\(\int \frac {(a+b \log (c x^n)) \operatorname {PolyLog}(2,e x)}{x^2} \, dx\) [218]

Optimal result

Integrand size = 19, antiderivative size = 142 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=2 b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-2 b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \operatorname {PolyLog}(2,e x)-\frac {b n \operatorname {PolyLog}(2,e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) (e x \log (x)+(1-e x) \log (1-e x)-\operatorname {PolyLog}(2,e x))}{x}+\frac {b n \left (e x \log ^2(x)-4 (-1+e x) \log (1-e x)+\log (x) (4 e x+(2-2 e x) \log (1-e x))-2 (1+e x+\log (x)) \operatorname {PolyLog}(2,e x)\right )}{2 x} \] Input:

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

Output:

((a - b*n*Log[x] + b*Log[c*x^n])*(e*x*Log[x] + (1 - e*x)*Log[1 - e*x] - Po 
lyLog[2, e*x]))/x + (b*n*(e*x*Log[x]^2 - 4*(-1 + e*x)*Log[1 - e*x] + Log[x 
]*(4*e*x + (2 - 2*e*x)*Log[1 - e*x]) - 2*(1 + e*x + Log[x])*PolyLog[2, e*x 
]))/(2*x)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2832, 25, 2823, 2009, 2842, 47, 14, 16}

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.

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

\(\Big \downarrow \) 2832

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2823

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2842

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 2009

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

rule 2823

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n])   u, x] - Simp[b*n   Int[1/x 
 u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q 
+ 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
 

rule 2832

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.)*PolyLog[k_, (e 
_.)*(x_)^(q_.)], x_Symbol] :> Simp[(-b)*n*(d*x)^(m + 1)*(PolyLog[k, e*x^q]/ 
(d*(m + 1)^2)), x] + (Simp[(d*x)^(m + 1)*PolyLog[k, e*x^q]*((a + b*Log[c*x^ 
n])/(d*(m + 1))), x] - Simp[q/(m + 1)   Int[(d*x)^m*PolyLog[k - 1, e*x^q]*( 
a + b*Log[c*x^n]), x], x] + Simp[b*n*(q/(m + 1)^2)   Int[(d*x)^m*PolyLog[k 
- 1, e*x^q], x], x]) /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
 

rule 2842

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

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.44

method	result	size
parallelrisch	\(\frac {-2 b \,e^{2} \ln \left (-e x +1\right ) \ln \left (c \,x^{n}\right ) x n -2 x \operatorname {polylog}\left (2, e x \right ) b \,e^{2} n^{2}-4 x \ln \left (-e x +1\right ) b \,e^{2} n^{2}+b \,e^{2} \ln \left (c \,x^{n}\right )^{2} x +4 x \ln \left (c \,x^{n}\right ) b \,e^{2} n -2 x \ln \left (-e x +1\right ) a \,e^{2} n +2 x \ln \left (c \,x^{n}\right ) a \,e^{2}-2 \ln \left (c \,x^{n}\right ) \operatorname {polylog}\left (2, e x \right ) b e n +2 b \ln \left (-e x +1\right ) \ln \left (c \,x^{n}\right ) e n -2 \operatorname {polylog}\left (2, e x \right ) b e \,n^{2}+4 \ln \left (-e x +1\right ) b e \,n^{2}-2 \operatorname {polylog}\left (2, e x \right ) a e n +2 \ln \left (-e x +1\right ) a e n}{2 x e n}\)	\(204\)

Input:

int((a+b*ln(c*x^n))*polylog(2,e*x)/x^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*(-2*b*e^2*ln(-e*x+1)*ln(c*x^n)*x*n-2*x*polylog(2,e*x)*b*e^2*n^2-4*x*ln 
(-e*x+1)*b*e^2*n^2+b*e^2*ln(c*x^n)^2*x+4*x*ln(c*x^n)*b*e^2*n-2*x*ln(-e*x+1 
)*a*e^2*n+2*x*ln(c*x^n)*a*e^2-2*ln(c*x^n)*polylog(2,e*x)*b*e*n+2*b*ln(-e*x 
+1)*ln(c*x^n)*e*n-2*polylog(2,e*x)*b*e*n^2+4*ln(-e*x+1)*b*e*n^2-2*polylog( 
2,e*x)*a*e*n+2*ln(-e*x+1)*a*e*n)/x/e/n
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\frac {b e n x \log \left (x\right )^{2} - 2 \, {\left (b e n x + b n + a\right )} {\rm Li}_2\left (e x\right ) + 2 \, {\left (2 \, b n - {\left (2 \, b e n + a e\right )} x + a\right )} \log \left (-e x + 1\right ) - 2 \, {\left (b {\rm Li}_2\left (e x\right ) + {\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + 2 \, {\left (b e x \log \left (c\right ) - b n {\rm Li}_2\left (e x\right ) + {\left (2 \, b e n + a e\right )} x - {\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right )}{2 \, x} \] Input:

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

Output:

1/2*(b*e*n*x*log(x)^2 - 2*(b*e*n*x + b*n + a)*dilog(e*x) + 2*(2*b*n - (2*b 
*e*n + a*e)*x + a)*log(-e*x + 1) - 2*(b*dilog(e*x) + (b*e*x - b)*log(-e*x 
+ 1))*log(c) + 2*(b*e*x*log(c) - b*n*dilog(e*x) + (2*b*e*n + a*e)*x - (b*e 
*n*x - b*n)*log(-e*x + 1))*log(x))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

integrate((b*log(c*x^n) + a)*dilog(e*x)/x^2, x)
 

Mupad [F(-1)]

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

int((polylog(2, e*x)*(a + b*log(c*x^n)))/x^2,x)
                                                                                    
                                                                                    
 

Output:

int((polylog(2, e*x)*(a + b*log(c*x^n)))/x^2, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\left (\int \frac {\mathit {polylog}\left (2, e x \right )}{x^{2}}d x \right ) a +\left (\int \frac {\mathrm {log}\left (x^{n} c \right ) \mathit {polylog}\left (2, e x \right )}{x^{2}}d x \right ) b \] Input:

int((a+b*log(c*x^n))*polylog(2,e*x)/x^2,x)
 

Output:

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