3.48 \(\int \frac{\text{PolyLog}(2,a x^q)}{x} \, dx\)

Optimal. Leaf size=11 \[ \frac{\text{PolyLog}\left (3,a x^q\right )}{q} \]

[Out]

PolyLog[3, a*x^q]/q

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Rubi [A]  time = 0.0102243, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6589} \[ \frac{\text{PolyLog}\left (3,a x^q\right )}{q} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q]/x,x]

[Out]

PolyLog[3, a*x^q]/q

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\left (a x^q\right )}{x} \, dx &=\frac{\text{Li}_3\left (a x^q\right )}{q}\\ \end{align*}

Mathematica [A]  time = 0.0017468, size = 11, normalized size = 1. \[ \frac{\text{PolyLog}\left (3,a x^q\right )}{q} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^q]/x,x]

[Out]

PolyLog[3, a*x^q]/q

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Maple [A]  time = 0.044, size = 12, normalized size = 1.1 \begin{align*}{\frac{{\it polylog} \left ( 3,a{x}^{q} \right ) }{q}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(3,a*x^q)/q

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{6} \, q^{2} \log \left (x\right )^{3} + \frac{1}{2} \, q \log \left (-a x^{q} + 1\right ) \log \left (x\right )^{2} - q^{2} \int \frac{\log \left (x\right )^{2}}{2 \,{\left (a x x^{q} - x\right )}}\,{d x} +{\rm Li}_2\left (a x^{q}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*q^2*log(x)^3 + 1/2*q*log(-a*x^q + 1)*log(x)^2 - q^2*integrate(1/2*log(x)^2/(a*x*x^q - x), x) + dilog(a*x^
q)*log(x)

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Fricas [A]  time = 2.83852, size = 28, normalized size = 2.55 \begin{align*} \frac{{\rm polylog}\left (3, a x^{q}\right )}{q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(3, a*x^q)/q

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{2}\left (a x^{q}\right )}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(polylog(2, a*x**q)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x^q)/x, x)