Integrand size = 13, antiderivative size = 401 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))+\frac {1}{2} \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )+\frac {1}{2} \left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2+\left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )+\log (x) \operatorname {PolyLog}(2,c (a+b x))+\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a (1-c (a+b x))}\right )-\log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \operatorname {PolyLog}\left (2,-\frac {b c x}{1-c (a+b x)}\right )+\left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \operatorname {PolyLog}(2,1-c (a+b x))-\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+\operatorname {PolyLog}\left (3,-\frac {b x}{a (1-c (a+b x))}\right )-\operatorname {PolyLog}\left (3,-\frac {b c x}{1-c (a+b x)}\right )-\operatorname {PolyLog}(3,1-c (a+b x)) \]
ln(x)*ln(1+b*x/a)*ln(1-c*(b*x+a))+1/2*(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+1/2*( ln(c*(b*x+a))-ln(1+b*x/a))*(ln(x)+ln(-a*(1-c*(b*x+a))/b/x))^2+(ln(1-c*(b*x +a))-ln(-a*(1-c*(b*x+a))/b/x))*polylog(2,-b*x/a)+ln(x)*polylog(2,c*(b*x+a) )+ln(-a*(1-c*(b*x+a))/b/x)*polylog(2,-b*x/a/(1-c*(b*x+a)))-ln(-a*(1-c*(b*x +a))/b/x)*polylog(2,-b*c*x/(1-c*(b*x+a)))+(ln(x)+ln(-a*(1-c*(b*x+a))/b/x)) *polylog(2,1-c*(b*x+a))-polylog(3,-b*x/a)+polylog(3,-b*x/a/(1-c*(b*x+a)))- polylog(3,-b*c*x/(1-c*(b*x+a)))-polylog(3,1-c*(b*x+a))
Time = 0.09 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.05 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-a c-b c x)+\frac {1}{2} \left (-\log (c (a+b x))+\log \left (1+\frac {b x}{a}\right )\right ) \log (1-a c-b c x) (-2 \log (x)+\log (1-a c-b c x))+\left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \log (1-a c-b c x) \log \left (\frac {a (-1+a c+b c x)}{b x}\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{b c x}\right )-\log \left (\frac {(1-a c) (a+b x)}{b x}\right )+\log \left (1+\frac {b x}{a}\right )\right ) \log ^2\left (\frac {a (-1+a c+b c x)}{b x}\right )+\left (\log (1-a c-b c x)-\log \left (\frac {a (-1+a c+b c x)}{b x}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )+\left (\log (x)+\log \left (\frac {a (-1+a c+b c x)}{b x}\right )\right ) \operatorname {PolyLog}(2,1-a c-b c x)+\log \left (\frac {a (-1+a c+b c x)}{b x}\right ) \left (-\operatorname {PolyLog}\left (2,\frac {a (-1+a c+b c x)}{b x}\right )+\operatorname {PolyLog}\left (2,\frac {-1+a c+b c x}{b c x}\right )\right )+\log (x) \operatorname {PolyLog}(2,a c+b c x)-\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )-\operatorname {PolyLog}(3,1-a c-b c x)+\operatorname {PolyLog}\left (3,\frac {a (-1+a c+b c x)}{b x}\right )-\operatorname {PolyLog}\left (3,\frac {-1+a c+b c x}{b c x}\right ) \]
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)] + ((Log[(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)]) + Log[x]*PolyLog[2, a*c + 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)]
Time = 0.48 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7151, 2890, 2885}
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.
| \(\displaystyle \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx\) |
| \(\Big \downarrow \) 7151 |
| \(\displaystyle b \int \frac {\log (x) \log (-a c-b x c+1)}{a+b x}dx+\log (x) \operatorname {PolyLog}(2,c (a+b x))\) |
| \(\Big \downarrow \) 2890 |
| \(\displaystyle \int \frac {\log \left (\frac {a+b x}{b}-\frac {a}{b}\right ) \log (1-c (a+b x))}{a+b x}d(a+b x)+\log (x) \operatorname {PolyLog}(2,c (a+b x))\) |
| \(\Big \downarrow \) 2885 |
| \(\displaystyle \operatorname {PolyLog}\left (3,-\frac {b x}{a (1-c (a+b x))}\right )-\operatorname {PolyLog}\left (3,-\frac {b c x}{1-c (a+b x)}\right )-\operatorname {PolyLog}(3,1-c (a+b x))+\operatorname {PolyLog}\left (2,-\frac {b x}{a (1-c (a+b x))}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )-\operatorname {PolyLog}\left (2,-\frac {b c x}{1-c (a+b x)}\right ) \log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log (x) \operatorname {PolyLog}(2,c (a+b x))+\operatorname {PolyLog}\left (2,1-\frac {a+b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )+\operatorname {PolyLog}(2,1-c (a+b x)) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log \left (\frac {a+b x}{b}-\frac {a}{b}\right )\right )+\frac {1}{2} \left (\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac {a+b x}{a}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )-\frac {1}{2} \left (\log \left (\frac {a+b x}{a}\right )-\log (c (a+b x))\right ) \left (\log \left (-\frac {a (1-c (a+b x))}{b x}\right )+\log \left (\frac {a+b x}{b}-\frac {a}{b}\right )\right )^2+\log \left (\frac {a+b x}{a}\right ) \log \left (\frac {a+b x}{b}-\frac {a}{b}\right ) \log (1-c (a+b x))-\operatorname {PolyLog}\left (3,1-\frac {a+b x}{a}\right )\) |
Log[(a + b*x)/a]*Log[-(a/b) + (a + b*x)/b]*Log[1 - c*(a + b*x)] + ((Log[(a + 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)/2 - ((Log[( a + b*x)/a] - Log[c*(a + b*x)])*(Log[-(a/b) + (a + b*x)/b] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/2 + Log[x]*PolyLog[2, c*(a + b*x)] + (Log[1 - c *(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - (a + b*x )/a] + Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))] - Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))] + (Log[-(a/b) + (a + b*x)/b] + Log[-((a*(1 - c*(a + b*x)) )/(b*x))])*PolyLog[2, 1 - c*(a + b*x)] - PolyLog[3, 1 - (a + b*x)/a] + Pol yLog[3, -((b*x)/(a*(1 - c*(a + b*x))))] - PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))] - PolyLog[3, 1 - c*(a + b*x)]
3.2.27.3.1 Defintions of rubi rules used
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)] - Lo g[(-d)*(x/c)])*(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])^2, x] + Si mp[(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[Log[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.) *((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Simp[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]
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] + Simp[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]
\[\int \frac {\operatorname {polylog}\left (2, c \left (b x +a \right )\right )}{x}d x\]
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\int { \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x} \,d x } \]
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x}\, dx \]
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\int { \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x} \,d x } \]
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\int { \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x} \, dx=\int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x} \,d x \]