Integrand size = 23, antiderivative size = 72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (1+k,e x^q\right )}{q}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2+k,e x^q\right )}{q^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3+k,e x^q\right )}{q^3} \]
(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
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\frac {q^2 \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (1+k,e x^q\right )+2 b n \left (-q \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2+k,e x^q\right )+b n \operatorname {PolyLog}\left (3+k,e x^q\right )\right )}{q^3} \]
(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
Time = 0.34 (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 |
\(\displaystyle \frac {\operatorname {PolyLog}\left (k+1,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )^2}{q}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (k+2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{q}-\frac {b n \operatorname {PolyLog}\left (k+3,e x^q\right )}{q^2}\right )}{q}\) |
((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
3.2.100.3.1 Defintions of rubi rules used
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]
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]
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \operatorname {Li}_{k}\left (e \,x^{q}\right )}{x}d x\]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} {\rm Li}_{k}(e x^{q})}{x} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2} \operatorname {Li}_{k}\left (e x^{q}\right )}{x}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} {\rm Li}_{k}(e x^{q})}{x} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} {\rm Li}_{k}(e x^{q})}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (k,e x^q\right )}{x} \, dx=\int \frac {\mathrm {polylog}\left (k,e\,x^q\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]