3.3.6 \(\int \frac {\log ^2(x) \text {Li}_n(a x)}{x} \, dx\) [206]

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

[Out]

ln(x)^2*polylog(1+n,a*x)-2*ln(x)*polylog(2+n,a*x)+2*polylog(3+n,a*x)

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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {2430, 6724} \begin {gather*} 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) \end {gather*}

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 6724

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*}

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Mathematica [A]
time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (x \right )^{2} \polylog \left (n , a x \right )}{x}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x \right )}^{2} \operatorname {Li}_{n}\left (a x\right )}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\ln \left (x\right )}^2\,\mathrm {polylog}\left (n,a\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*polylog(n, a*x))/x,x)

[Out]

int((log(x)^2*polylog(n, a*x))/x, x)

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