3.6 \(\int \frac{\text{PolyLog}(2,a x)}{x} \, dx\)

Optimal. Leaf size=5 \[ \text{PolyLog}(3,a x) \]

[Out]

PolyLog[3, a*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0090331, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6589} \[ \text{PolyLog}(3,a x) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/x,x]

[Out]

PolyLog[3, a*x]

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{\text{Li}_2(a x)}{x} \, dx &=\text{Li}_3(a x)\\ \end{align*}

Mathematica [A]  time = 0.0012581, size = 5, normalized size = 1. \[ \text{PolyLog}(3,a x) \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x]/x,x]

[Out]

PolyLog[3, a*x]

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 6, normalized size = 1.2 \begin{align*}{\it polylog} \left ( 3,ax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x)/x,x)

[Out]

polylog(3,a*x)

________________________________________________________________________________________

Maxima [A]  time = 0.975433, size = 7, normalized size = 1.4 \begin{align*}{\rm Li}_{3}(a x) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x,x, algorithm="maxima")

[Out]

polylog(3, a*x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x,x, algorithm="fricas")

[Out]

integral(dilog(a*x)/x, x)

________________________________________________________________________________________

Sympy [A]  time = 1.71222, size = 3, normalized size = 0.6 \begin{align*} \operatorname{Li}_{3}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x,x)

[Out]

polylog(3, a*x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x,x, algorithm="giac")

[Out]

integrate(dilog(a*x)/x, x)