3.55 \(\int \frac{\text{PolyLog}(3,a x^q)}{x} \, dx\)

Optimal. Leaf size=11 \[ \frac{\text{PolyLog}\left (4,a x^q\right )}{q} \]

[Out]

PolyLog[4, a*x^q]/q

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Rubi [A]  time = 0.0097364, 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} \[ \frac{\text{PolyLog}\left (4,a x^q\right )}{q} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^q]/x,x]

[Out]

PolyLog[4, 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}_3\left (a x^q\right )}{x} \, dx &=\frac{\text{Li}_4\left (a x^q\right )}{q}\\ \end{align*}

Mathematica [A]  time = 0.0014392, size = 11, normalized size = 1. \[ \frac{\text{PolyLog}\left (4,a x^q\right )}{q} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^q]/x,x]

[Out]

PolyLog[4, a*x^q]/q

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Maple [A]  time = 0.041, size = 12, normalized size = 1.1 \begin{align*}{\frac{{\it polylog} \left ( 4,a{x}^{q} \right ) }{q}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(4,a*x^q)/q

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{24} \, q^{3} \log \left (x\right )^{4} - \frac{1}{6} \, q^{2} \log \left (-a x^{q} + 1\right ) \log \left (x\right )^{3} + q^{3} \int \frac{\log \left (x\right )^{3}}{6 \,{\left (a x x^{q} - x\right )}}\,{d x} - \frac{1}{2} \, q{\rm Li}_2\left (a x^{q}\right ) \log \left (x\right )^{2} + \log \left (x\right ){\rm Li}_{3}(a x^{q}) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*q^3*log(x)^4 - 1/6*q^2*log(-a*x^q + 1)*log(x)^3 + q^3*integrate(1/6*log(x)^3/(a*x*x^q - x), x) - 1/2*q*di
log(a*x^q)*log(x)^2 + log(x)*polylog(3, a*x^q)

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Fricas [C]  time = 2.72366, size = 209, normalized size = 19. \begin{align*} -\frac{q^{2}{\rm \%iint}\left (a, q, x, -\frac{\log \left (-a x^{q} + 1\right )}{a}, -\log \left (-a x^{q} + 1\right ) \log \left (x\right ), -\frac{q \log \left (-a x^{q} + 1\right )}{x}\right ) \log \left (x\right )^{2} - q^{2}{\rm Li}_2\left (a x^{q}\right ) \log \left (x\right )^{2} - 2 \,{\rm polylog}\left (4, a x^{q}\right )}{2 \, q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(q^2*\%iint(a, q, x, -log(-a*x^q + 1)/a, -log(-a*x^q + 1)*log(x), -q*log(-a*x^q + 1)/x)*log(x)^2 - q^2*dil
og(a*x^q)*log(x)^2 - 2*polylog(4, a*x^q))/q

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(polylog(3, a*x**q)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(polylog(3, a*x^q)/x, x)