Integrand size = 21, antiderivative size = 661 \[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=\frac {53 b x}{192 c^3}+\frac {11 a x}{27 c^2}+\frac {49 (3 b+4 a c) x}{432 c^3}+\frac {29 b x^2}{384 c^2}+\frac {5 a x^2}{54 c}+\frac {13 (3 b+4 a c) x^2}{864 c^2}+\frac {2 a x^3}{81}+\frac {17 b x^3}{576 c}+\frac {(3 b+4 a c) x^3}{324 c}+\frac {3 b x^4}{256}+\frac {29 b \log (1-c x)}{192 c^4}+\frac {5 a \log (1-c x)}{27 c^3}+\frac {13 (3 b+4 a c) \log (1-c x)}{432 c^4}-\frac {b x^2 \log (1-c x)}{16 c^2}-\frac {a x^2 \log (1-c x)}{9 c}-\frac {(3 b+4 a c) x^2 \log (1-c x)}{48 c^2}-\frac {2}{27} a x^3 \log (1-c x)-\frac {b x^3 \log (1-c x)}{24 c}-\frac {(3 b+4 a c) x^3 \log (1-c x)}{108 c}-\frac {3}{64} b x^4 \log (1-c x)+\frac {b (1-c x) \log (1-c x)}{8 c^4}+\frac {2 a (1-c x) \log (1-c x)}{9 c^3}+\frac {(3 b+4 a c) (1-c x) \log (1-c x)}{12 c^4}-\frac {b \log ^2(1-c x)}{16 c^4}-\frac {a \log ^2(1-c x)}{9 c^3}+\frac {1}{9} a x^3 \log ^2(1-c x)+\frac {1}{16} b x^4 \log ^2(1-c x)-\frac {(3 b+4 a c) \log (c x) \log ^2(1-c x)}{12 c^4}-\frac {(3 b+4 a c) x \operatorname {PolyLog}(2,c x)}{12 c^3}-\frac {(3 b+4 a c) x^2 \operatorname {PolyLog}(2,c x)}{24 c^2}-\frac {(3 b+4 a c) x^3 \operatorname {PolyLog}(2,c x)}{36 c}-\frac {1}{16} b x^4 \operatorname {PolyLog}(2,c x)-\frac {(3 b+4 a c) \log (1-c x) \operatorname {PolyLog}(2,c x)}{12 c^4}+\frac {1}{12} \left (4 a x^3+3 b x^4\right ) \log (1-c x) \operatorname {PolyLog}(2,c x)-\frac {(3 b+4 a c) \log (1-c x) \operatorname {PolyLog}(2,1-c x)}{6 c^4}+\frac {(3 b+4 a c) \operatorname {PolyLog}(3,1-c x)}{6 c^4} \]
-1/12*(4*a*c+3*b)*x*polylog(2,c*x)/c^3-1/24*(4*a*c+3*b)*x^2*polylog(2,c*x) /c^2-1/36*(4*a*c+3*b)*x^3*polylog(2,c*x)/c-1/16*b*x^2*ln(-c*x+1)/c^2-1/9*a *x^2*ln(-c*x+1)/c-1/48*(4*a*c+3*b)*x^2*ln(-c*x+1)/c^2-1/24*b*x^3*ln(-c*x+1 )/c-1/108*(4*a*c+3*b)*x^3*ln(-c*x+1)/c+1/8*b*(-c*x+1)*ln(-c*x+1)/c^4+2/9*a *(-c*x+1)*ln(-c*x+1)/c^3+1/12*(4*a*c+3*b)*(-c*x+1)*ln(-c*x+1)/c^4-1/12*(4* a*c+3*b)*ln(c*x)*ln(-c*x+1)^2/c^4-1/12*(4*a*c+3*b)*ln(-c*x+1)*polylog(2,c* x)/c^4-1/6*(4*a*c+3*b)*ln(-c*x+1)*polylog(2,-c*x+1)/c^4+2/81*a*x^3+3/256*b *x^4+53/192*b*x/c^3+11/27*a*x/c^2+49/432*(4*a*c+3*b)*x/c^3+29/384*b*x^2/c^ 2+5/54*a*x^2/c+13/864*(4*a*c+3*b)*x^2/c^2+17/576*b*x^3/c+1/324*(4*a*c+3*b) *x^3/c-1/16*b*x^4*polylog(2,c*x)+1/6*(4*a*c+3*b)*polylog(3,-c*x+1)/c^4+29/ 192*b*ln(-c*x+1)/c^4+5/27*a*ln(-c*x+1)/c^3+13/432*(4*a*c+3*b)*ln(-c*x+1)/c ^4-2/27*a*x^3*ln(-c*x+1)-3/64*b*x^4*ln(-c*x+1)-1/16*b*ln(-c*x+1)^2/c^4-1/9 *a*ln(-c*x+1)^2/c^3+1/9*a*x^3*ln(-c*x+1)^2+1/16*b*x^4*ln(-c*x+1)^2+1/12*(3 *b*x^4+4*a*x^3)*ln(-c*x+1)*polylog(2,c*x)
Time = 0.57 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.64 \[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=\frac {4260 b c x+5952 a c^2 x+834 b c^2 x^2+1056 a c^3 x^2+268 b c^3 x^3+256 a c^4 x^3+81 b c^4 x^4+4260 b \log (1-c x)+5952 a c \log (1-c x)-2592 b c x \log (1-c x)-3840 a c^2 x \log (1-c x)-864 b c^2 x^2 \log (1-c x)-1344 a c^3 x^2 \log (1-c x)-480 b c^3 x^3 \log (1-c x)-768 a c^4 x^3 \log (1-c x)-324 b c^4 x^4 \log (1-c x)-432 b \log ^2(1-c x)-768 a c \log ^2(1-c x)+768 a c^4 x^3 \log ^2(1-c x)+432 b c^4 x^4 \log ^2(1-c x)-1728 b \log (c x) \log ^2(1-c x)-2304 a c \log (c x) \log ^2(1-c x)+48 \left (-c x \left (8 a c \left (6+3 c x+2 c^2 x^2\right )+3 b \left (12+6 c x+4 c^2 x^2+3 c^3 x^3\right )\right )+12 \left (4 a c \left (-1+c^3 x^3\right )+3 b \left (-1+c^4 x^4\right )\right ) \log (1-c x)\right ) \operatorname {PolyLog}(2,c x)-1152 (3 b+4 a c) \log (1-c x) \operatorname {PolyLog}(2,1-c x)+3456 b \operatorname {PolyLog}(3,1-c x)+4608 a c \operatorname {PolyLog}(3,1-c x)}{6912 c^4} \]
(4260*b*c*x + 5952*a*c^2*x + 834*b*c^2*x^2 + 1056*a*c^3*x^2 + 268*b*c^3*x^ 3 + 256*a*c^4*x^3 + 81*b*c^4*x^4 + 4260*b*Log[1 - c*x] + 5952*a*c*Log[1 - c*x] - 2592*b*c*x*Log[1 - c*x] - 3840*a*c^2*x*Log[1 - c*x] - 864*b*c^2*x^2 *Log[1 - c*x] - 1344*a*c^3*x^2*Log[1 - c*x] - 480*b*c^3*x^3*Log[1 - c*x] - 768*a*c^4*x^3*Log[1 - c*x] - 324*b*c^4*x^4*Log[1 - c*x] - 432*b*Log[1 - c *x]^2 - 768*a*c*Log[1 - c*x]^2 + 768*a*c^4*x^3*Log[1 - c*x]^2 + 432*b*c^4* x^4*Log[1 - c*x]^2 - 1728*b*Log[c*x]*Log[1 - c*x]^2 - 2304*a*c*Log[c*x]*Lo g[1 - c*x]^2 + 48*(-(c*x*(8*a*c*(6 + 3*c*x + 2*c^2*x^2) + 3*b*(12 + 6*c*x + 4*c^2*x^2 + 3*c^3*x^3))) + 12*(4*a*c*(-1 + c^3*x^3) + 3*b*(-1 + c^4*x^4) )*Log[1 - c*x])*PolyLog[2, c*x] - 1152*(3*b + 4*a*c)*Log[1 - c*x]*PolyLog[ 2, 1 - c*x] + 3456*b*PolyLog[3, 1 - c*x] + 4608*a*c*PolyLog[3, 1 - c*x])/( 6912*c^4)
Time = 1.26 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {7158, 2009}
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.
| \(\displaystyle \int x^2 (a+b x) \operatorname {PolyLog}(2,c x) \log (1-c x) \, dx\) |
| \(\Big \downarrow \) 7158 |
| \(\displaystyle c \int \left (-\frac {b \operatorname {PolyLog}(2,c x) x^3}{4 c}-\frac {(3 b+4 a c) \operatorname {PolyLog}(2,c x) x^2}{12 c^2}-\frac {(3 b+4 a c) \operatorname {PolyLog}(2,c x) x}{12 c^3}-\frac {(3 b+4 a c) \operatorname {PolyLog}(2,c x)}{12 c^4}+\frac {(3 b+4 a c) \operatorname {PolyLog}(2,c x)}{12 c^4 (1-c x)}\right )dx+\int \left (\frac {1}{4} b \log ^2(1-c x) x^3+\frac {1}{3} a \log ^2(1-c x) x^2\right )dx+\frac {1}{12} \left (4 a x^3+3 b x^4\right ) \operatorname {PolyLog}(2,c x) \log (1-c x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c \left (\frac {(4 a c+3 b) \operatorname {PolyLog}(3,1-c x)}{6 c^5}-\frac {(4 a c+3 b) \operatorname {PolyLog}(2,c x) \log (1-c x)}{12 c^5}-\frac {(4 a c+3 b) \operatorname {PolyLog}(2,1-c x) \log (1-c x)}{6 c^5}-\frac {(4 a c+3 b) \log (c x) \log ^2(1-c x)}{12 c^5}+\frac {13 (4 a c+3 b) \log (1-c x)}{432 c^5}+\frac {(1-c x) (4 a c+3 b) \log (1-c x)}{12 c^5}-\frac {x (4 a c+3 b) \operatorname {PolyLog}(2,c x)}{12 c^4}+\frac {49 x (4 a c+3 b)}{432 c^4}-\frac {x^2 (4 a c+3 b) \operatorname {PolyLog}(2,c x)}{24 c^3}+\frac {13 x^2 (4 a c+3 b)}{864 c^3}-\frac {x^2 (4 a c+3 b) \log (1-c x)}{48 c^3}-\frac {x^3 (4 a c+3 b) \operatorname {PolyLog}(2,c x)}{36 c^2}+\frac {x^3 (4 a c+3 b)}{324 c^2}-\frac {x^3 (4 a c+3 b) \log (1-c x)}{108 c^2}+\frac {b \log (1-c x)}{64 c^5}+\frac {b x}{64 c^4}+\frac {b x^2}{128 c^3}+\frac {b x^3}{192 c^2}-\frac {b x^4 \operatorname {PolyLog}(2,c x)}{16 c}+\frac {b x^4}{256 c}-\frac {b x^4 \log (1-c x)}{64 c}\right )+\frac {1}{12} \left (4 a x^3+3 b x^4\right ) \operatorname {PolyLog}(2,c x) \log (1-c x)-\frac {a \log ^2(1-c x)}{9 c^3}+\frac {2 a (1-c x) \log (1-c x)}{9 c^3}+\frac {5 a \log (1-c x)}{27 c^3}+\frac {11 a x}{27 c^2}+\frac {1}{9} a x^3 \log ^2(1-c x)-\frac {2}{27} a x^3 \log (1-c x)+\frac {5 a x^2}{54 c}-\frac {a x^2 \log (1-c x)}{9 c}+\frac {2 a x^3}{81}-\frac {b \log ^2(1-c x)}{16 c^4}+\frac {b (1-c x) \log (1-c x)}{8 c^4}+\frac {13 b \log (1-c x)}{96 c^4}+\frac {25 b x}{96 c^3}+\frac {13 b x^2}{192 c^2}-\frac {b x^2 \log (1-c x)}{16 c^2}+\frac {1}{16} b x^4 \log ^2(1-c x)-\frac {1}{32} b x^4 \log (1-c x)+\frac {7 b x^3}{288 c}-\frac {b x^3 \log (1-c x)}{24 c}+\frac {b x^4}{128}\) |
(25*b*x)/(96*c^3) + (11*a*x)/(27*c^2) + (13*b*x^2)/(192*c^2) + (5*a*x^2)/( 54*c) + (2*a*x^3)/81 + (7*b*x^3)/(288*c) + (b*x^4)/128 + (13*b*Log[1 - c*x ])/(96*c^4) + (5*a*Log[1 - c*x])/(27*c^3) - (b*x^2*Log[1 - c*x])/(16*c^2) - (a*x^2*Log[1 - c*x])/(9*c) - (2*a*x^3*Log[1 - c*x])/27 - (b*x^3*Log[1 - c*x])/(24*c) - (b*x^4*Log[1 - c*x])/32 + (b*(1 - c*x)*Log[1 - c*x])/(8*c^4 ) + (2*a*(1 - c*x)*Log[1 - c*x])/(9*c^3) - (b*Log[1 - c*x]^2)/(16*c^4) - ( a*Log[1 - c*x]^2)/(9*c^3) + (a*x^3*Log[1 - c*x]^2)/9 + (b*x^4*Log[1 - c*x] ^2)/16 + ((4*a*x^3 + 3*b*x^4)*Log[1 - c*x]*PolyLog[2, c*x])/12 + c*((b*x)/ (64*c^4) + (49*(3*b + 4*a*c)*x)/(432*c^4) + (b*x^2)/(128*c^3) + (13*(3*b + 4*a*c)*x^2)/(864*c^3) + (b*x^3)/(192*c^2) + ((3*b + 4*a*c)*x^3)/(324*c^2) + (b*x^4)/(256*c) + (b*Log[1 - c*x])/(64*c^5) + (13*(3*b + 4*a*c)*Log[1 - c*x])/(432*c^5) - ((3*b + 4*a*c)*x^2*Log[1 - c*x])/(48*c^3) - ((3*b + 4*a *c)*x^3*Log[1 - c*x])/(108*c^2) - (b*x^4*Log[1 - c*x])/(64*c) + ((3*b + 4* a*c)*(1 - c*x)*Log[1 - c*x])/(12*c^5) - ((3*b + 4*a*c)*Log[c*x]*Log[1 - c* x]^2)/(12*c^5) - ((3*b + 4*a*c)*x*PolyLog[2, c*x])/(12*c^4) - ((3*b + 4*a* c)*x^2*PolyLog[2, c*x])/(24*c^3) - ((3*b + 4*a*c)*x^3*PolyLog[2, c*x])/(36 *c^2) - (b*x^4*PolyLog[2, c*x])/(16*c) - ((3*b + 4*a*c)*Log[1 - c*x]*PolyL og[2, c*x])/(12*c^5) - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(6 *c^5) + ((3*b + 4*a*c)*PolyLog[3, 1 - c*x])/(6*c^5))
3.2.84.3.1 Defintions of rubi rules used
Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(Px_)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{u = IntHide[Px, x]}, Simp[u *(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x] + (Simp[b Int[Exp andIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x], x] - Simp[e*h*n Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x], x])] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x]
\[\int x^{2} \left (b x +a \right ) \ln \left (-c x +1\right ) \operatorname {polylog}\left (2, c x \right )d x\]
\[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=\int { {\left (b x + a\right )} x^{2} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ) \,d x } \]
Timed out. \[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.63 \[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=-\frac {1}{6912} \, c {\left (\frac {576 \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} {\left (4 \, a c + 3 \, b\right )}}{c^{5}} - \frac {81 \, b c^{4} x^{4} + 4 \, {\left (64 \, a c^{4} + 67 \, b c^{3}\right )} x^{3} + 6 \, {\left (176 \, a c^{3} + 139 \, b c^{2}\right )} x^{2} + 12 \, {\left (496 \, a c^{2} + 355 \, b c\right )} x - 48 \, {\left (9 \, b c^{4} x^{4} + 4 \, {\left (4 \, a c^{4} + 3 \, b c^{3}\right )} x^{3} + 6 \, {\left (4 \, a c^{3} + 3 \, b c^{2}\right )} x^{2} + 12 \, {\left (4 \, a c^{2} + 3 \, b c\right )} x + 12 \, {\left (4 \, a c + 3 \, b\right )} \log \left (-c x + 1\right )\right )} {\rm Li}_2\left (c x\right ) - 4 \, {\left (54 \, b c^{4} x^{4} + 4 \, {\left (32 \, a c^{4} + 21 \, b c^{3}\right )} x^{3} + 6 \, {\left (40 \, a c^{3} + 27 \, b c^{2}\right )} x^{2} - 1488 \, a c + 12 \, {\left (64 \, a c^{2} + 45 \, b c\right )} x - 1065 \, b\right )} \log \left (-c x + 1\right )}{c^{5}}\right )} + \frac {1}{1728} \, {\left (\frac {32 \, {\left (18 \, c^{3} x^{3} {\rm Li}_2\left (c x\right ) - 2 \, c^{3} x^{3} - 3 \, c^{2} x^{2} - 6 \, c x + 6 \, {\left (c^{3} x^{3} - 1\right )} \log \left (-c x + 1\right )\right )} a}{c^{3}} + \frac {9 \, {\left (48 \, c^{4} x^{4} {\rm Li}_2\left (c x\right ) - 3 \, c^{4} x^{4} - 4 \, c^{3} x^{3} - 6 \, c^{2} x^{2} - 12 \, c x + 12 \, {\left (c^{4} x^{4} - 1\right )} \log \left (-c x + 1\right )\right )} b}{c^{4}}\right )} \log \left (-c x + 1\right ) \]
-1/6912*c*(576*(log(c*x)*log(-c*x + 1)^2 + 2*dilog(-c*x + 1)*log(-c*x + 1) - 2*polylog(3, -c*x + 1))*(4*a*c + 3*b)/c^5 - (81*b*c^4*x^4 + 4*(64*a*c^4 + 67*b*c^3)*x^3 + 6*(176*a*c^3 + 139*b*c^2)*x^2 + 12*(496*a*c^2 + 355*b*c )*x - 48*(9*b*c^4*x^4 + 4*(4*a*c^4 + 3*b*c^3)*x^3 + 6*(4*a*c^3 + 3*b*c^2)* x^2 + 12*(4*a*c^2 + 3*b*c)*x + 12*(4*a*c + 3*b)*log(-c*x + 1))*dilog(c*x) - 4*(54*b*c^4*x^4 + 4*(32*a*c^4 + 21*b*c^3)*x^3 + 6*(40*a*c^3 + 27*b*c^2)* x^2 - 1488*a*c + 12*(64*a*c^2 + 45*b*c)*x - 1065*b)*log(-c*x + 1))/c^5) + 1/1728*(32*(18*c^3*x^3*dilog(c*x) - 2*c^3*x^3 - 3*c^2*x^2 - 6*c*x + 6*(c^3 *x^3 - 1)*log(-c*x + 1))*a/c^3 + 9*(48*c^4*x^4*dilog(c*x) - 3*c^4*x^4 - 4* c^3*x^3 - 6*c^2*x^2 - 12*c*x + 12*(c^4*x^4 - 1)*log(-c*x + 1))*b/c^4)*log( -c*x + 1)
\[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=\int { {\left (b x + a\right )} x^{2} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right ) \,d x } \]
Timed out. \[ \int x^2 (a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x) \, dx=\int x^2\,\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )\,\left (a+b\,x\right ) \,d x \]