3.55 \(\int \frac {\text {Li}_3(a x^q)}{x} \, dx\)

Optimal. Leaf size=11 \[ \frac {\text {Li}_4\left (a x^q\right )}{q} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {\text {Li}_4\left (a x^q\right )}{q} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^q]/x,x]

[Out]

PolyLog[4, a*x^q]/q

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fricas [A]  time = 0.46, size = 11, normalized size = 1.00 \[ \frac {{\rm polylog}\left (4, a x^{q}\right )}{q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(4, a*x^q)/q

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{q})}{x}\,{d x} \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 1.09 \[ \frac {\polylog \left (4, a \,x^{q}\right )}{q} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{24} \, q^{3} \log \relax (x)^{4} - \frac {1}{6} \, q^{2} \log \left (-a x^{q} + 1\right ) \log \relax (x)^{3} + q^{3} \int \frac {\log \relax (x)^{3}}{6 \, {\left (a x x^{q} - x\right )}}\,{d x} - \frac {1}{2} \, q {\rm Li}_2\left (a x^{q}\right ) \log \relax (x)^{2} + \log \relax (x) {\rm Li}_{3}(a x^{q}) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.09 \[ \int \frac {\mathrm {polylog}\left (3,a\,x^q\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, a*x^q)/x,x)

[Out]

int(polylog(3, a*x^q)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{q}\right )}{x}\, dx \]

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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