Optimal. Leaf size=103 \[ \frac {q^2 \text {Int}\left (\frac {\text {Li}_{k-2}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )},x\right )}{2 b^2 n^2}-\frac {q \text {Li}_{k-1}\left (e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\text {Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx &=-\frac {\text {Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {q \int \frac {\text {Li}_{-1+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac {q \text {Li}_{-1+k}\left (e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {q^2 \int \frac {\text {Li}_{-2+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b^2 n^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\text {Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm polylog}\left (k, e x^{q}\right )}{b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x \log \left (c x^{n}\right ) + a^{3} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (k , e \,x^{q}\right )}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (k,e\,x^q\right )}{x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{k}\left (e x^{q}\right )}{x \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________