3.2.79 \(\int (g+h \log (f (d+e x)^n)) \text {PolyLog}(2,c (a+b x)) \, dx\) [179]

Optimal. Leaf size=1653 \[ -g x+3 h n x-\frac {g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {2 h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}-\frac {h (d+e x) \log \left (f (d+e x)^n\right )}{e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac {(1-a c) h \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac {a h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac {a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac {a h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac {a g \text {PolyLog}(2,c (a+b x))}{b}-\frac {a h n \text {PolyLog}(2,c (a+b x))}{b}-\frac {a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {PolyLog}(2,c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {PolyLog}(2,c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {PolyLog}(2,1-a c-b c x)}{e}+\frac {d h n \text {PolyLog}\left (2,\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{e}-h n x \text {PolyLog}(2,a c+b c x)+\frac {d h n \log (-d-e x) \text {PolyLog}(2,a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {PolyLog}\left (2,-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {PolyLog}\left (2,\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {a h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {(1-a c) h n \text {PolyLog}\left (2,\frac {b c (d+e x)}{b c d+e-a c e}\right )}{b c}-\frac {a h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {PolyLog}(2,1-c (a+b x))}{b}+\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {PolyLog}\left (2,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {PolyLog}\left (2,\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {d h n \text {PolyLog}(3,1-a c-b c x)}{e}-\frac {d h n \text {PolyLog}\left (3,-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {PolyLog}\left (3,\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {a h n \text {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {d h n \text {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {a h n \text {PolyLog}(3,1-c (a+b x))}{b}+\frac {a h n \text {PolyLog}\left (3,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \text {PolyLog}\left (3,\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 2.14, antiderivative size = 1653, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 17, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {6735, 2465, 2436, 2332, 2441, 2440, 2438, 6820, 45, 2463, 6874, 2479, 2490, 2487, 2485, 6730, 6732} \begin {gather*} \frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d-a c e+e}{b c (d+e x)}\right )-\log \left (\frac {(b c d-a c e+e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (-a c-b x c+1)}\right )}{2 e}-\frac {d h n \text {Li}_2\left (-\frac {e (-a c-b x c+1)}{b c (d+e x)}\right ) \log \left (\frac {b (d+e x)}{(b d-a e) (-a c-b x c+1)}\right )}{e}+\frac {d h n \text {Li}_2\left (\frac {(b d-a e) (-a c-b x c+1)}{b (d+e x)}\right ) \log \left (\frac {b (d+e x)}{(b d-a e) (-a c-b x c+1)}\right )}{e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (-a c-b x c+1)+\log \left (\frac {b (d+e x)}{(b d-a e) (-a c-b x c+1)}\right )\right )^2}{2 e}-\frac {a h n \left (\log (c (a+b x))+\log \left (\frac {b c d-a c e+e}{b c (d+e x)}\right )-\log \left (\frac {(b c d-a c e+e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}+\frac {a h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}-g x+3 h n x-\frac {g (-a c-b x c+1) \log (-a c-b x c+1)}{b c}+\frac {2 h n (-a c-b x c+1) \log (-a c-b x c+1)}{b c}+\frac {d h n \log (c (a+b x)) \log (-a c-b x c+1) \log (-d-e x)}{e}+\frac {d h n \log (-a c-b x c+1) \log \left (\frac {b c (d+e x)}{b c d-a c e+e}\right )}{e}-\frac {h (d+e x) \log \left (f (d+e x)^n\right )}{e}+h x \log (-a c-b x c+1) \log \left (f (d+e x)^n\right )-\frac {(1-a c) h \log \left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac {a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac {a g \text {Li}_2(c (a+b x))}{b}-\frac {a h n \text {Li}_2(c (a+b x))}{b}-\frac {a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (-a c-b x c+1)}\right )\right ) \text {Li}_2(-a c-b x c+1)}{e}+\frac {d h n \text {Li}_2\left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{e}-h n x \text {Li}_2(a c+b x c)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b x c)}{e}+\frac {d h n \left (\log (-a c-b x c+1)+\log \left (\frac {b (d+e x)}{(b d-a e) (-a c-b x c+1)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {a h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {(1-a c) h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d-a c e+e}\right )}{b c}-\frac {a h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{b}+\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {d h n \text {Li}_3(-a c-b x c+1)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (-a c-b x c+1)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (-a c-b x c+1)}{b (d+e x)}\right )}{e}+\frac {a h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {a h n \text {Li}_3(1-c (a+b x))}{b}+\frac {a h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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 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 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 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2479

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

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 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 6730

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)], x_Symbol] :> Simp[x*PolyLog[n, c*(a + b*x)^p], x] + (-Dist[
p, Int[PolyLog[n - 1, c*(a + b*x)^p], x], x] + Dist[a*p, Int[PolyLog[n - 1, c*(a + b*x)^p]/(a + b*x), x], x])
/; FreeQ[{a, b, c, p}, x] && GtQ[n, 0]

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 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 \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x)) \, dx &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+b \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right ) \, dx-(e h n) \int \left (\frac {1}{e}-\frac {d}{e (d+e x)}\right ) \text {Li}_2(c (a+b x)) \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+b \int \frac {x \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a+b x} \, dx-(e h n) \int \frac {x \text {Li}_2(a c+b c x)}{d+e x} \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+b \int \left (\frac {g x \log (1-a c-b c x)}{a+b x}+\frac {h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{a+b x}\right ) \, dx-(e h n) \int \left (\frac {\text {Li}_2(a c+b c x)}{e}+\frac {d \text {Li}_2(a c+b c x)}{e (-d-e x)}\right ) \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+(b g) \int \frac {x \log (1-a c-b c x)}{a+b x} \, dx+(b h) \int \frac {x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{a+b x} \, dx-(h n) \int \text {Li}_2(a c+b c x) \, dx-(d h n) \int \frac {\text {Li}_2(a c+b c x)}{-d-e x} \, dx\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}+(b g) \int \left (\frac {\log (1-a c-b c x)}{b}-\frac {a \log (1-a c-b c x)}{b (a+b x)}\right ) \, dx+(b h) \int \left (\frac {\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{b}-\frac {a \log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{b (a+b x)}\right ) \, dx-(h n) \int \log (1-a c-b c x) \, dx+(a c h n) \int \frac {\log (1-a c-b c x)}{a c+b c x} \, dx+\frac {(b c d h n) \int \frac {\log (1-a c-b c x) \log (-d-e x)}{a c+b c x} \, dx}{e}\\ &=x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}+g \int \log (1-a c-b c x) \, dx-(a g) \int \frac {\log (1-a c-b c x)}{a+b x} \, dx+h \int \log (1-a c-b c x) \log \left (f (d+e x)^n\right ) \, dx-(a h) \int \frac {\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{a+b x} \, dx+\frac {(a h n) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,a c+b c x\right )}{b}+\frac {(h n) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}+\frac {(d h n) \text {Subst}\left (\int \frac {\log \left (-\frac {-a b c^2-b c (1-a c)}{b c}-x\right ) \log \left (-\frac {b c d-a c e}{b c}-\frac {e x}{b c}\right )}{x} \, dx,x,a c+b c x\right )}{e}\\ &=h n x+\frac {h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac {a h n \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2(1-a c-b c x)}{e}-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {d h n \text {Li}_3(1-a c-b c x)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {(a g) \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{b}-\frac {g \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}-\frac {(a h) \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 )}{b}+(b c h) \int \frac {x \log \left (f (d+e x)^n\right )}{1-a c-b c x} \, dx-(e h n) \int \frac {x \log (1-a c-b c x)}{d+e x} \, dx\\ &=-g x+h n x-\frac {g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )+\frac {a g \text {Li}_2(c (a+b x))}{b}-\frac {a h n \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2(1-a c-b c x)}{e}-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {d h n \text {Li}_3(1-a c-b c x)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+(b c h) \int \left (-\frac {\log \left (f (d+e x)^n\right )}{b c}+\frac {(-1+a c) \log \left (f (d+e x)^n\right )}{b c (-1+a c+b c x)}\right ) \, dx-\frac {(a h n) \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 )}{b}-(e h n) \int \left (\frac {\log (1-a c-b c x)}{e}-\frac {d \log (1-a c-b c x)}{e (d+e x)}\right ) \, dx+\frac {\left (a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\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 )}{b}\\ &=-g x+h n x-\frac {g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac {a h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac {a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac {a h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac {a g \text {Li}_2(c (a+b x))}{b}-\frac {a h n \text {Li}_2(c (a+b x))}{b}-\frac {a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2(1-a c-b c x)}{e}-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {a h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {a h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{b}+\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {d h n \text {Li}_3(1-a c-b c x)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {a h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {a h n \text {Li}_3(1-c (a+b x))}{b}+\frac {a h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-h \int \log \left (f (d+e x)^n\right ) \, dx-((1-a c) h) \int \frac {\log \left (f (d+e x)^n\right )}{-1+a c+b c x} \, dx-(h n) \int \log (1-a c-b c x) \, dx+(d h n) \int \frac {\log (1-a c-b c x)}{d+e x} \, dx\\ &=-g x+h n x-\frac {g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac {(1-a c) h \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac {a h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac {a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac {a h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac {a g \text {Li}_2(c (a+b x))}{b}-\frac {a h n \text {Li}_2(c (a+b x))}{b}-\frac {a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2(1-a c-b c x)}{e}-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {a h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {a h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{b}+\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {d h n \text {Li}_3(1-a c-b c x)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {a h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {a h n \text {Li}_3(1-c (a+b x))}{b}+\frac {a h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {h \text {Subst}\left (\int \log \left (f x^n\right ) \, dx,x,d+e x\right )}{e}+\frac {(h n) \text {Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{b c}+\frac {(b c d h n) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{e}+\frac {((1-a c) e h n) \int \frac {\log \left (\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )}{d+e x} \, dx}{b c}\\ &=-g x+3 h n x-\frac {g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {2 h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}-\frac {h (d+e x) \log \left (f (d+e x)^n\right )}{e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac {(1-a c) h \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac {a h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac {a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac {a h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac {a g \text {Li}_2(c (a+b x))}{b}-\frac {a h n \text {Li}_2(c (a+b x))}{b}-\frac {a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2(1-a c-b c x)}{e}-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {a h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {a h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{b}+\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {d h n \text {Li}_3(1-a c-b c x)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {a h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {a h n \text {Li}_3(1-c (a+b x))}{b}+\frac {a h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}+\frac {((1-a c) h n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b c x}{-b c d+(-1+a c) e}\right )}{x} \, dx,x,d+e x\right )}{b c}-\frac {(d h n) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{e}\\ &=-g x+3 h n x-\frac {g (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {2 h n (1-a c-b c x) \log (1-a c-b c x)}{b c}+\frac {d h n \log (c (a+b x)) \log (1-a c-b c x) \log (-d-e x)}{e}+\frac {d h n \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{e}+\frac {d h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )}{2 e}-\frac {d h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right )^2}{2 e}-\frac {h (d+e x) \log \left (f (d+e x)^n\right )}{e}+h x \log (1-a c-b c x) \log \left (f (d+e x)^n\right )-\frac {(1-a c) h \log \left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right ) \log \left (f (d+e x)^n\right )}{b c}-\frac {a h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 b}-\frac {a h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{b}+\frac {a h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 b}+\frac {a g \text {Li}_2(c (a+b x))}{b}-\frac {a h n \text {Li}_2(c (a+b x))}{b}-\frac {a h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{b}+x \left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))+\frac {d h n \left (\log (-d-e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2(1-a c-b c x)}{e}+\frac {d h n \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{e}-h n x \text {Li}_2(a c+b c x)+\frac {d h n \log (-d-e x) \text {Li}_2(a c+b c x)}{e}-\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {d h n \left (\log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{(b d-a e) (1-a c-b c x)}\right )\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {a h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {(1-a c) h n \text {Li}_2\left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{b c}-\frac {a h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{b}+\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}-\frac {d h n \text {Li}_3(1-a c-b c x)}{e}-\frac {d h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{e}+\frac {d h n \text {Li}_3\left (\frac {(b d-a e) (1-a c-b c x)}{b (d+e x)}\right )}{e}+\frac {a h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{b}-\frac {d h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {a h n \text {Li}_3(1-c (a+b x))}{b}+\frac {a h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{b}-\frac {a h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{b}\\ \end {align*}

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Mathematica [A]
time = 3.13, size = 1546, normalized size = 0.94 \begin {gather*} \frac {\left (g-h n \log (d+e x)+h \log \left (f (d+e x)^n\right )\right ) (-b c x+(-1+a c+b c x) \log (1-a c-b c x)+c (a+b x) \text {PolyLog}(2,c (a+b x)))}{b c}+\frac {h n \left ((-e x+(d+e x) \log (d+e x)) \text {PolyLog}(2,c (a+b x))+\frac {-e+a c e+2 b c e x-b c d \log (d+e x)-b c e x \log (d+e x)+\log (1-a c-b c x) \left (-e (-1+a c+b c x)+e (-1+a c+b c x) \log (d+e x)+(b c d+e-a c e) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )\right )+e (-1+a c+b c x+(1-a c-b c x+a c \log (c (a+b x))) \log (1-a c-b c x)+a c \text {PolyLog}(2,1-a c-b c x))+(b c d+e-a c e) \text {PolyLog}\left (2,\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )+b c d \left (\log (c (a+b x)) \log (1-a c-b c x) \log (d+e x)+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \left (-2 \log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{b d-a e}\right )\right )+\left (-\log (c (a+b x))+\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )+\frac {1}{2} \log ^2\left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\log (c (a+b x))-\log \left (\frac {(b c d+e-a c e) (a+b x)}{(b d-a e) (-1+a c+b c x)}\right )+\log \left (\frac {b c d+e-a c e}{e-a c e-b c e x}\right )\right )+\left (\log (d+e x)-\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \text {PolyLog}(2,1-a c-b c x)+\left (\log (1-a c-b c x)+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \text {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\text {PolyLog}\left (2,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )-\text {PolyLog}\left (2,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right )-\text {PolyLog}(3,1-a c-b c x)-\text {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )-\text {PolyLog}\left (3,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )+\text {PolyLog}\left (3,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right )-a c e \left (\log (c (a+b x)) \log (1-a c-b c x) \log (d+e x)+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \left (-2 \log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{b d-a e}\right )\right )+\left (-\log (c (a+b x))+\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )+\frac {1}{2} \log ^2\left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\log (c (a+b x))-\log \left (\frac {(b c d+e-a c e) (a+b x)}{(b d-a e) (-1+a c+b c x)}\right )+\log \left (\frac {b c d+e-a c e}{e-a c e-b c e x}\right )\right )+\left (\log (d+e x)-\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \text {PolyLog}(2,1-a c-b c x)+\left (\log (1-a c-b c x)+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \text {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\text {PolyLog}\left (2,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )-\text {PolyLog}\left (2,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right )-\text {PolyLog}(3,1-a c-b c x)-\text {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )-\text {PolyLog}\left (3,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )+\text {PolyLog}\left (3,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right )}{b c}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )\, dx\]

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)

[Out]

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

[Out]

(d*h*n*log(x*e + d) + h*x*e*log((x*e + d)^n) - (h*n*e - h*e*log(f) - g*e)*x)*dilog(b*c*x + a*c)*e^(-1) + integ
rate((b*h*x*e*log(-b*c*x - a*c + 1)*log((x*e + d)^n) + (b*d*h*n*log(x*e + d) - (b*h*n*e - b*h*e*log(f) - b*g*e
)*x)*log(-b*c*x - a*c + 1))/(b*x*e + a*e), 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, 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)

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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)

[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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \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 ) \,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)

[Out]

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

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