Integrand size = 20, antiderivative size = 340 \[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=\frac {7 c^2 h}{36 x}-\frac {3}{4} c^3 h \log (x)+\frac {19}{36} c^3 h \log (1-c x)-\frac {c h \log (1-c x)}{12 x^2}-\frac {c^2 h \log (1-c x)}{3 x}+\frac {1}{3} c^3 h \log (c x) \log ^2(1-c x)+\frac {\log (1-c x) (g+h \log (1-c x))}{9 x^3}-\frac {c (g+2 h \log (1-c x))}{18 x^2}-\frac {c^2 (1-c x) (g+2 h \log (1-c x))}{9 x}+\frac {1}{9} c^3 (g+2 h \log (1-c x)) \log \left (1-\frac {1}{1-c x}\right )+\frac {c h \operatorname {PolyLog}(2,c x)}{6 x^2}+\frac {c^2 h \operatorname {PolyLog}(2,c x)}{3 x}+\frac {1}{3} c^3 h \log (1-c x) \operatorname {PolyLog}(2,c x)-\frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{3 x^3}-\frac {2}{9} c^3 h \operatorname {PolyLog}\left (2,\frac {1}{1-c x}\right )+\frac {2}{3} c^3 h \log (1-c x) \operatorname {PolyLog}(2,1-c x)-\frac {1}{3} c^3 h \operatorname {PolyLog}(3,c x)-\frac {2}{3} c^3 h \operatorname {PolyLog}(3,1-c x) \]
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*ln(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*poly log(3,-c*x+1)
Time = 0.24 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.89 \[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=-\frac {g \left (c x (1+2 c x)-2 c^3 x^3 \log (x)+2 \left (-1+c^3 x^3\right ) \log (1-c x)+6 \operatorname {PolyLog}(2,c x)\right )}{18 x^3}+\frac {h \left (7 c^2 x^2-4 c^3 x^3-15 c^3 x^3 \log (x)-12 c^3 x^3 \log (c x)-7 c x \log (1-c x)-20 c^2 x^2 \log (1-c x)+27 c^3 x^3 \log (1-c x)+8 c^3 x^3 \log (c x) \log (1-c x)+4 \log ^2(1-c x)-4 c^3 x^3 \log ^2(1-c x)+12 c^3 x^3 \log (c x) \log ^2(1-c x)+6 \left (c x (1+2 c x)+2 \left (-1+c^3 x^3\right ) \log (1-c x)\right ) \operatorname {PolyLog}(2,c x)+8 c^3 x^3 (1+3 \log (1-c x)) \operatorname {PolyLog}(2,1-c x)-12 c^3 x^3 \operatorname {PolyLog}(3,c x)-24 c^3 x^3 \operatorname {PolyLog}(3,1-c x)\right )}{36 x^3} \]
-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] - 2 4*c^3*x^3*PolyLog[3, 1 - c*x]))/(36*x^3)
Time = 1.45 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {7157, 2009, 2883, 2858, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{x^4} \, dx\) |
| \(\Big \downarrow \) 7157 |
| \(\displaystyle -\frac {1}{3} c h \int \left (\frac {\operatorname {PolyLog}(2,c x) c^3}{1-c x}+\frac {\operatorname {PolyLog}(2,c x) c^2}{x}+\frac {\operatorname {PolyLog}(2,c x) c}{x^2}+\frac {\operatorname {PolyLog}(2,c x)}{x^3}\right )dx-\frac {1}{3} \int \frac {\log (1-c x) (g+h \log (1-c x))}{x^4}dx-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{3} \int \frac {\log (1-c x) (g+h \log (1-c x))}{x^4}dx-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2883 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} c \int \frac {g+2 h \log (1-c x)}{x^3 (1-c x)}dx+\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} \int \frac {g+2 h \log (1-c x)}{x^3 (1-c x)}d(1-c x)\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \int \frac {g+2 h \log (1-c x)}{c^3 x^3 (1-c x)}d(1-c x)\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c^3 x^3}d(1-c x)+\int \frac {g+2 h \log (1-c x)}{c^2 x^2 (1-c x)}d(1-c x)\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c^2 x^2 (1-c x)}d(1-c x)-h \int \frac {1}{c^2 x^2 (1-c x)}d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c^2 x^2 (1-c x)}d(1-c x)-h \int \left (\frac {1}{1-c x}+\frac {1}{c x}+\frac {1}{c^2 x^2}\right )d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c^2 x^2 (1-c x)}d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}-h \left (\frac {1}{c x}-\log (c x)+\log (1-c x)\right )\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c^2 x^2}d(1-c x)+\int \frac {g+2 h \log (1-c x)}{c x (1-c x)}d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}-h \left (\frac {1}{c x}-\log (c x)+\log (1-c x)\right )\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c x (1-c x)}d(1-c x)-2 h \int \frac {1}{c x}d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}+\frac {(1-c x) (2 h \log (1-c x)+g)}{c x}-h \left (\frac {1}{c x}-\log (c x)+\log (1-c x)\right )\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\int \frac {g+2 h \log (1-c x)}{c x (1-c x)}d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}+\frac {(1-c x) (2 h \log (1-c x)+g)}{c x}+2 h \log (c x)-h \left (\frac {1}{c x}-\log (c x)+\log (1-c x)\right )\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (2 h \int \frac {\log \left (1-\frac {1}{1-c x}\right )}{1-c x}d(1-c x)+\frac {2 h \log (1-c x)+g}{2 c^2 x^2}+\frac {(1-c x) (2 h \log (1-c x)+g)}{c x}-\log \left (1-\frac {1}{1-c x}\right ) (2 h \log (1-c x)+g)+2 h \log (c x)-h \left (\frac {1}{c x}-\log (c x)+\log (1-c x)\right )\right )\right )-\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {1}{3} c h \left (c^2 \operatorname {PolyLog}(3,c x)+2 c^2 \operatorname {PolyLog}(3,1-c x)-c^2 \operatorname {PolyLog}(2,c x) \log (1-c x)-2 c^2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)+c^2 (-\log (c x)) \log ^2(1-c x)+\frac {5}{4} c^2 \log (x)-\frac {5}{4} c^2 \log (1-c x)-\frac {\operatorname {PolyLog}(2,c x)}{2 x^2}-\frac {c \operatorname {PolyLog}(2,c x)}{x}+\frac {\log (1-c x)}{4 x^2}-\frac {c}{4 x}+\frac {c \log (1-c x)}{x}\right )+\frac {1}{3} \left (\frac {\log (1-c x) (h \log (1-c x)+g)}{3 x^3}-\frac {1}{3} c^3 \left (\frac {2 h \log (1-c x)+g}{2 c^2 x^2}+\frac {(1-c x) (2 h \log (1-c x)+g)}{c x}-\log \left (1-\frac {1}{1-c x}\right ) (2 h \log (1-c x)+g)+2 h \operatorname {PolyLog}\left (2,\frac {1}{1-c x}\right )+2 h \log (c x)-h \left (\frac {1}{c x}-\log (c x)+\log (1-c x)\right )\right )\right )-\frac {\operatorname {PolyLog}(2,c x) (h \log (1-c x)+g)}{3 x^3}\) |
-1/3*((g + h*Log[1 - c*x])*PolyLog[2, c*x])/x^3 + ((Log[1 - c*x]*(g + h*Lo g[1 - c*x]))/(3*x^3) - (c^3*(2*h*Log[c*x] - h*(1/(c*x) - Log[c*x] + Log[1 - c*x]) + (g + 2*h*Log[1 - c*x])/(2*c^2*x^2) + ((1 - c*x)*(g + 2*h*Log[1 - c*x]))/(c*x) - (g + 2*h*Log[1 - c*x])*Log[1 - (1 - c*x)^(-1)] + 2*h*PolyL og[2, (1 - c*x)^(-1)]))/3)/3 - (c*h*(-1/4*c/x + (5*c^2*Log[x])/4 - (5*c^2* Log[1 - c*x])/4 + Log[1 - c*x]/(4*x^2) + (c*Log[1 - c*x])/x - c^2*Log[c*x] *Log[1 - c*x]^2 - PolyLog[2, c*x]/(2*x^2) - (c*PolyLog[2, c*x])/x - c^2*Lo g[1 - c*x]*PolyLog[2, c*x] - 2*c^2*Log[1 - c*x]*PolyLog[2, 1 - c*x] + c^2* PolyLog[3, c*x] + 2*c^2*PolyLog[3, 1 - c*x]))/3
3.2.76.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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] - Simp[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]
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] - Simp[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] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
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] + Simp[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]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(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]
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] - Simp[ 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]
Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(x_)^(m_.)*PolyLo g[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] + (Simp[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] - Simp[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]
\[\int \frac {\left (g +h \ln \left (-c x +1\right )\right ) \operatorname {polylog}\left (2, c x \right )}{x^{4}}d x\]
\[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=\int { \frac {{\left (h \log \left (-c x + 1\right ) + g\right )} {\rm Li}_2\left (c x\right )}{x^{4}} \,d x } \]
\[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=\int \frac {\left (g + h \log {\left (- c x + 1 \right )}\right ) \operatorname {Li}_{2}\left (c x\right )}{x^{4}}\, dx \]
\[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=\int { \frac {{\left (h \log \left (-c x + 1\right ) + g\right )} {\rm Li}_2\left (c x\right )}{x^{4}} \,d x } \]
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(dilog(c*x)*log(-c*x + 1)/x^4, x)
\[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=\int { \frac {{\left (h \log \left (-c x + 1\right ) + g\right )} {\rm Li}_2\left (c x\right )}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {(g+h \log (1-c x)) \operatorname {PolyLog}(2,c x)}{x^4} \, dx=\int \frac {\left (g+h\,\ln \left (1-c\,x\right )\right )\,\mathrm {polylog}\left (2,c\,x\right )}{x^4} \,d x \]