3.116 \(\int \frac{\text{PolyLog}(n,a x)}{x} \, dx\)

Optimal. Leaf size=7 \[ \text{PolyLog}(n+1,a x) \]

[Out]

PolyLog[1 + n, a*x]

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Rubi [A]  time = 0.0089991, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6589} \[ \text{PolyLog}(n+1,a x) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[n, a*x]/x,x]

[Out]

PolyLog[1 + n, a*x]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin{align*} \int \frac{\text{Li}_n(a x)}{x} \, dx &=\text{Li}_{1+n}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0011286, size = 7, normalized size = 1. \[ \text{PolyLog}(n+1,a x) \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[n, a*x]/x,x]

[Out]

PolyLog[1 + n, a*x]

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Maple [A]  time = 0.044, size = 8, normalized size = 1.1 \begin{align*}{\it polylog} \left ( 1+n,ax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(n,a*x)/x,x)

[Out]

polylog(1+n,a*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,a*x)/x,x, algorithm="maxima")

[Out]

integrate(polylog(n, a*x)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm polylog}\left (n, a x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,a*x)/x,x, algorithm="fricas")

[Out]

integral(polylog(n, a*x)/x, x)

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Sympy [A]  time = 0.528206, size = 5, normalized size = 0.71 \begin{align*} \operatorname{Li}_{n + 1}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,a*x)/x,x)

[Out]

polylog(n + 1, a*x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,a*x)/x,x, algorithm="giac")

[Out]

integrate(polylog(n, a*x)/x, x)