3.127 \(\int \frac{\text{PolyLog}(2,c (a+b x))}{x} \, dx\)

Optimal. Leaf size=401 \[ \text{PolyLog}\left (3,-\frac{b x}{a (1-c (a+b x))}\right )-\text{PolyLog}\left (3,-\frac{b c x}{1-c (a+b x)}\right )-\text{PolyLog}(3,1-c (a+b x))+\log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{PolyLog}\left (2,-\frac{b x}{a (1-c (a+b x))}\right )-\log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{PolyLog}\left (2,-\frac{b c x}{1-c (a+b x)}\right )+\text{PolyLog}\left (2,-\frac{b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right )+\log (x) \text{PolyLog}(2,c (a+b x))+\left (\log \left (-\frac{a (1-c (a+b x))}{b x}\right )+\log (x)\right ) \text{PolyLog}(2,1-c (a+b x))-\text{PolyLog}\left (3,-\frac{b x}{a}\right )+\frac{1}{2} \left (\log \left (\frac{1-a c}{1-c (a+b x)}\right )-\log \left (\frac{(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (-\frac{a (1-c (a+b x))}{b x}\right )+\frac{1}{2} \left (\log (c (a+b x))-\log \left (\frac{b x}{a}+1\right )\right ) \left (\log \left (-\frac{a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2+\log (x) \log \left (\frac{b x}{a}+1\right ) \log (1-c (a+b x)) \]

[Out]

Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)] + ((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)/2 + ((Log[c*(a + b*x)] - Log
[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/2 + (Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(
a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)] + Log[x]*PolyLog[2, c*(a + b*x)] + 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[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)] - PolyLog[3, -((b
*x)/a)] + PolyLog[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)]

________________________________________________________________________________________

Rubi [A]  time = 0.355029, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6597, 2440, 2435} \[ \text{PolyLog}\left (3,-\frac{b x}{a (1-c (a+b x))}\right )-\text{PolyLog}\left (3,-\frac{b c x}{1-c (a+b x)}\right )-\text{PolyLog}(3,1-c (a+b x))+\log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{PolyLog}\left (2,-\frac{b x}{a (1-c (a+b x))}\right )-\log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{PolyLog}\left (2,-\frac{b c x}{1-c (a+b x)}\right )+\text{PolyLog}\left (2,-\frac{b x}{a}\right ) \left (\log (1-c (a+b x))-\log \left (-\frac{a (1-c (a+b x))}{b x}\right )\right )+\log (x) \text{PolyLog}(2,c (a+b x))+\left (\log \left (-\frac{a (1-c (a+b x))}{b x}\right )+\log (x)\right ) \text{PolyLog}(2,1-c (a+b x))-\text{PolyLog}\left (3,-\frac{b x}{a}\right )+\frac{1}{2} \left (\log \left (\frac{1-a c}{1-c (a+b x)}\right )-\log \left (\frac{(1-a c) (a+b x)}{a (1-c (a+b x))}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (-\frac{a (1-c (a+b x))}{b x}\right )+\frac{1}{2} \left (\log (c (a+b x))-\log \left (\frac{b x}{a}+1\right )\right ) \left (\log \left (-\frac{a (1-c (a+b x))}{b x}\right )+\log (x)\right )^2+\log (x) \log \left (\frac{b x}{a}+1\right ) \log (1-c (a+b x)) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, c*(a + b*x)]/x,x]

[Out]

Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)] + ((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)/2 + ((Log[c*(a + b*x)] - Log
[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/2 + (Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(
a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)] + Log[x]*PolyLog[2, c*(a + b*x)] + 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[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)] - PolyLog[3, -((b
*x)/a)] + PolyLog[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)]

Rule 6597

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 2440

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 2435

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*(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)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/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]

Rubi steps

\begin{align*} \int \frac{\text{Li}_2(c (a+b x))}{x} \, dx &=\log (x) \text{Li}_2(c (a+b x))+b \int \frac{\log (x) \log (1-a c-b c x)}{a+b x} \, dx\\ &=\log (x) \text{Li}_2(c (a+b x))+\operatorname{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 )\\ &=\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 ) \text{Li}_2\left (-\frac{b x}{a}\right )+\log (x) \text{Li}_2(c (a+b x))+\log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\frac{b x}{a (1-c (a+b x))}\right )-\log \left (-\frac{a (1-c (a+b x))}{b x}\right ) \text{Li}_2\left (-\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 ) \text{Li}_2(1-c (a+b x))-\text{Li}_3\left (-\frac{b x}{a}\right )+\text{Li}_3\left (-\frac{b x}{a (1-c (a+b x))}\right )-\text{Li}_3\left (-\frac{b c x}{1-c (a+b x)}\right )-\text{Li}_3(1-c (a+b x))\\ \end{align*}

Mathematica [A]  time = 0.145861, size = 422, normalized size = 1.05 \[ -\text{PolyLog}(3,-a c-b c x+1)+\text{PolyLog}\left (3,\frac{a (a c+b c x-1)}{b x}\right )-\text{PolyLog}\left (3,\frac{a c+b c x-1}{b c x}\right )+\log \left (\frac{a (a c+b c x-1)}{b x}\right ) \left (\text{PolyLog}\left (2,\frac{a c+b c x-1}{b c x}\right )-\text{PolyLog}\left (2,\frac{a (a c+b c x-1)}{b x}\right )\right )+\text{PolyLog}\left (2,-\frac{b x}{a}\right ) \left (\log (-a c-b c x+1)-\log \left (\frac{a (a c+b c x-1)}{b x}\right )\right )+\left (\log \left (\frac{a (a c+b c x-1)}{b x}\right )+\log (x)\right ) \text{PolyLog}(2,-a c-b c x+1)+\log (x) \text{PolyLog}(2,a c+b c x)-\text{PolyLog}\left (3,-\frac{b x}{a}\right )+\frac{1}{2} \left (\log \left (\frac{1-a c}{b c x}\right )-\log \left (-\frac{(a c-1) (a+b x)}{b x}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (\frac{a (a c+b c x-1)}{b x}\right )+\left (\log (c (a+b x))-\log \left (\frac{b x}{a}+1\right )\right ) \log (-a c-b c x+1) \log \left (\frac{a (a c+b c x-1)}{b x}\right )+\log (x) \log \left (\frac{b x}{a}+1\right ) \log (-a c-b c x+1)+\frac{1}{2} \left (\log \left (\frac{b x}{a}+1\right )-\log (c (a+b x))\right ) \log (-a c-b c x+1) (\log (-a c-b c x+1)-2 \log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, c*(a + b*x)]/x,x]

[Out]

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]*PolyL
og[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)]

________________________________________________________________________________________

Maple [F]  time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it polylog} \left ( 2,c \left ( bx+a \right ) \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,c*(b*x+a))/x,x)

[Out]

int(polylog(2,c*(b*x+a))/x,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/x,x, algorithm="maxima")

[Out]

integrate(dilog((b*x + a)*c)/x, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm Li}_2\left (b c x + a c\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/x,x, algorithm="fricas")

[Out]

integral(dilog(b*c*x + a*c)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/x,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(dilog((b*x + a)*c)/x, x)