\(\int (\frac {q \operatorname {PolyLog}(-1+k,e x^q)}{b n x (a+b \log (c x^n))}-\frac {\operatorname {PolyLog}(k,e x^q)}{x (a+b \log (c x^n))^2}) \, dx\) [213]

Optimal result

Integrand size = 57, antiderivative size = 26 \[ \int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx=\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )} \] Output:

polylog(k,e*x^q)/b/n/(a+b*ln(c*x^n))
 

Mathematica [F]

\[ \int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx=\int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {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.

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

\[ \int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx=\int { \frac {q {\rm Li}_{k - 1}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n x} - \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x} \,d x } \] Input:

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

Output:

integral(-(b*n*polylog(k, e*x^q) - (b*q*log(c*x^n) + a*q)*polylog(k - 1, e 
*x^q))/(b^3*n*x*log(c*x^n)^2 + 2*a*b^2*n*x*log(c*x^n) + a^2*b*n*x), x)
 

Sympy [F]

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

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

Output:

(Integral(a*q*polylog(k - 1, e*x**q)/(a**2*x + 2*a*b*x*log(c*x**n) + b**2* 
x*log(c*x**n)**2), x) + Integral(-b*n*polylog(k, e*x**q)/(a**2*x + 2*a*b*x 
*log(c*x**n) + b**2*x*log(c*x**n)**2), x) + Integral(b*q*log(c*x**n)*polyl 
og(k - 1, e*x**q)/(a**2*x + 2*a*b*x*log(c*x**n) + b**2*x*log(c*x**n)**2), 
x))/(b*n)
 

Maxima [F]

\[ \int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx=\int { \frac {q {\rm Li}_{k - 1}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n x} - \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx=\int { \frac {q {\rm Li}_{k - 1}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n x} - \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (\frac {q \operatorname {PolyLog}\left (-1+k,e x^q\right )}{b n x \left (a+b \log \left (c x^n\right )\right )}-\frac {\operatorname {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2}\right ) \, dx=\frac {\left (\int \frac {\mathit {polylog}\left (k -1, x^{q} e \right )}{\mathrm {log}\left (x^{n} c \right )^{2} b^{2} x +2 \,\mathrm {log}\left (x^{n} c \right ) a b x +a^{2} x}d x \right ) a q -\left (\int \frac {\mathit {polylog}\left (k , x^{q} e \right )}{\mathrm {log}\left (x^{n} c \right )^{2} b^{2} x +2 \,\mathrm {log}\left (x^{n} c \right ) a b x +a^{2} x}d x \right ) b n +\left (\int \frac {\mathrm {log}\left (x^{n} c \right ) \mathit {polylog}\left (k -1, x^{q} e \right )}{\mathrm {log}\left (x^{n} c \right )^{2} b^{2} x +2 \,\mathrm {log}\left (x^{n} c \right ) a b x +a^{2} x}d x \right ) b q}{b n} \] Input:

int(q*polylog(-1+k,e*x^q)/b/n/x/(a+b*log(c*x^n))-polylog(k,e*x^q)/x/(a+b*l 
og(c*x^n))^2,x)
 

Output:

(int(polylog(k - 1,x**q*e)/(log(x**n*c)**2*b**2*x + 2*log(x**n*c)*a*b*x + 
a**2*x),x)*a*q - int(polylog(k,x**q*e)/(log(x**n*c)**2*b**2*x + 2*log(x**n 
*c)*a*b*x + a**2*x),x)*b*n + int((log(x**n*c)*polylog(k - 1,x**q*e))/(log( 
x**n*c)**2*b**2*x + 2*log(x**n*c)*a*b*x + a**2*x),x)*b*q)/(b*n)