\(\int \frac {(a+b \log (c x^n))^2 \operatorname {PolyLog}(k,e x^q)}{x} \, dx\) [206]

Optimal result

Integrand size = 23, antiderivative size = 72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (1+k,e x^q\right )}{q}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2+k,e x^q\right )}{q^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3+k,e x^q\right )}{q^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\frac {q^2 \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (1+k,e x^q\right )+2 b n \left (-q \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2+k,e x^q\right )+b n \operatorname {PolyLog}\left (3+k,e x^q\right )\right )}{q^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2830, 2830, 7143}

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.

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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