3.173 \(\int \frac{(g+h \log (1-c x)) \text{PolyLog}(2,c x)}{x} \, dx\)

Optimal. Leaf size=20 \[ g \text{PolyLog}(3,c x)-\frac{1}{2} h \text{PolyLog}(2,c x)^2 \]

[Out]

-(h*PolyLog[2, c*x]^2)/2 + g*PolyLog[3, c*x]

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Rubi [A]  time = 0.0575455, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6602, 6589, 6601} \[ g \text{PolyLog}(3,c x)-\frac{1}{2} h \text{PolyLog}(2,c x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(h*PolyLog[2, c*x]^2)/2 + g*PolyLog[3, c*x]

Rule 6602

Int[((Log[1 + (e_.)*(x_)]*(h_.) + (g_))*PolyLog[2, (c_.)*(x_)])/(x_), x_Symbol] :> Dist[g, Int[PolyLog[2, c*x]
/x, x], x] + Dist[h, Int[(Log[1 + e*x]*PolyLog[2, c*x])/x, x], x] /; FreeQ[{c, e, g, h}, x] && EqQ[c + e, 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]

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{(g+h \log (1-c x)) \text{Li}_2(c x)}{x} \, dx &=g \int \frac{\text{Li}_2(c x)}{x} \, dx+h \int \frac{\log (1-c x) \text{Li}_2(c x)}{x} \, dx\\ &=-\frac{1}{2} h \text{Li}_2(c x){}^2+g \text{Li}_3(c x)\\ \end{align*}

Mathematica [A]  time = 0.0110358, size = 20, normalized size = 1. \[ g \text{PolyLog}(3,c x)-\frac{1}{2} h \text{PolyLog}(2,c x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(h*PolyLog[2, c*x]^2)/2 + g*PolyLog[3, c*x]

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Maple [A]  time = 0.155, size = 19, normalized size = 1. \begin{align*} -{\frac{h \left ({\it polylog} \left ( 2,cx \right ) \right ) ^{2}}{2}}+g{\it polylog} \left ( 3,cx \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*h*polylog(2,c*x)^2+g*polylog(3,c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((h*dilog(c*x)*log(-c*x + 1) + g*dilog(c*x))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((g + h*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{{\left (h \log \left (-c x + 1\right ) + g\right )}{\rm Li}_2\left (c x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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