3.3.17 \(\int \frac {(a+b \log (c x^n)) \text {Li}_3(e x)}{x} \, dx\) [217]

Optimal. Leaf size=26 \[ \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(e x)-b n \text {Li}_5(e x) \]

[Out]

(a+b*ln(c*x^n))*polylog(4,e*x)-b*n*polylog(5,e*x)

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2430, 6724} \begin {gather*} \text {PolyLog}(4,e x) \left (a+b \log \left (c x^n\right )\right )-b n \text {PolyLog}(5,e x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*Log[c*x^n])*PolyLog[4, e*x] - b*n*PolyLog[5, e*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 {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)}{x} \, dx &=\left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(e x)-(b n) \int \frac {\text {Li}_4(e x)}{x} \, dx\\ &=\left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(e x)-b n \text {Li}_5(e x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.15 \begin {gather*} a \text {Li}_4(e x)+b \log \left (c x^n\right ) \text {Li}_4(e x)-b n \text {Li}_5(e x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*PolyLog[4, e*x] + b*Log[c*x^n]*PolyLog[4, e*x] - b*n*PolyLog[5, e*x]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, e x \right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/6*(2*b*n*log(x)^3 - 3*b*log(x)^2*log(x^n) - 3*(b*log(c) + a)*log(x)^2)*dilog(x*e) - 1/2*(b*n*log(x)^2 - 2*b*
log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*polylog(3, x*e) - 1/6*integrate((3*b*log(-x*e + 1)*log(x)^2*log(x^n
) - (2*b*n*log(x)^3 - 3*(b*log(c) + a)*log(x)^2)*log(-x*e + 1))/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((a+b*log(c*x^n))*polylog(3,e*x)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 5.11, size = 26, normalized size = 1.00 \begin {gather*} a \operatorname {Li}_{4}\left (e x\right ) + b \left (- n \operatorname {Li}_{5}\left (e x\right ) + \log {\left (c x^{n} \right )} \operatorname {Li}_{4}\left (e x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*polylog(4, e*x) + b*(-n*polylog(5, e*x) + log(c*x**n)*polylog(4, e*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((a+b*log(c*x^n))*polylog(3,e*x)/x,x, algorithm="giac")

[Out]

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

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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