \, dx-\frac {1}{3} \left (c^3 h\right ) \int \frac {\text {Li}_2(c x)}{x} \, dx+\frac {1}{3} \left (c^4 h\right ) \int \frac {\text {Li}_2(c x)}{-1+c x} \, dx\\ &=\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}+\frac {c h \text {Li}_2(c x)}{6 x^2}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {Li}_2(c x)-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {1}{9} \text {Subst}\left (\int \frac {g+h \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-c x\right )+\frac {1}{9} (c h) \int \left (\frac {\log (1-c x)}{x^3}+\frac {c \log (1-c x)}{x^2}+\frac {c^2 \log (1-c x)}{x}-\frac {c^3 \log (1-c x)}{-1+c x}\right ) \, dx+\frac {1}{6} (c h) \int \frac {\log (1-c x)}{x^3} \, dx+\frac {1}{3} \left (c^2 h\right ) \int \frac {\log (1-c x)}{x^2} \, dx+\frac {1}{3} \left (c^3 h\right ) \int \frac {\log ^2(1-c x)}{x} \, dx\\ &=-\frac {c h \log (1-c x)}{12 x^2}-\frac {c^2 h \log (1-c x)}{3 x}+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}+\frac {c h \text {Li}_2(c x)}{6 x^2}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {Li}_2(c x)-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {1}{9} \text {Subst}\left (\int \frac {g+h \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-c x\right )-\frac {1}{9} c \text {Subst}\left (\int \frac {g+h \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x\right )+\frac {1}{9} (c h) \int \frac {\log (1-c x)}{x^3} \, dx-\frac {1}{12} \left (c^2 h\right ) \int \frac {1}{x^2 (1-c x)} \, dx+\frac {1}{9} \left (c^2 h\right ) \int \frac {\log (1-c x)}{x^2} \, dx+\frac {1}{9} \left (c^3 h\right ) \int \frac {\log (1-c x)}{x} \, dx-\frac {1}{3} \left (c^3 h\right ) \int \frac {1}{x (1-c x)} \, dx-\frac {1}{9} \left (c^4 h\right ) \int \frac {\log (1-c x)}{-1+c x} \, dx+\frac {1}{3} \left (2 c^4 h\right ) \int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx\\ &=-\frac {5 c h \log (1-c x)}{36 x^2}-\frac {4 c^2 h \log (1-c x)}{9 x}+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac {c (g+h \log (1-c x))}{18 x^2}+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {1}{9} c^3 h \text {Li}_2(c x)+\frac {c h \text {Li}_2(c x)}{6 x^2}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {Li}_2(c x)-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {1}{9} c \text {Subst}\left (\int \frac {g+h \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x\right )-\frac {1}{9} c^2 \text {Subst}\left (\int \frac {g+h \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-c x\right )+\frac {1}{18} (c h) \text {Subst}\left (\int \frac {1}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-c x\right )-\frac {1}{18} \left (c^2 h\right ) \int \frac {1}{x^2 (1-c x)} \, dx-\frac {1}{12} \left (c^2 h\right ) \int \left (\frac {1}{x^2}+\frac {c}{x}-\frac {c^2}{-1+c x}\right ) \, dx-\frac {1}{9} \left (c^3 h\right ) \int \frac {1}{x (1-c x)} \, dx-\frac {1}{9} \left (c^3 h\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c x\right )-\frac {1}{3} \left (c^3 h\right ) \int \frac {1}{x} \, dx-\frac {1}{3} \left (2 c^3 h\right ) \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 )-\frac {1}{3} \left (c^4 h\right ) \int \frac {1}{1-c x} \, dx\\ &=\frac {c^2 h}{12 x}-\frac {5}{12} c^3 h \log (x)+\frac {5}{12} c^3 h \log (1-c x)-\frac {5 c h \log (1-c x)}{36 x^2}-\frac {4 c^2 h \log (1-c x)}{9 x}-\frac {1}{18} c^3 h \log ^2(1-c x)+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac {c (g+h \log (1-c x))}{18 x^2}-\frac {c^2 (1-c x) (g+h \log (1-c x))}{9 x}+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {1}{9} c^3 h \text {Li}_2(c x)+\frac {c h \text {Li}_2(c x)}{6 x^2}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {Li}_2(c x)-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}+\frac {2}{3} c^3 h \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {1}{9} c^2 \text {Subst}\left (\int \frac {g+h \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x\right )-\frac {1}{9} c^3 \text {Subst}\left (\int \frac {g+h \log (x)}{x} \, dx,x,1-c x\right )+\frac {1}{18} (c h) \text {Subst}\left (\int \left (\frac {c^2}{(-1+x)^2}-\frac {c^2}{-1+x}+\frac {c^2}{x}\right ) \, dx,x,1-c x\right )-\frac {1}{18} \left (c^2 h\right ) \int \left (\frac {1}{x^2}+\frac {c}{x}-\frac {c^2}{-1+c x}\right ) \, dx+\frac {1}{9} \left (c^2 h\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x\right )-\frac {1}{9} \left (c^3 h\right ) \int \frac {1}{x} \, dx-\frac {1}{3} \left (2 c^3 h\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x\right )-\frac {1}{9} \left (c^4 h\right ) \int \frac {1}{1-c x} \, dx\\ &=\frac {7 c^2 h}{36 x}+\frac {1}{9} c^3 g \log (x)-\frac {3}{4} c^3 h \log (x)+\frac {23}{36} c^3 h \log (1-c x)-\frac {5 c h \log (1-c x)}{36 x^2}-\frac {4 c^2 h \log (1-c x)}{9 x}-\frac {1}{18} c^3 h \log ^2(1-c x)+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac {c (g+h \log (1-c x))}{18 x^2}-\frac {c^2 (1-c x) (g+h \log (1-c x))}{9 x}+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {c^3 (g+h \log (1-c x))^2}{18 h}-\frac {1}{9} c^3 h \text {Li}_2(c x)+\frac {c h \text {Li}_2(c x)}{6 x^2}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {Li}_2(c x)-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}+\frac {2}{3} c^3 h \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {2}{3} c^3 h \text {Li}_3(1-c x)-\frac {1}{9} \left (c^2 h\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-c x\right )\\ &=\frac {7 c^2 h}{36 x}+\frac {1}{9} c^3 g \log (x)-\frac {3}{4} c^3 h \log (x)+\frac {23}{36} c^3 h \log (1-c x)-\frac {5 c h \log (1-c x)}{36 x^2}-\frac {4 c^2 h \log (1-c x)}{9 x}-\frac {1}{18} c^3 h \log ^2(1-c x)+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)-\frac {c (g+h \log (1-c x))}{18 x^2}-\frac {c^2 (1-c x) (g+h \log (1-c x))}{9 x}+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {c^3 (g+h \log (1-c x))^2}{18 h}-\frac {2}{9} c^3 h \text {Li}_2(c x)+\frac {c h \text {Li}_2(c x)}{6 x^2}+\frac {c^2 h \text {Li}_2(c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \text {Li}_2(c x)-\frac {(g+h \log (1-c x)) \text {Li}_2(c x)}{3 x^3}+\frac {2}{3} c^3 h \log (1-c x) \text {Li}_2(1-c x)-\frac {1}{3} c^3 h \text {Li}_3(c x)-\frac {2}{3} c^3 h \text {Li}_3(1-c x)\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 301, normalized size = 0.89 \begin {gather*} -\frac {g \left (c x (1+2 c x)-2 c^3 x^3 \log (x)+2 \left (-1+c^3 x^3\right ) \log (1-c x)+6 \text {PolyLog}(2,c x)\right )}{18 x^3}+\frac {h \left (7 c^2 x^2-4 c^3 x^3-15 c^3 x^3 \log (x)-12 c^3 x^3 \log (c x)-7 c x \log (1-c x)-20 c^2 x^2 \log (1-c x)+27 c^3 x^3 \log (1-c x)+8 c^3 x^3 \log (c x) \log (1-c x)+4 \log ^2(1-c x)-4 c^3 x^3 \log ^2(1-c x)+12 c^3 x^3 \log (c x) \log ^2(1-c x)+6 \left (c x (1+2 c x)+2 \left (-1+c^3 x^3\right ) \log (1-c x)\right ) \text {PolyLog}(2,c x)+8 c^3 x^3 (1+3 \log (1-c x)) \text {PolyLog}(2,1-c x)-12 c^3 x^3 \text {PolyLog}(3,c x)-24 c^3 x^3 \text {PolyLog}(3,1-c x)\right )}{36 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*(g*(c*x*(1 + 2*c*x) - 2*c^3*x^3*Log[x] + 2*(-1 + c^3*x^3)*Log[1 - c*x] + 6*PolyLog[2, c*x]))/x^3 + (h*(7
*c^2*x^2 - 4*c^3*x^3 - 15*c^3*x^3*Log[x] - 12*c^3*x^3*Log[c*x] - 7*c*x*Log[1 - c*x] - 20*c^2*x^2*Log[1 - c*x]
+ 27*c^3*x^3*Log[1 - c*x] + 8*c^3*x^3*Log[c*x]*Log[1 - c*x] + 4*Log[1 - c*x]^2 - 4*c^3*x^3*Log[1 - c*x]^2 + 12
*c^3*x^3*Log[c*x]*Log[1 - c*x]^2 + 6*(c*x*(1 + 2*c*x) + 2*(-1 + c^3*x^3)*Log[1 - c*x])*PolyLog[2, c*x] + 8*c^3
*x^3*(1 + 3*Log[1 - c*x])*PolyLog[2, 1 - c*x] - 12*c^3*x^3*PolyLog[3, c*x] - 24*c^3*x^3*PolyLog[3, 1 - c*x]))/
(36*x^3)

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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\left (g +h \ln \left (-c x +1\right )\right ) \polylog \left (2, c x \right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g+h*ln(-c*x+1))*polylog(2,c*x)/x^4,x)

[Out]

int((g+h*ln(-c*x+1))*polylog(2,c*x)/x^4,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^4,x, algorithm="maxima")

[Out]

1/18*(2*c^3*log(x) - (2*c^2*x^2 + c*x + 2*(c^3*x^3 - 1)*log(-c*x + 1) + 6*dilog(c*x))/x^3)*g + h*integrate(dil
og(c*x)*log(-c*x + 1)/x^4, 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^4,x, algorithm="fricas")

[Out]

integral((h*dilog(c*x)*log(-c*x + 1) + g*dilog(c*x))/x^4, 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^{4}}\, 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**4,x)

[Out]

Integral((g + h*log(-c*x + 1))*polylog(2, c*x)/x**4, 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^4,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (g+h\,\ln \left (1-c\,x\right )\right )\,\mathrm {polylog}\left (2,c\,x\right )}{x^4} \,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^4,x)

[Out]

int(((g + h*log(1 - c*x))*polylog(2, c*x))/x^4, x)

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