3.2.83 \(\int \frac {(g+h \log (f (d+e x)^n)) \text {PolyLog}(2,c (a+b x))}{x^4} \, dx\) [183]

Optimal. Leaf size=3733 \[ \text {result too large to display} \]

[Out]

-1/3*b^3*g*polylog(2,1-b*c*x/(-a*c+1))/a^3-1/3*b^3*g*polylog(2,c*(b*x+a))/a^3+1/6*b^2*e*h*n*polylog(2,c*(b*x+a
))/a^2/d+1/3*b*e^2*h*n*polylog(2,c*(b*x+a))/a/d^2+1/3*b*e*h*n*ln(-b*c*x-a*c+1)/a/d/x+1/2*b^2*e*h*n*ln(b*c*x/(-
a*c+1))*ln(-b*c*x-a*c+1)/a^2/d+1/2*b*e^2*h*n*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)/a/d^2-1/3*b^2*e*h*n*ln(-b*c*x
-a*c+1)*ln(b*c*(e*x+d)/(-a*c*e+b*c*d+e))/a^2/d-1/6*b*e^2*h*n*ln(-b*c*x-a*c+1)*ln(b*c*(e*x+d)/(-a*c*e+b*c*d+e))
/a/d^2+1/3*b^3*h*n*polylog(3,-e*(-b*c*x-a*c+1)/b/c/(e*x+d))/a^3-1/3*b^3*h*n*polylog(3,b*(e*x+d)/(-a*e+b*d))/a^
3+1/3*e^3*h*n*polylog(3,b*(e*x+d)/(-a*e+b*d))/d^3+1/3*b^3*h*n*polylog(3,1+e*x/d)/a^3+1/3*e^3*h*n*polylog(3,-b*
x/a/(1-c*(b*x+a)))/d^3-1/3*e^3*h*n*polylog(3,-b*c*x/(1-c*(b*x+a)))/d^3-1/3*b^3*h*n*polylog(3,1-c*(b*x+a))/a^3-
1/3*b^3*h*n*polylog(3,-e*(1-c*(b*x+a))/b/c/(e*x+d))/a^3+1/3*e^3*h*n*polylog(3,-e*(1-c*(b*x+a))/b/c/(e*x+d))/d^
3+1/3*b^3*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/a^3-1/3*e^3*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a))
/b/(e*x+d))/d^3-1/3*e^3*h*n*polylog(3,-b*x/a)/d^3+1/3*b^3*h*n*polylog(3,1-b*c*x/(-a*c+1))/a^3-1/3*b^3*h*n*poly
log(3,d*(-b*c*x-a*c+1)/(-a*c+1)/(e*x+d))/a^3+1/2*b^2*c*e*h*n*ln(x)/a/(-a*c+1)/d-1/3*b^2*c*e*h*n*ln(-b*c*x-a*c+
1)/a/(-a*c+1)/d-1/6*b^2*c*e*h*n*ln(e*x+d)/a/(-a*c+1)/d-1/3*(g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^3-1/3*
b^3*h*n*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)*ln(e*x+d)/a^3+1/3*e^3*h*n*ln(x)*ln(1+b*x/a)*ln(1-c*(b*x+a))/d^3+1/
3*b^3*h*n*ln(c*(b*x+a))*ln(e*x+d)*ln(1-c*(b*x+a))/a^3-1/3*e^3*h*n*ln(c*(b*x+a))*ln(e*x+d)*ln(1-c*(b*x+a))/d^3-
1/3*b^3*g*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)/a^3+1/6*b*ln(-b*c*x-a*c+1)*(g+h*ln(f*(e*x+d)^n))/a/x^2-1/3*b^2*l
n(-b*c*x-a*c+1)*(g+h*ln(f*(e*x+d)^n))/a^2/x+1/3*b^3*h*(n*ln(e*x+d)-ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/a^3+1
/3*b^3*h*(n*ln(e*x+d)-ln(f*(e*x+d)^n))*polylog(2,1-b*c*x/(-a*c+1))/a^3-1/3*e^3*h*n*ln(b*(e*x+d)/(-a*e+b*d)/(1-
c*(b*x+a)))*polylog(2,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/d^3-1/6*b^3*h*n*(ln(b*c*x/(-a*c+1))+ln((-a*c*e+b*c*d
+e)/b/c/(e*x+d))-ln((-a*c*e+b*c*d+e)*x/(-a*c+1)/(e*x+d)))*ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1))^2/a^3+1/6*b^3*
h*n*(ln(b*c*x/(-a*c+1))-ln(-e*x/d))*(ln(-b*c*x-a*c+1)+ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1)))^2/a^3+1/3*b^3*h*l
n(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)*(n*ln(e*x+d)-ln(f*(e*x+d)^n))/a^3-1/6*b^2*c*(g+h*ln(f*(e*x+d)^n))/a/(-a*c+1
)/x+1/6*b^3*c^2*ln(-e*x/d)*(g+h*ln(f*(e*x+d)^n))/a/(-a*c+1)^2-1/3*b^3*c*ln(-e*x/d)*(g+h*ln(f*(e*x+d)^n))/a^2/(
-a*c+1)-1/6*b^3*c^2*ln(e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))*(g+h*ln(f*(e*x+d)^n))/a/(-a*c+1)^2+1/3*b^3*c*ln(e*(-
b*c*x-a*c+1)/(-a*c*e+b*c*d+e))*(g+h*ln(f*(e*x+d)^n))/a^2/(-a*c+1)+1/6*b^3*h*n*(ln(c*(b*x+a))+ln((-a*c*e+b*c*d+
e)/b/c/(e*x+d))-ln((-a*c*e+b*c*d+e)*(b*x+a)/b/(e*x+d)))*ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))^2/a^3-1/6*e^3*h
*n*(ln(c*(b*x+a))+ln((-a*c*e+b*c*d+e)/b/c/(e*x+d))-ln((-a*c*e+b*c*d+e)*(b*x+a)/b/(e*x+d)))*ln(b*(e*x+d)/(-a*e+
b*d)/(1-c*(b*x+a)))^2/d^3-1/6*b^3*h*n*(ln(c*(b*x+a))-ln(-e*(b*x+a)/(-a*e+b*d)))*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*
(b*x+a)))+ln(1-c*(b*x+a)))^2/a^3+1/6*e^3*h*n*(ln(c*(b*x+a))-ln(-e*(b*x+a)/(-a*e+b*d)))*(ln(b*(e*x+d)/(-a*e+b*d
)/(1-c*(b*x+a)))+ln(1-c*(b*x+a)))^2/d^3+1/6*e^3*h*n*(ln(1+b*x/a)+ln((-a*c+1)/(1-c*(b*x+a)))-ln((-a*c+1)*(b*x+a
)/a/(1-c*(b*x+a))))*ln(-a*(1-c*(b*x+a))/b/x)^2/d^3+1/6*e^3*h*n*(ln(c*(b*x+a))-ln(1+b*x/a))*(ln(x)+ln(-a*(1-c*(
b*x+a))/b/x))^2/d^3+1/3*e^3*h*n*(ln(1-c*(b*x+a))-ln(-a*(1-c*(b*x+a))/b/x))*polylog(2,-b*x/a)/d^3+1/3*e^3*h*n*l
n(x)*polylog(2,c*(b*x+a))/d^3-1/3*e^3*h*n*ln(e*x+d)*polylog(2,c*(b*x+a))/d^3-1/3*b^3*h*n*(ln(e*x+d)-ln((-a*c+1
)*(e*x+d)/d/(-b*c*x-a*c+1)))*polylog(2,1-b*c*x/(-a*c+1))/a^3-1/3*b^3*h*n*ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1))
*polylog(2,d*(-b*c*x-a*c+1)/(-a*c+1)/(e*x+d))/a^3+1/3*b^3*h*n*ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1))*polylog(2,
-e*(-b*c*x-a*c+1)/b/c/(e*x+d))/a^3+1/3*b^3*h*n*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))+ln(1-c*(b*x+a)))*polylo
g(2,b*(e*x+d)/(-a*e+b*d))/a^3-1/3*e^3*h*n*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))+ln(1-c*(b*x+a)))*polylog(2,b
*(e*x+d)/(-a*e+b*d))/d^3-1/3*b^3*h*n*(ln(-b*c*x-a*c+1)+ln((-a*c+1)*(e*x+d)/d/(-b*c*x-a*c+1)))*polylog(2,1+e*x/
d)/a^3+1/3*e^3*h*n*ln(-a*(1-c*(b*x+a))/b/x)*polylog(2,-b*x/a/(1-c*(b*x+a)))/d^3-1/3*e^3*h*n*ln(-a*(1-c*(b*x+a)
)/b/x)*polylog(2,-b*c*x/(1-c*(b*x+a)))/d^3+1/3*b^3*h*n*(ln(e*x+d)-ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a))))*poly
log(2,1-c*(b*x+a))/a^3-1/3*e^3*h*n*(ln(e*x+d)-ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a))))*polylog(2,1-c*(b*x+a))/d
^3+1/3*e^3*h*n*(ln(x)+ln(-a*(1-c*(b*x+a))/b/x))*polylog(2,1-c*(b*x+a))/d^3-1/3*b^3*h*n*ln(b*(e*x+d)/(-a*e+b*d)
/(1-c*(b*x+a)))*polylog(2,-e*(1-c*(b*x+a))/b/c/(e*x+d))/a^3+1/3*e^3*h*n*ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))
*polylog(2,-e*(1-c*(b*x+a))/b/c/(e*x+d))/d^3+1/3*b^3*h*n*ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))*polylog(2,(-a*
e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/a^3-1/3*b^2*e*h*n*polylog(2,e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))/a^2/d-1/6*b*e^2
*h*n*polylog(2,e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))/a/d^2+1/2*b^2*e*h*n*polylog(2,1-b*c*x/(-a*c+1))/a^2/d+1/2*b*
e^2*h*n*polylog(2,1-b*c*x/(-a*c+1))/a/d^2-1/6*b^3*c^2*h*n*polylog(2,b*c*(e*x+d)/(-a*c*e+b*c*d+e))/a/(-a*c+1)^2
+1/3*b^3*c*h*n*polylog(2,b*c*(e*x+d)/(-a*c*e+b*c*d+e))/a^2/(-a*c+1)+1/6*b^3*c^2*h*n*polylog(2,1+e*x/d)/a/(-a*c
+1)^2-1/3*b^3*c*h*n*polylog(2,1+e*x/d)/a^2/(-a*c+1)-1/6*e*h*n*polylog(2,c*(b*x+a))/d/x^2+1/3*e^2*h*n*polylog(2
,c*(b*x+a))/d^2/x

________________________________________________________________________________________

Rubi [A]
time = 2.87, antiderivative size = 3733, normalized size of antiderivative = 1.00, number of steps used = 78, number of rules used = 18, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6738, 2489, 46, 2463, 2442, 36, 29, 31, 2441, 2352, 2440, 2438, 2488, 2487, 2485, 2490, 6733, 6732} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x^4,x]

[Out]

(b^2*c*e*h*n*Log[x])/(2*a*(1 - a*c)*d) - (b^2*c*e*h*n*Log[1 - a*c - b*c*x])/(3*a*(1 - a*c)*d) + (b*e*h*n*Log[1
- a*c - b*c*x])/(3*a*d*x) - (b^3*g*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(3*a^3) + (b^2*e*h*n*Log[(b*c
*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a^2*d) + (b*e^2*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a*
d^2) - (b^2*c*e*h*n*Log[d + e*x])/(6*a*(1 - a*c)*d) - (b^3*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*Log
[d + e*x])/(3*a^3) - (b^2*e*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*a^2*d) - (b*
e^2*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*a*d^2) - (b^3*h*n*(Log[(b*c*x)/(1 -
a*c)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*x)/((1 - a*c)*(d + e*x))])*Log[((1
- a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]^2)/(6*a^3) + (b^3*h*n*(Log[(b*c*x)/(1 - a*c)] - Log[-((e*x)/d)])*(Lo
g[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])^2)/(6*a^3) + (b^3*h*Log[(b*c*x)/(1 - a*
c)]*Log[1 - a*c - b*c*x]*(n*Log[d + e*x] - Log[f*(d + e*x)^n]))/(3*a^3) - (b^2*c*(g + h*Log[f*(d + e*x)^n]))/(
6*a*(1 - a*c)*x) + (b^3*c^2*Log[-((e*x)/d)]*(g + h*Log[f*(d + e*x)^n]))/(6*a*(1 - a*c)^2) - (b^3*c*Log[-((e*x)
/d)]*(g + h*Log[f*(d + e*x)^n]))/(3*a^2*(1 - a*c)) + (b*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(6*a*
x^2) - (b^2*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(3*a^2*x) - (b^3*c^2*Log[(e*(1 - a*c - b*c*x))/(b
*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(6*a*(1 - a*c)^2) + (b^3*c*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e
- a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(3*a^2*(1 - a*c)) + (b^3*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e
)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c
*(a + b*x)))]^2)/(6*a^3) - (e^3*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d
+ e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(6*d^3) + (e^3*
h*n*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/(3*d^3) + (b^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(
a + b*x)])/(3*a^3) - (e^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(3*d^3) - (b^3*h*n*(Log[c*(a
+ b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*
(a + b*x)])^2)/(6*a^3) + (e^3*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((
b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(6*d^3) + (e^3*h*n*(Log[1 + (b*x)/a] + Log[(1 - a*c)
/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)
/(6*d^3) + (e^3*h*n*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(6
*d^3) + (e^3*h*n*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/(3*d^3)
- (b^3*g*PolyLog[2, c*(a + b*x)])/(3*a^3) + (b^2*e*h*n*PolyLog[2, c*(a + b*x)])/(6*a^2*d) + (b*e^2*h*n*PolyLog
[2, c*(a + b*x)])/(3*a*d^2) - (e*h*n*PolyLog[2, c*(a + b*x)])/(6*d*x^2) + (e^2*h*n*PolyLog[2, c*(a + b*x)])/(3
*d^2*x) + (e^3*h*n*Log[x]*PolyLog[2, c*(a + b*x)])/(3*d^3) - (e^3*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/(3
*d^3) + (b^3*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(3*a^3) - ((g + h*Log[f*(d + e*x
)^n])*PolyLog[2, c*(a + b*x)])/(3*x^3) - (b^2*e*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(3*
a^2*d) - (b*e^2*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(6*a*d^2) - (b^3*g*PolyLog[2, 1 - (
b*c*x)/(1 - a*c)])/(3*a^3) + (b^2*e*h*n*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2*d) + (b*e^2*h*n*PolyLog[2, 1
- (b*c*x)/(1 - a*c)])/(2*a*d^2) - (b^3*h*n*(Log[d + e*x] - Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*
PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(3*a^3) + (b^3*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, 1 - (b*c*
x)/(1 - a*c)])/(3*a^3) - (b^3*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, (d*(1 - a*c - b*
c*x))/((1 - a*c)*(d + e*x))])/(3*a^3) + (b^3*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, -
((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/(3*a^3) + (b^3*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x))
)] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(3*a^3) - (e^3*h*n*(Log[(b*(d + e*x))/((b*d
- a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(3*d^3) - (b^3*c^2*h
*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*a*(1 - a*c)^2) + (b^3*c*h*n*PolyLog[2, (b*c*(d + e*x))/
(b*c*d + e - a*c*e)])/(3*a^2*(1 - a*c)) + (b^3*c^2*h*n*PolyLog[2, 1 + (e*x)/d])/(6*a*(1 - a*c)^2) - (b^3*c*h*n
*PolyLog[2, 1 + (e*x)/d])/(3*a^2*(1 - a*c)) - (b^3*h*n*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1
- a*c - b*c*x))])*PolyLog[2, 1 + (e*x)/d])/(3*a^3) + (e^3*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2,
-((b*x)/(a*(1 - c*(a + b*x))))])/(3*d^3) - (e^3*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(
1 - c*(a + b*x)))])/(3*d^3) + (b^3*h*n*(Log[d +...

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 2352

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

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 2440

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

Rule 2441

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

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 2463

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 2485

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[(-b)*(x/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-(b*c - a*d))*(x/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d
*x))])*Log[a*((c + d*x)/(c*(a + b*x)))]^2, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-d)*(x/c)])*(Log[a + b*x] +
Log[a*((c + d*x)/(c*(a + b*x)))])^2, x] + Simp[(Log[c + d*x] - Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1
+ b*(x/a)], x] + Simp[(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 + d*(x/c)], x] + Simp[Lo
g[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[Log[a*((c + d*x)/(c*(a + b*x
)))]*PolyLog[2, d*((a + b*x)/(b*(c + d*x)))], x] - Simp[PolyLog[3, 1 + b*(x/a)], x] - Simp[PolyLog[3, 1 + d*(x
/c)], x] + Simp[PolyLog[3, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[PolyLog[3, d*((a + b*x)/(b*(c + d*x)))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2487

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[Log[i + j*x]*(Log[c*(d + e*x)^n]/x), x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2488

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.) + (f_))
)/(x_), x_Symbol] :> Dist[f, Int[(a + b*Log[c*(d + e*x)^n])/x, x], x] + Dist[g, Int[Log[h*(i + j*x)^m]*((a + b
*Log[c*(d + e*x)^n])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]

Rule 2489

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1
)), x] + (-Dist[g*j*(m/(r + 1)), Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Dist[b*e*n*(
p/(r + 1)), Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*((f + g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /
; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && N
eQ[r, -1]

Rule 2490

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-(e*k - d*l)/l + e*(x/l))^n])*(
f + g*Log[h*(-(j*k - i*l)/l + j*(x/l))^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l,
m, n}, x] && IntegerQ[r]

Rule 6732

Int[PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*(PolyLog[2, c*
(a + b*x)]/e), x] + Dist[b/e, Int[Log[d + e*x]*(Log[1 - a*c - b*c*x]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d,
e}, x] && NeQ[c*(b*d - a*e) + e, 0]

Rule 6733

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 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 {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x^4} \, dx &=-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 x^3}-\frac {1}{3} b \int \left (\frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a x^3}-\frac {b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a^2 x^2}+\frac {b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a^3 x}-\frac {b^3 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a^3 (a+b x)}\right ) \, dx+\frac {1}{3} (e h n) \int \left (\frac {\text {Li}_2(c (a+b x))}{d x^3}-\frac {e \text {Li}_2(c (a+b x))}{d^2 x^2}+\frac {e^2 \text {Li}_2(c (a+b x))}{d^3 x}-\frac {e^3 \text {Li}_2(c (a+b x))}{d^3 (d+e x)}\right ) \, dx\\ &=-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 x^3}-\frac {b \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x^3} \, dx}{3 a}+\frac {b^2 \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x^2} \, dx}{3 a^2}-\frac {b^3 \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x} \, dx}{3 a^3}+\frac {b^4 \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a+b x} \, dx}{3 a^3}+\frac {(e h n) \int \frac {\text {Li}_2(c (a+b x))}{x^3} \, dx}{3 d}-\frac {\left (e^2 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{x^2} \, dx}{3 d^2}+\frac {\left (e^3 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{x} \, dx}{3 d^3}-\frac {\left (e^4 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{d+e x} \, dx}{3 d^3}\\ &=\frac {b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac {b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac {e h n \text {Li}_2(c (a+b x))}{6 d x^2}+\frac {e^2 h n \text {Li}_2(c (a+b x))}{3 d^2 x}+\frac {e^3 h n \log (x) \text {Li}_2(c (a+b x))}{3 d^3}-\frac {e^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 d^3}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 x^3}+\frac {b^3 \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \left (g+h \log \left (f \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )^n\right )\right )}{x} \, dx,x,a+b x\right )}{3 a^3}+\frac {\left (b^2 c\right ) \int \frac {g+h \log \left (f (d+e x)^n\right )}{x^2 (1-a c-b c x)} \, dx}{6 a}-\frac {\left (b^3 c\right ) \int \frac {g+h \log \left (f (d+e x)^n\right )}{x (1-a c-b c x)} \, dx}{3 a^2}-\frac {\left (b^3 g\right ) \int \frac {\log (1-a c-b c x)}{x} \, dx}{3 a^3}-\frac {\left (b^3 h\right ) \int \frac {\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{x} \, dx}{3 a^3}-\frac {(b e h n) \int \frac {\log (1-a c-b c x)}{x^2 (d+e x)} \, dx}{6 a}+\frac {\left (b^2 e h n\right ) \int \frac {\log (1-a c-b c x)}{x (d+e x)} \, dx}{3 a^2}-\frac {(b e h n) \int \frac {\log (1-a c-b c x)}{x^2 (a+b x)} \, dx}{6 d}+\frac {\left (b e^2 h n\right ) \int \frac {\log (1-a c-b c x)}{x (a+b x)} \, dx}{3 d^2}+\frac {\left (b e^3 h n\right ) \int \frac {\log (x) \log (1-a c-b c x)}{a+b x} \, dx}{3 d^3}-\frac {\left (b e^3 h n\right ) \int \frac {\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{3 d^3}\\ &=-\frac {b^3 g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}+\frac {b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{6 a x^2}-\frac {b^2 \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{3 a^2 x}-\frac {e h n \text {Li}_2(c (a+b x))}{6 d x^2}+\frac {e^2 h n \text {Li}_2(c (a+b x))}{3 d^2 x}+\frac {e^3 h n \log (x) \text {Li}_2(c (a+b x))}{3 d^3}-\frac {e^3 h n \log (d+e x) \text {Li}_2(c (a+b x))}{3 d^3}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{3 x^3}+\frac {\left (b^2 c\right ) \int \left (-\frac {g+h \log \left (f (d+e x)^n\right )}{(-1+a c) x^2}+\frac {b c \left (g+h \log \left (f (d+e x)^n\right )\right )}{(-1+a c)^2 x}-\frac {b^2 c^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{(-1+a c)^2 (-1+a c+b c x)}\right ) \, dx}{6 a}-\frac {\left (b^3 c\right ) \int \left (-\frac {g+h \log \left (f (d+e x)^n\right )}{(-1+a c) x}+\frac {b c \left (g+h \log \left (f (d+e x)^n\right )\right )}{(-1+a c) (-1+a c+b c x)}\right ) \, dx}{3 a^2}+\frac {\left (b^3 g\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{3 a^3}-\frac {\left (b^4 c g\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{3 a^3}+\frac {\left (b^3 h\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (f \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )^n\right )}{x} \, dx,x,a+b x\right )}{3 a^3}-\frac {\left (b^3 h n\right ) \int \frac {\log (1-a c-b c x) \log (d+e x)}{x} \, dx}{3 a^3}-\frac {(b e h n) \int \left (\frac {\log (1-a c-b c x)}{d x^2}-\frac {e \log (1-a c-b c x)}{d^2 x}+\frac {e^2 \log (1-a c-b c x)}{d^2 (d+e x)}\right ) \, dx}{6 a}+\frac {\left (b^2 e h n\right ) \int \left (\frac {\log (1-a c-b c x)}{d x}-\frac {e \log (1-a c-b c x)}{d (d+e x)}\right ) \, dx}{3 a^2}-\frac {(b e h n) \int \left (\frac {\log (1-a c-b c x)}{a x^2}-\frac {b \log (1-a c-b c x)}{a^2 x}+\frac {b^2 \log (1-a c-b c x)}{a^2 (a+b x)}\right ) \, dx}{6 d}+\frac {\left (b e^2 h n\right ) \int \left (\frac {\log (1-a c-b c x)}{a x}-\frac {b \log (1-a c-b c x)}{a (a+b x)}\right ) \, dx}{3 d^2}+\frac {\left (e^3 h n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {a}{b}+\frac {x}{b}\right ) \log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{3 d^3}-\frac {\left (e^3 h n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )}{x} \, dx,x,a+b x\right )}{3 d^3}+\frac {\left (b^3 h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac {\log (1-a c-b c x)}{x} \, dx}{3 a^3}\\ &=\text {too large to display} \end {align*}

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Mathematica [A]
time = 18.15, size = 3331, normalized size = 0.89 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x^4,x]

[Out]

(g + h*(-(n*Log[d + e*x]) + Log[f*(d + e*x)^n]))*(-1/6*(b*((-2*a*b^2*c*(Log[x] - Log[1 - a*c - b*c*x]))/(-1 +
a*c) - (a^2*Log[1 - a*c - b*c*x])/x^2 + (2*a*b*Log[1 - a*c - b*c*x])/x + 2*b^2*Log[(b*c*x)/(1 - a*c)]*Log[1 -
a*c - b*c*x] - (a^2*b*c*(-1 + a*c + b*c*x*Log[x] - b*c*x*Log[1 - a*c - b*c*x]))/((-1 + a*c)^2*x) + 2*b^2*PolyL
og[2, c*(a + b*x)] + 2*b^2*PolyLog[2, (-1 + a*c + b*c*x)/(-1 + a*c)]))/a^3 - PolyLog[2, a*c + b*c*x]/(3*x^3))
+ (h*n*((-2*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/x^3 + (e*(-(d*(d - 2*e*x)) + 2*e^2*x^2*Log[x] - 2*e^2*x^2*Lo
g[d + e*x])*PolyLog[2, c*(a + b*x)])/(d^3*x^2) + (2*b^2*(-((Log[1 - a*c - b*c*x]*Log[d + e*x])/x) + (e*(Log[x]
*Log[1 - a*c - b*c*x] - Log[x]*Log[1 + (b*c*x)/(-1 + a*c)] - Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d +
e - a*c*e)] - PolyLog[2, (b*c*x)/(1 - a*c)] - PolyLog[2, (e*(-1 + a*c + b*c*x))/(-(b*c*d) - e + a*c*e)]))/d +
(b*c*(Log[d + e*x]*(Log[-((e*x)/d)] - Log[1 - (b*c*(d + e*x))/(b*c*d + e - a*c*e)]) - PolyLog[2, (b*c*(d + e*
x))/(b*c*d + e - a*c*e)] + PolyLog[2, 1 + (e*x)/d]))/(-1 + a*c)))/a^2 - (b*(-((Log[1 - a*c - b*c*x]*Log[d + e*
x])/x^2) + (e*((d*(b*c*x*Log[x] - (-1 + a*c + b*c*x)*Log[1 - a*c - b*c*x]))/((-1 + a*c)*x) + e*(Log[x]*(-Log[1
- a*c - b*c*x] + Log[1 + (b*c*x)/(-1 + a*c)]) + PolyLog[2, (b*c*x)/(1 - a*c)]) + e*(Log[1 - a*c - b*c*x]*Log[
(b*c*(d + e*x))/(b*c*d + e - a*c*e)] + PolyLog[2, (e*(-1 + a*c + b*c*x))/(-(b*c*d) + (-1 + a*c)*e)])))/d^2 + (
b*c*(-(e*x*Log[e*x]) + a*c*e*x*Log[e*x] + d*Log[d + e*x] - a*c*d*Log[d + e*x] + e*x*Log[d + e*x] - a*c*e*x*Log
[d + e*x] - b*c*d*x*Log[-((e*x)/d)]*Log[d + e*x] + b*c*d*x*Log[d + e*x]*Log[1 - (b*c*(d + e*x))/(b*c*d + e - a
*c*e)] + b*c*d*x*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)] - b*c*d*x*PolyLog[2, 1 + (e*x)/d]))/((-1 + a*
c)^2*d*x)))/a + (2*b^3*(Log[c*(a + b*x)]*Log[1 - a*c - b*c*x]*Log[d + e*x] + ((Log[c*(a + b*x)] - Log[(e*(a +
b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*(-2*Log[1 - a*c - b*c*x] + Log[(b*(d + e*x))/(b*d - a*e)
]))/2 + (-Log[c*(a + b*x)] + Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*Log[-((b*(d + e
*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))] + (Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]^2*(Log[c*(a
+ b*x)] - Log[((b*c*d + e - a*c*e)*(a + b*x))/((b*d - a*e)*(-1 + a*c + b*c*x))] + Log[(b*c*d + e - a*c*e)/(e -
a*c*e - b*c*e*x)]))/2 + (Log[d + e*x] - Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, 1
- a*c - b*c*x] + (Log[1 - a*c - b*c*x] + Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, (b
*(d + e*x))/(b*d - a*e)] + Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]*(PolyLog[2, (b*c*(d + e*x))/
(e*(-1 + a*c + b*c*x))] - PolyLog[2, -((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]) - PolyLog[3, 1 - a*c
- b*c*x] - PolyLog[3, (b*(d + e*x))/(b*d - a*e)] - PolyLog[3, (b*c*(d + e*x))/(e*(-1 + a*c + b*c*x))] + PolyLo
g[3, -((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]))/a^3 + (e*((2*b*d*e*(Log[x]*Log[1 - a*c - b*c*x] - Lo
g[c*(a + b*x)]*Log[1 - a*c - b*c*x] - Log[x]*Log[1 + (b*c*x)/(-1 + a*c)] - PolyLog[2, (b*c*x)/(1 - a*c)] - Pol
yLog[2, 1 - a*c - b*c*x]))/a - (b*d^2*((a*(b*c*x*Log[x] - (-1 + a*c + b*c*x)*Log[1 - a*c - b*c*x]))/((-1 + a*c
)*x) + b*(Log[x]*(-Log[1 - a*c - b*c*x] + Log[1 + (b*c*x)/(-1 + a*c)]) + PolyLog[2, (b*c*x)/(1 - a*c)]) + b*(L
og[c*(a + b*x)]*Log[1 - a*c - b*c*x] + PolyLog[2, 1 - a*c - b*c*x])))/a^2 + 2*e^2*(Log[x]*Log[1 + (b*x)/a]*Log
[1 - a*c - b*c*x] + ((-Log[c*(a + b*x)] + Log[1 + (b*x)/a])*Log[1 - a*c - b*c*x]*(-2*Log[x] + Log[1 - a*c - b*
c*x]))/2 + (Log[c*(a + b*x)] - Log[1 + (b*x)/a])*Log[1 - a*c - b*c*x]*Log[(a*(-1 + a*c + b*c*x))/(b*x)] + ((Lo
g[(1 - a*c)/(b*c*x)] - Log[((1 - a*c)*(a + b*x))/(b*x)] + Log[1 + (b*x)/a])*Log[(a*(-1 + a*c + b*c*x))/(b*x)]^
2)/2 + (Log[1 - a*c - b*c*x] - Log[(a*(-1 + a*c + b*c*x))/(b*x)])*PolyLog[2, -((b*x)/a)] + (Log[x] + Log[(a*(-
1 + a*c + b*c*x))/(b*x)])*PolyLog[2, 1 - a*c - b*c*x] + Log[(a*(-1 + a*c + b*c*x))/(b*x)]*(-PolyLog[2, (a*(-1
+ a*c + b*c*x))/(b*x)] + PolyLog[2, (-1 + a*c + b*c*x)/(b*c*x)]) - PolyLog[3, -((b*x)/a)] - PolyLog[3, 1 - a*c
- b*c*x] + PolyLog[3, (a*(-1 + a*c + b*c*x))/(b*x)] - PolyLog[3, (-1 + a*c + b*c*x)/(b*c*x)]) - 2*e^2*(Log[c*
(a + b*x)]*Log[1 - a*c - b*c*x]*Log[d + e*x] + ((Log[c*(a + b*x)] - Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*
(d + e*x))/(b*d - a*e)]*(-2*Log[1 - a*c - b*c*x] + Log[(b*(d + e*x))/(b*d - a*e)]))/2 + (-Log[c*(a + b*x)] + L
og[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c +
b*c*x)))] + (Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]^2*(Log[c*(a + b*x)] - Log[((b*c*d + e - a*
c*e)*(a + b*x))/((b*d - a*e)*(-1 + a*c + b*c*x))] + Log[(b*c*d + e - a*c*e)/(e - a*c*e - b*c*e*x)]))/2 + (Log[
d + e*x] - Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, 1 - a*c - b*c*x] + (Log[1 - a*c
- b*c*x] + Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)] + Log
[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x...

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^4,x)

[Out]

int((g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/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(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^4,x, algorithm="maxima")

[Out]

integrate((h*log((x*e + d)^n*f) + g)*dilog((b*x + a)*c)/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(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^4,x, algorithm="fricas")

[Out]

integral((h*dilog(b*c*x + a*c)*log((x*e + d)^n*f) + g*dilog(b*c*x + a*c))/x^4, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g+h*ln(f*(e*x+d)**n))*polylog(2,c*(b*x+a))/x**4,x)

[Out]

Timed out

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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(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^4,x, algorithm="giac")

[Out]

integrate((h*log((e*x + d)^n*f) + g)*dilog((b*x + a)*c)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((polylog(2, c*(a + b*x))*(g + h*log(f*(d + e*x)^n)))/x^4,x)

[Out]

int((polylog(2, c*(a + b*x))*(g + h*log(f*(d + e*x)^n)))/x^4, x)

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