3.3.19 \(\int \frac {(a+b \log (c x^n)) \operatorname {PolyLog}(3,e x)}{x^3} \, dx\) [219]

3.3.19.1 Optimal result

Integrand size = 19, antiderivative size = 238 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=-\frac {5 b e n}{16 x}+\frac {3}{16} b e^2 n \log (x)-\frac {1}{16} b e^2 n \log ^2(x)-\frac {e \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {1}{8} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{16} b e^2 n \log (1-e x)+\frac {3 b n \log (1-e x)}{16 x^2}-\frac {1}{8} e^2 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{8 x^2}-\frac {1}{8} b e^2 n \operatorname {PolyLog}(2,e x)-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{2 x^2} \]

output

-5/16*b*e*n/x+3/16*b*e^2*n*ln(x)-1/16*b*e^2*n*ln(x)^2-1/8*e*(a+b*ln(c*x^n) 
)/x+1/8*e^2*ln(x)*(a+b*ln(c*x^n))-3/16*b*e^2*n*ln(-e*x+1)+3/16*b*n*ln(-e*x 
+1)/x^2-1/8*e^2*(a+b*ln(c*x^n))*ln(-e*x+1)+1/8*(a+b*ln(c*x^n))*ln(-e*x+1)/ 
x^2-1/8*b*e^2*n*polylog(2,e*x)-1/4*b*n*polylog(2,e*x)/x^2-1/4*(a+b*ln(c*x^ 
n))*polylog(2,e*x)/x^2-1/4*b*n*polylog(3,e*x)/x^2-1/2*(a+b*ln(c*x^n))*poly 
log(3,e*x)/x^2
 

3.3.19.2 Mathematica [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=\int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx \]

input

Integrate[((a + b*Log[c*x^n])*PolyLog[3, e*x])/x^3,x]
 

output

Integrate[((a + b*Log[c*x^n])*PolyLog[3, e*x])/x^3, x]
 

3.3.19.3 Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.47, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {2832, 2832, 25, 2823, 2009, 2842, 54, 2009, 7145, 25, 2842, 54, 2009}

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.

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

\(\Big \downarrow \) 2832

\(\displaystyle \frac {1}{2} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^3}dx+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2832

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int -\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^3}dx+\frac {1}{4} b n \int -\frac {\log (1-e x)}{x^3}dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}\right )+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^3}dx-\frac {1}{4} b n \int \frac {\log (1-e x)}{x^3}dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}\right )+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2823

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (b n \int \left (-\frac {\log (x) e^2}{2 x}+\frac {\log (1-e x) e^2}{2 x}+\frac {e}{2 x^2}-\frac {\log (1-e x)}{2 x^3}\right )dx+\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}\right )-\frac {1}{4} b n \int \frac {\log (1-e x)}{x^3}dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}\right )+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b n \int \frac {\log (1-e x)}{x^3}dx+\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}\right )+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b n \left (-\frac {1}{2} e \int \frac {1}{x^2 (1-e x)}dx-\frac {\log (1-e x)}{2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}\right )+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {1}{2} \left (-\frac {1}{4} b n \left (-\frac {1}{2} e \int \left (-\frac {e^2}{e x-1}+\frac {e}{x}+\frac {1}{x^2}\right )dx-\frac {\log (1-e x)}{2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}\right )+\frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} b n \int \frac {\operatorname {PolyLog}(2,e x)}{x^3}dx+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {1}{4} b n \left (-\frac {\log (1-e x)}{2 x^2}-\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )\right )-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 7145

\(\displaystyle \frac {1}{4} b n \left (\frac {1}{2} \int -\frac {\log (1-e x)}{x^3}dx-\frac {\operatorname {PolyLog}(2,e x)}{2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {1}{4} b n \left (-\frac {\log (1-e x)}{2 x^2}-\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )\right )-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} b n \left (-\frac {1}{2} \int \frac {\log (1-e x)}{x^3}dx-\frac {\operatorname {PolyLog}(2,e x)}{2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {1}{4} b n \left (-\frac {\log (1-e x)}{2 x^2}-\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )\right )-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {1}{4} b n \left (\frac {1}{2} \left (\frac {1}{2} e \int \frac {1}{x^2 (1-e x)}dx+\frac {\log (1-e x)}{2 x^2}\right )-\frac {\operatorname {PolyLog}(2,e x)}{2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {1}{4} b n \left (-\frac {\log (1-e x)}{2 x^2}-\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )\right )-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {1}{4} b n \left (\frac {1}{2} \left (\frac {1}{2} e \int \left (-\frac {e^2}{e x-1}+\frac {e}{x}+\frac {1}{x^2}\right )dx+\frac {\log (1-e x)}{2 x^2}\right )-\frac {\operatorname {PolyLog}(2,e x)}{2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {1}{4} b n \left (-\frac {\log (1-e x)}{2 x^2}-\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )\right )-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} e^2 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {e \left (a+b \log \left (c x^n\right )\right )}{2 x}+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+b n \left (-\frac {1}{2} e^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} e^2 \log ^2(x)+\frac {1}{4} e^2 \log (x)-\frac {1}{4} e^2 \log (1-e x)+\frac {\log (1-e x)}{4 x^2}-\frac {3 e}{4 x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(2,e x)}{4 x^2}-\frac {1}{4} b n \left (-\frac {\log (1-e x)}{2 x^2}-\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )\right )-\frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b n \operatorname {PolyLog}(3,e x)}{4 x^2}+\frac {1}{4} b n \left (\frac {1}{2} \left (\frac {\log (1-e x)}{2 x^2}+\frac {1}{2} e \left (e \log (x)-e \log (1-e x)-\frac {1}{x}\right )\right )-\frac {\operatorname {PolyLog}(2,e x)}{2 x^2}\right )\)

input

Int[((a + b*Log[c*x^n])*PolyLog[3, e*x])/x^3,x]
 

output

(b*n*((Log[1 - e*x]/(2*x^2) + (e*(-x^(-1) + e*Log[x] - e*Log[1 - e*x]))/2) 
/2 - PolyLog[2, e*x]/(2*x^2)))/4 + (-1/4*(b*n*(-1/2*Log[1 - e*x]/x^2 - (e* 
(-x^(-1) + e*Log[x] - e*Log[1 - e*x]))/2)) - (b*n*PolyLog[2, e*x])/(4*x^2) 
 - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/(2*x^2) + (-1/2*(e*(a + b*Log[c*x^ 
n]))/x + (e^2*Log[x]*(a + b*Log[c*x^n]))/2 - (e^2*(a + b*Log[c*x^n])*Log[1 
 - e*x])/2 + ((a + b*Log[c*x^n])*Log[1 - e*x])/(2*x^2) + b*n*((-3*e)/(4*x) 
 + (e^2*Log[x])/4 - (e^2*Log[x]^2)/4 - (e^2*Log[1 - e*x])/4 + Log[1 - e*x] 
/(4*x^2) - (e^2*PolyLog[2, e*x])/2))/2)/2 - (b*n*PolyLog[3, e*x])/(4*x^2) 
- ((a + b*Log[c*x^n])*PolyLog[3, e*x])/(2*x^2)
 

3.3.19.3.1 Defintions of rubi rules used

rule 25

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

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

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]
 

rule 7145

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo 
l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[p 
*(q/(m + 1))   Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, 
 b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
 

3.3.19.4 Maple [F]

\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \operatorname {Li}_{3}\left (e x \right )}{x^{3}}d x\]

input

int((a+b*ln(c*x^n))*polylog(3,e*x)/x^3,x)
 

output

int((a+b*ln(c*x^n))*polylog(3,e*x)/x^3,x)
 

3.3.19.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=\frac {b e^{2} n x^{2} \log \left (x\right )^{2} - {\left (5 \, b e n + 2 \, a e\right )} x - 2 \, {\left (b e^{2} n x^{2} + 2 \, b n + 2 \, a\right )} {\rm Li}_2\left (e x\right ) - {\left ({\left (3 \, b e^{2} n + 2 \, a e^{2}\right )} x^{2} - 3 \, b n - 2 \, a\right )} \log \left (-e x + 1\right ) - 2 \, {\left (b e x + 2 \, b {\rm Li}_2\left (e x\right ) + {\left (b e^{2} x^{2} - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + {\left (2 \, b e^{2} x^{2} \log \left (c\right ) - 2 \, b e n x + {\left (3 \, b e^{2} n + 2 \, a e^{2}\right )} x^{2} - 4 \, b n {\rm Li}_2\left (e x\right ) - 2 \, {\left (b e^{2} n x^{2} - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right ) - 4 \, {\left (2 \, b n \log \left (x\right ) + b n + 2 \, b \log \left (c\right ) + 2 \, a\right )} {\rm polylog}\left (3, e x\right )}{16 \, x^{2}} \]

input

integrate((a+b*log(c*x^n))*polylog(3,e*x)/x^3,x, algorithm="fricas")
 

output

1/16*(b*e^2*n*x^2*log(x)^2 - (5*b*e*n + 2*a*e)*x - 2*(b*e^2*n*x^2 + 2*b*n 
+ 2*a)*dilog(e*x) - ((3*b*e^2*n + 2*a*e^2)*x^2 - 3*b*n - 2*a)*log(-e*x + 1 
) - 2*(b*e*x + 2*b*dilog(e*x) + (b*e^2*x^2 - b)*log(-e*x + 1))*log(c) + (2 
*b*e^2*x^2*log(c) - 2*b*e*n*x + (3*b*e^2*n + 2*a*e^2)*x^2 - 4*b*n*dilog(e* 
x) - 2*(b*e^2*n*x^2 - b*n)*log(-e*x + 1))*log(x) - 4*(2*b*n*log(x) + b*n + 
 2*b*log(c) + 2*a)*polylog(3, e*x))/x^2
 

3.3.19.6 Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )}{x^{3}}\, dx \]

input

integrate((a+b*ln(c*x**n))*polylog(3,e*x)/x**3,x)
 

output

Integral((a + b*log(c*x**n))*polylog(3, e*x)/x**3, x)
 

3.3.19.7 Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x^{3}} \,d x } \]

input

integrate((a+b*log(c*x^n))*polylog(3,e*x)/x^3,x, algorithm="maxima")
 

output

1/8*(e^2*log(x) - (e*x + (e^2*x^2 - 1)*log(-e*x + 1) + 2*dilog(e*x) + 4*po 
lylog(3, e*x))/x^2)*a - 1/16*b*((4*(n + log(c) + log(x^n))*dilog(e*x) - (2 
*e^2*n*x^2*log(x) + 3*n + 2*log(c))*log(-e*x + 1) - 2*(e^2*x^2*log(x) - e* 
x - (e^2*x^2 - 1)*log(-e*x + 1))*log(x^n) + 4*(n + 2*log(c) + 2*log(x^n))* 
polylog(3, e*x))/x^2 + 16*integrate(-1/16*(2*e^2*n*x - 5*e*n - 2*e*log(c) 
- 2*(2*e^3*n*x^2 - e^2*n*x)*log(x))/(e*x^3 - x^2), x))
 

3.3.19.8 Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x^{3}} \,d x } \]

input

integrate((a+b*log(c*x^n))*polylog(3,e*x)/x^3,x, algorithm="giac")
 

output

integrate((b*log(c*x^n) + a)*polylog(3, e*x)/x^3, x)
 

3.3.19.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x^3} \, dx=\text {Hanged} \]

input

int((polylog(3, e*x)*(a + b*log(c*x^n)))/x^3,x)
 

output

\text{Hanged}