3.14 \(\int \text{PolyLog}(3,a x) \, dx\)

Optimal. Leaf size=34 \[ x (-\text{PolyLog}(2,a x))+x \text{PolyLog}(3,a x)+\frac{(1-a x) \log (1-a x)}{a}+x \]

[Out]

x + ((1 - a*x)*Log[1 - a*x])/a - x*PolyLog[2, a*x] + x*PolyLog[3, a*x]

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Rubi [A]  time = 0.0108174, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6586, 2389, 2295} \[ x (-\text{PolyLog}(2,a x))+x \text{PolyLog}(3,a x)+\frac{(1-a x) \log (1-a x)}{a}+x \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x],x]

[Out]

x + ((1 - a*x)*Log[1 - a*x])/a - x*PolyLog[2, a*x] + x*PolyLog[3, a*x]

Rule 6586

Int[PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[x*PolyLog[n, a*(b*x^p)^q], x] - Dist[p*q, I
nt[PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int \text{Li}_3(a x) \, dx &=x \text{Li}_3(a x)-\int \text{Li}_2(a x) \, dx\\ &=-x \text{Li}_2(a x)+x \text{Li}_3(a x)-\int \log (1-a x) \, dx\\ &=-x \text{Li}_2(a x)+x \text{Li}_3(a x)+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-a x)}{a}\\ &=x+\frac{(1-a x) \log (1-a x)}{a}-x \text{Li}_2(a x)+x \text{Li}_3(a x)\\ \end{align*}

Mathematica [A]  time = 0.0116823, size = 39, normalized size = 1.15 \[ x \left (-\text{PolyLog}(2,a x)+\text{PolyLog}(3,a x)+\frac{\log (1-a x)}{a x}-\log (1-a x)+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x],x]

[Out]

x*(1 - Log[1 - a*x] + Log[1 - a*x]/(a*x) - PolyLog[2, a*x] + PolyLog[3, a*x])

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Maple [A]  time = 0.092, size = 41, normalized size = 1.2 \begin{align*}{\frac{1}{a} \left ( ax+{\frac{ \left ( -2\,ax+2 \right ) \ln \left ( -ax+1 \right ) }{2}}-ax{\it polylog} \left ( 2,ax \right ) +ax{\it polylog} \left ( 3,ax \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x),x)

[Out]

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

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Maxima [A]  time = 0.990846, size = 53, normalized size = 1.56 \begin{align*} -\frac{a x{\rm Li}_2\left (a x\right ) - a x{\rm Li}_{3}(a x) - a x +{\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x, algorithm="maxima")

[Out]

-(a*x*dilog(a*x) - a*x*polylog(3, a*x) - a*x + (a*x - 1)*log(-a*x + 1))/a

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Fricas [C]  time = 2.5983, size = 151, normalized size = 4.44 \begin{align*} -\frac{a x{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - a x{\rm polylog}\left (3, a x\right ) - a x +{\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x, algorithm="fricas")

[Out]

-(a*x*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - a*x*polylog(3, a*x) - a*x + (a*x - 1)*log(-a*x + 1))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Li}_{3}\left (a x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x)

[Out]

Integral(polylog(3, a*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{3}(a x)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x, algorithm="giac")

[Out]

integrate(polylog(3, a*x), x)