3.2.41 \(\int \frac {\operatorname {PolyLog}(2,c (a+b x))}{d+e x} \, dx\) [141]

3.2.41.1 Optimal result

Integrand size = 17, antiderivative size = 591 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{d+e x} \, dx=\frac {\left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 e}+\frac {\log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{e}-\frac {\left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 e}+\frac {\log (d+e x) \operatorname {PolyLog}(2,c (a+b x))}{e}+\frac {\left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \operatorname {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {\left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \operatorname {PolyLog}(2,1-c (a+b x))}{e}-\frac {\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \operatorname {PolyLog}\left (2,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \operatorname {PolyLog}\left (2,\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{e}-\frac {\operatorname {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {\operatorname {PolyLog}(3,1-c (a+b x))}{e}-\frac {\operatorname {PolyLog}\left (3,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac {\operatorname {PolyLog}\left (3,\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{e} \]

output

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

3.2.41.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.05 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{d+e x} \, dx=\frac {\log (c (a+b x)) \log (1-a c-b c x) \log (d+e x)+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \left (-2 \log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{b d-a e}\right )\right )+\left (-\log (c (a+b x))+\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )+\frac {1}{2} \log ^2\left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\log (c (a+b x))-\log \left (\frac {(b c d+e-a c e) (a+b x)}{(b d-a e) (-1+a c+b c x)}\right )+\log \left (\frac {b c d+e-a c e}{e-a c e-b c e x}\right )\right )+\log (d+e x) \operatorname {PolyLog}(2,c (a+b x))+\left (\log (d+e x)-\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \operatorname {PolyLog}(2,1-a c-b c x)+\left (\log (1-a c-b c x)+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\operatorname {PolyLog}\left (2,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )-\operatorname {PolyLog}\left (2,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right )-\operatorname {PolyLog}(3,1-a c-b c x)-\operatorname {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )-\operatorname {PolyLog}\left (3,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )+\operatorname {PolyLog}\left (3,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )}{e} \]

input

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

output

(Log[c*(a + b*x)]*Log[1 - a*c - b*c*x]*Log[d + e*x] + ((Log[c*(a + b*x)] - 
 Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*(-2*Log 
[1 - a*c - b*c*x] + Log[(b*(d + e*x))/(b*d - a*e)]))/2 + (-Log[c*(a + b*x) 
] + Log[(e*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*Log[ 
-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))] + (Log[-((b*(d + e*x))/ 
((b*d - a*e)*(-1 + a*c + b*c*x)))]^2*(Log[c*(a + b*x)] - Log[((b*c*d + e - 
 a*c*e)*(a + b*x))/((b*d - a*e)*(-1 + a*c + b*c*x))] + Log[(b*c*d + e - a* 
c*e)/(e - a*c*e - b*c*e*x)]))/2 + Log[d + e*x]*PolyLog[2, c*(a + b*x)] + ( 
Log[d + e*x] - Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*Pol 
yLog[2, 1 - a*c - b*c*x] + (Log[1 - a*c - b*c*x] + Log[-((b*(d + e*x))/((b 
*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)] + L 
og[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))]*(PolyLog[2, (b*c*(d 
+ e*x))/(e*(-1 + a*c + b*c*x))] - PolyLog[2, -((b*(d + e*x))/((b*d - a*e)* 
(-1 + a*c + b*c*x)))]) - PolyLog[3, 1 - a*c - b*c*x] - PolyLog[3, (b*(d + 
e*x))/(b*d - a*e)] - PolyLog[3, (b*c*(d + e*x))/(e*(-1 + a*c + b*c*x))] + 
PolyLog[3, -((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])/e
 

3.2.41.3 Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, 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.

\(\displaystyle \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{d+e x} \, dx\)

\(\Big \downarrow \) 7151

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

\(\Big \downarrow \) 2890

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

\(\Big \downarrow \) 2885

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

input

Int[PolyLog[2, c*(a + b*x)]/(d + e*x),x]
 

output

(Log[d + e*x]*PolyLog[2, c*(a + b*x)])/e + (((Log[c*(a + b*x)] + Log[(b*c* 
d + e - a*c*e)/(c*(b*(d - (a*e)/b) + e*(a + b*x)))] - Log[((b*c*d + e - a* 
c*e)*(a + b*x))/(b*(d - (a*e)/b) + e*(a + b*x))])*Log[(b*(d - (a*e)/b) + e 
*(a + b*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/2 - ((Log[c*(a + b*x)] - L 
og[-((e*(a + b*x))/(b*d - a*e))])*(Log[1 - c*(a + b*x)] + Log[(b*(d - (a*e 
)/b) + e*(a + b*x))/((b*d - a*e)*(1 - c*(a + b*x)))])^2)/2 + Log[c*(a + b* 
x)]*Log[1 - c*(a + b*x)]*Log[d - (a*e)/b + (e*(a + b*x))/b] - (Log[(b*(d - 
 (a*e)/b) + e*(a + b*x))/((b*d - a*e)*(1 - c*(a + b*x)))] - Log[d - (a*e)/ 
b + (e*(a + b*x))/b])*PolyLog[2, 1 - c*(a + b*x)] - Log[(b*(d - (a*e)/b) + 
 e*(a + b*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + 
 b*x)))/(c*(b*(d - (a*e)/b) + e*(a + b*x))))] + Log[(b*(d - (a*e)/b) + e*( 
a + b*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, (b*(d - (a*e)/b)*(1 
- c*(a + b*x)))/(b*(d - (a*e)/b) + e*(a + b*x))] + (Log[1 - c*(a + b*x)] + 
 Log[(b*(d - (a*e)/b) + e*(a + b*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*Pol 
yLog[2, 1 + (e*(a + b*x))/(b*d - a*e)] - PolyLog[3, 1 - c*(a + b*x)] - Pol 
yLog[3, -((e*(1 - c*(a + b*x)))/(c*(b*(d - (a*e)/b) + e*(a + b*x))))] + Po 
lyLog[3, (b*(d - (a*e)/b)*(1 - c*(a + b*x)))/(b*(d - (a*e)/b) + e*(a + b*x 
))] - PolyLog[3, 1 + (e*(a + b*x))/(b*d - a*e)])/e
 

3.2.41.3.1 Defintions of rubi rules used

rule 2885

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]
 

rule 2890

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]
 

rule 7151

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]
 

3.2.41.4 Maple [F]

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

input

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

output

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

3.2.41.5 Fricas [F]

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

input

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

output

integral(dilog(b*c*x + a*c)/(e*x + d), x)
 

3.2.41.6 Sympy [F]

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

input

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

output

Integral(polylog(2, a*c + b*c*x)/(d + e*x), x)
 

3.2.41.7 Maxima [F]

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

input

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

output

integrate(dilog((b*x + a)*c)/(e*x + d), x)
 

3.2.41.8 Giac [F]

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

input

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

output

integrate(dilog((b*x + a)*c)/(e*x + d), x)
 

3.2.41.9 Mupad [F(-1)]

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

input

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

output

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