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

Optimal. Leaf size=26 \[ \text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}(4,e x) \]

[Out]

(a + b*Log[c*x^n])*PolyLog[3, e*x] - b*n*PolyLog[4, e*x]

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Rubi [A]  time = 0.0293442, antiderivative size = 26, normalized size of antiderivative = 1., 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 = {2383, 6589} \[ \text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}(4,e x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*Log[c*x^n])*PolyLog[3, e*x] - b*n*PolyLog[4, e*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{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x} \, dx &=\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)-(b n) \int \frac{\text{Li}_3(e x)}{x} \, dx\\ &=\left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(e x)-b n \text{Li}_4(e x)\\ \end{align*}

Mathematica [A]  time = 0.0026455, size = 30, normalized size = 1.15 \[ a \text{PolyLog}(3,e x)+b \text{PolyLog}(3,e x) \log \left (c x^n\right )-b n \text{PolyLog}(4,e x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*PolyLog[3, e*x] + b*Log[c*x^n]*PolyLog[3, e*x] - b*n*PolyLog[4, e*x]

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Maple [F]  time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\it polylog} \left ( 2,ex \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (b n \log \left (x\right )^{2} - 2 \, b \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )}{\rm Li}_2\left (e x\right ) + \frac{1}{2} \, \int \frac{2 \, b \log \left (-e x + 1\right ) \log \left (x\right ) \log \left (x^{n}\right ) -{\left (b n \log \left (x\right )^{2} - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )} \log \left (-e x + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*dilog(e*x) + 1/2*integrate((2*b*log(-e*x +
1)*log(x)*log(x^n) - (b*n*log(x)^2 - 2*(b*log(c) + a)*log(x))*log(-e*x + 1))/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b{\rm Li}_2\left (e x\right ) \log \left (c x^{n}\right ) + a{\rm Li}_2\left (e x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 123.353, size = 26, normalized size = 1. \begin{align*} a \operatorname{Li}_{3}\left (e x\right ) + b \left (- n \operatorname{Li}_{4}\left (e x\right ) + \log{\left (c x^{n} \right )} \operatorname{Li}_{3}\left (e x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*polylog(2,e*x)/x,x)

[Out]

a*polylog(3, e*x) + b*(-n*polylog(4, e*x) + log(c*x**n)*polylog(3, e*x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}{\rm Li}_2\left (e x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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