3.206 \(\int \frac{\log ^2(x) \text{PolyLog}(n,a x)}{x} \, dx\)

Optimal. Leaf size=33 \[ 2 \text{PolyLog}(n+3,a x)+\log ^2(x) \text{PolyLog}(n+1,a x)-2 \log (x) \text{PolyLog}(n+2,a x) \]

[Out]

Log[x]^2*PolyLog[1 + n, a*x] - 2*Log[x]*PolyLog[2 + n, a*x] + 2*PolyLog[3 + n, a*x]

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Rubi [A]  time = 0.0443116, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2383, 6589} \[ 2 \text{PolyLog}(n+3,a x)+\log ^2(x) \text{PolyLog}(n+1,a x)-2 \log (x) \text{PolyLog}(n+2,a x) \]

Antiderivative was successfully verified.

[In]

Int[(Log[x]^2*PolyLog[n, a*x])/x,x]

[Out]

Log[x]^2*PolyLog[1 + n, a*x] - 2*Log[x]*PolyLog[2 + n, a*x] + 2*PolyLog[3 + n, a*x]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(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]

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{\log ^2(x) \text{Li}_n(a x)}{x} \, dx &=\log ^2(x) \text{Li}_{1+n}(a x)-2 \int \frac{\log (x) \text{Li}_{1+n}(a x)}{x} \, dx\\ &=\log ^2(x) \text{Li}_{1+n}(a x)-2 \log (x) \text{Li}_{2+n}(a x)+2 \int \frac{\text{Li}_{2+n}(a x)}{x} \, dx\\ &=\log ^2(x) \text{Li}_{1+n}(a x)-2 \log (x) \text{Li}_{2+n}(a x)+2 \text{Li}_{3+n}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0027551, size = 33, normalized size = 1. \[ 2 \text{PolyLog}(n+3,a x)+\log ^2(x) \text{PolyLog}(n+1,a x)-2 \log (x) \text{PolyLog}(n+2,a x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[x]^2*PolyLog[n, a*x])/x,x]

[Out]

Log[x]^2*PolyLog[1 + n, a*x] - 2*Log[x]*PolyLog[2 + n, a*x] + 2*PolyLog[3 + n, a*x]

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}{\it polylog} \left ( n,ax \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)^2*polylog(n,a*x)/x,x)

[Out]

int(ln(x)^2*polylog(n,a*x)/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )^{2}{\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)^2*polylog(n,a*x)/x,x, algorithm="maxima")

[Out]

integrate(log(x)^2*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{\log \left (x\right )^{2}{\rm polylog}\left (n, a x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)^2*polylog(n,a*x)/x,x, algorithm="fricas")

[Out]

integral(log(x)^2*polylog(n, a*x)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x \right )}^{2} \operatorname{Li}_{n}\left (a x\right )}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)**2*polylog(n,a*x)/x,x)

[Out]

Integral(log(x)**2*polylog(n, a*x)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )^{2}{\rm Li}_{n}(a x)}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)^2*polylog(n,a*x)/x,x, algorithm="giac")

[Out]

integrate(log(x)^2*polylog(n, a*x)/x, x)