3.165 \(\int \frac{\log (1-c x) \text{PolyLog}(2,c x)}{x} \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{2} \text{PolyLog}(2,c x)^2 \]

[Out]

-PolyLog[2, c*x]^2/2

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Rubi [A]  time = 0.0257549, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6589, 6601} \[ -\frac{1}{2} \text{PolyLog}(2,c x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(Log[1 - c*x]*PolyLog[2, c*x])/x,x]

[Out]

-PolyLog[2, c*x]^2/2

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]

Rule 6601

Int[(Log[1 + (e_.)*(x_)]*PolyLog[2, (c_.)*(x_)])/(x_), x_Symbol] :> -Simp[PolyLog[2, c*x]^2/2, x] /; FreeQ[{c,
e}, x] && EqQ[c + e, 0]

Rubi steps

\begin{align*} \int \frac{\log (1-c x) \text{Li}_2(c x)}{x} \, dx &=-\frac{1}{2} \text{Li}_2(c x){}^2\\ \end{align*}

Mathematica [A]  time = 0.0091507, size = 11, normalized size = 1. \[ -\frac{1}{2} \text{PolyLog}(2,c x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[1 - c*x]*PolyLog[2, c*x])/x,x]

[Out]

-PolyLog[2, c*x]^2/2

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Maple [A]  time = 0.046, size = 10, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\it polylog} \left ( 2,cx \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(-c*x+1)*polylog(2,c*x)/x,x)

[Out]

-1/2*polylog(2,c*x)^2

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Maxima [A]  time = 0.966354, size = 11, normalized size = 1. \begin{align*} -\frac{1}{2} \,{\rm Li}_2\left (c x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-c*x+1)*polylog(2,c*x)/x,x, algorithm="maxima")

[Out]

-1/2*dilog(c*x)^2

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Fricas [A]  time = 2.53436, size = 26, normalized size = 2.36 \begin{align*} -\frac{1}{2} \,{\rm Li}_2\left (c x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-c*x+1)*polylog(2,c*x)/x,x, algorithm="fricas")

[Out]

-1/2*dilog(c*x)^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (- c x + 1 \right )} \operatorname{Li}_{2}\left (c x\right )}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(-c*x+1)*polylog(2,c*x)/x,x)

[Out]

Integral(log(-c*x + 1)*polylog(2, c*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-c*x+1)*polylog(2,c*x)/x,x, algorithm="giac")

[Out]

integrate(dilog(c*x)*log(-c*x + 1)/x, x)