3.203 \(\int \frac{\text{PolyLog}(k,e x^q)}{x (a+b \log (c x^n))^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{q \text{Unintegrable}\left (\frac{\text{PolyLog}\left (k-1,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )},x\right )}{b n}-\frac{\text{PolyLog}\left (k,e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.070842, 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 = {} \[ \int \frac{\text{PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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 )^2} \, dx &=-\frac{\text{Li}_k\left (e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac{q \int \frac{\text{Li}_{-1+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0441017, size = 0, normalized size = 0. \[ \int \frac{\text{PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.025, 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 ) ^{2}}}\, 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))^2,x)

[Out]

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

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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 )}^{2} 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm polylog}\left (k, e x^{q}\right )}{b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} 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))^2,x, algorithm="fricas")

[Out]

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

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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 )^{2}}\, 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))**2,x)

[Out]

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

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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 )}^{2} 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))^2,x, algorithm="giac")

[Out]

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