3.204 \(\int \frac{\text{PolyLog}(k,e x^q)}{x (a+b \log (c x^n))^3} \, dx\)

Optimal. Leaf size=102 \[ \frac{q^2 \text{Unintegrable}\left (\frac{\text{PolyLog}\left (k-2,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )},x\right )}{2 b^2 n^2}-\frac{q \text{PolyLog}\left (k-1,e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{\text{PolyLog}\left (k,e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.111534, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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]

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

[Out]

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

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 = 0.0474568, size = 0, normalized size = 0. \[ \int \frac{\text{PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it polylog} \left ( k,e{x}^{q} \right ) }{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{k}\left (e x^{q}\right )}{x \left (a + b \log{\left (c x^{n} \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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