3.2.88 \(\int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx\) [188]

3.2.88.1 Optimal result

Integrand size = 21, antiderivative size = 131 \[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=\frac {a (1-c x) \log ^2(1-c x)}{x}+a c \log (c x) \log ^2(1-c x)-2 a c \operatorname {PolyLog}(2,c x)+a c \log (1-c x) \operatorname {PolyLog}(2,c x)-\frac {a \log (1-c x) \operatorname {PolyLog}(2,c x)}{x}-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2+2 a c \log (1-c x) \operatorname {PolyLog}(2,1-c x)-a c \operatorname {PolyLog}(3,c x)-2 a c \operatorname {PolyLog}(3,1-c x) \]

output

a*(-c*x+1)*ln(-c*x+1)^2/x+a*c*ln(c*x)*ln(-c*x+1)^2-2*a*c*polylog(2,c*x)+a* 
c*ln(-c*x+1)*polylog(2,c*x)-a*ln(-c*x+1)*polylog(2,c*x)/x-1/2*b*polylog(2, 
c*x)^2+2*a*c*ln(-c*x+1)*polylog(2,-c*x+1)-a*c*polylog(3,c*x)-2*a*c*polylog 
(3,-c*x+1)
 

3.2.88.2 Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=2 a c \log (c x) \log (1-c x)-a c \log ^2(1-c x)+\frac {a \log ^2(1-c x)}{x}+a c \log (c x) \log ^2(1-c x)+\frac {a (-1+c x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x}-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2+2 a c (1+\log (1-c x)) \operatorname {PolyLog}(2,1-c x)-a c \operatorname {PolyLog}(3,c x)-2 a c \operatorname {PolyLog}(3,1-c x) \]

input

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

output

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

3.2.88.3 Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {7159, 27, 7155, 7157, 2009, 2844, 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.

\(\displaystyle \int \frac {(a+b x) \operatorname {PolyLog}(2,c x) \log (1-c x)}{x^2} \, dx\)

\(\Big \downarrow \) 7159

\(\displaystyle \int \frac {a \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2}dx+b \int \frac {\log (1-c x) \operatorname {PolyLog}(2,c x)}{x}dx\)

\(\Big \downarrow \) 27

\(\displaystyle a \int \frac {\log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2}dx+b \int \frac {\log (1-c x) \operatorname {PolyLog}(2,c x)}{x}dx\)

\(\Big \downarrow \) 7155

\(\displaystyle a \int \frac {\log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2}dx-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2\)

\(\Big \downarrow \) 7157

\(\displaystyle a \left (-c \int \left (\frac {\operatorname {PolyLog}(2,c x)}{x}+\frac {c \operatorname {PolyLog}(2,c x)}{1-c x}\right )dx-\int \frac {\log ^2(1-c x)}{x^2}dx-\frac {\operatorname {PolyLog}(2,c x) \log (1-c x)}{x}\right )-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2\)

\(\Big \downarrow \) 2009

\(\displaystyle a \left (-\int \frac {\log ^2(1-c x)}{x^2}dx-c \left (\operatorname {PolyLog}(3,c x)+2 \operatorname {PolyLog}(3,1-c x)-\operatorname {PolyLog}(2,c x) \log (1-c x)-2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)-\log (c x) \log ^2(1-c x)\right )-\frac {\operatorname {PolyLog}(2,c x) \log (1-c x)}{x}\right )-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2\)

\(\Big \downarrow \) 2844

\(\displaystyle a \left (2 c \int \frac {\log (1-c x)}{x}dx-c \left (\operatorname {PolyLog}(3,c x)+2 \operatorname {PolyLog}(3,1-c x)-\operatorname {PolyLog}(2,c x) \log (1-c x)-2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)-\log (c x) \log ^2(1-c x)\right )-\frac {\operatorname {PolyLog}(2,c x) \log (1-c x)}{x}+\frac {(1-c x) \log ^2(1-c x)}{x}\right )-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2\)

\(\Big \downarrow \) 2838

\(\displaystyle a \left (-2 c \operatorname {PolyLog}(2,c x)-c \left (\operatorname {PolyLog}(3,c x)+2 \operatorname {PolyLog}(3,1-c x)-\operatorname {PolyLog}(2,c x) \log (1-c x)-2 \operatorname {PolyLog}(2,1-c x) \log (1-c x)-\log (c x) \log ^2(1-c x)\right )-\frac {\operatorname {PolyLog}(2,c x) \log (1-c x)}{x}+\frac {(1-c x) \log ^2(1-c x)}{x}\right )-\frac {1}{2} b \operatorname {PolyLog}(2,c x)^2\)

input

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

output

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

3.2.88.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2838

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 2844

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. 
)*(x_))^2, x_Symbol] :> Simp[(d + e*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f 
- d*g)*(f + g*x))), x] - Simp[b*e*n*(p/(e*f - d*g))   Int[(a + b*Log[c*(d + 
 e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] & 
& NeQ[e*f - d*g, 0] && GtQ[p, 0]
 

rule 7155

Int[(Log[1 + (e_.)*(x_)]*PolyLog[2, (c_.)*(x_)])/(x_), x_Symbol] :> Simp[-P 
olyLog[2, c*x]^2/2, x] /; FreeQ[{c, e}, x] && EqQ[c + e, 0]
 

rule 7157

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]
 

rule 7159

Int[((g_.) + Log[1 + (e_.)*(x_)]*(h_.))*(Px_)*(x_)^(m_)*PolyLog[2, (c_.)*(x 
_)], x_Symbol] :> Simp[Coeff[Px, x, -m - 1]   Int[(g + h*Log[1 + e*x])*(Pol 
yLog[2, c*x]/x), x], x] + Int[x^m*(Px - Coeff[Px, x, -m - 1]*x^(-m - 1))*(g 
 + h*Log[1 + e*x])*PolyLog[2, c*x], x] /; FreeQ[{c, e, g, h}, x] && PolyQ[P 
x, x] && ILtQ[m, 0] && EqQ[c + e, 0] && NeQ[Coeff[Px, x, -m - 1], 0]
 

3.2.88.4 Maple [F]

\[\int \frac {\left (b x +a \right ) \ln \left (-c x +1\right ) \operatorname {polylog}\left (2, c x \right )}{x^{2}}d x\]

input

int((b*x+a)*ln(-c*x+1)*polylog(2,c*x)/x^2,x)
 

output

int((b*x+a)*ln(-c*x+1)*polylog(2,c*x)/x^2,x)
 

3.2.88.5 Fricas [F]

\[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=\int { \frac {{\left (b x + a\right )} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}} \,d x } \]

input

integrate((b*x+a)*log(-c*x+1)*polylog(2,c*x)/x^2,x, algorithm="fricas")
 

output

integral((b*x + a)*dilog(c*x)*log(-c*x + 1)/x^2, x)
 

3.2.88.6 Sympy [F]

\[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=\int \frac {\left (a + b x\right ) \log {\left (- c x + 1 \right )} \operatorname {Li}_{2}\left (c x\right )}{x^{2}}\, dx \]

input

integrate((b*x+a)*ln(-c*x+1)*polylog(2,c*x)/x**2,x)
 

output

Integral((a + b*x)*log(-c*x + 1)*polylog(2, c*x)/x**2, x)
 

3.2.88.7 Maxima [F]

\[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=\int { \frac {{\left (b x + a\right )} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}} \,d x } \]

input

integrate((b*x+a)*log(-c*x+1)*polylog(2,c*x)/x^2,x, algorithm="maxima")
 

output

integrate((b*x + a)*dilog(c*x)*log(-c*x + 1)/x^2, x)
 

3.2.88.8 Giac [F]

\[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=\int { \frac {{\left (b x + a\right )} {\rm Li}_2\left (c x\right ) \log \left (-c x + 1\right )}{x^{2}} \,d x } \]

input

integrate((b*x+a)*log(-c*x+1)*polylog(2,c*x)/x^2,x, algorithm="giac")
 

output

integrate((b*x + a)*dilog(c*x)*log(-c*x + 1)/x^2, x)
 

3.2.88.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x) \log (1-c x) \operatorname {PolyLog}(2,c x)}{x^2} \, dx=\int \frac {\ln \left (1-c\,x\right )\,\mathrm {polylog}\left (2,c\,x\right )\,\left (a+b\,x\right )}{x^2} \,d x \]

input

int((log(1 - c*x)*polylog(2, c*x)*(a + b*x))/x^2,x)
 

output

int((log(1 - c*x)*polylog(2, c*x)*(a + b*x))/x^2, x)