3.35 \(\int \frac {\text {Li}_3(a x^2)}{x} \, dx\)

Optimal. Leaf size=11 \[ \frac {\text {Li}_4\left (a x^2\right )}{2} \]

[Out]

1/2*polylog(4,a*x^2)

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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} \[ \frac {1}{2} \text {PolyLog}\left (4,a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^2]/x,x]

[Out]

PolyLog[4, a*x^2]/2

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^2\right )}{x} \, dx &=\frac {\text {Li}_4\left (a x^2\right )}{2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {\text {Li}_4\left (a x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^2]/x,x]

[Out]

PolyLog[4, a*x^2]/2

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm polylog}\left (3, a x^{2}\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x,x, algorithm="fricas")

[Out]

integral(polylog(3, a*x^2)/x, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x,x, algorithm="giac")

[Out]

integrate(polylog(3, a*x^2)/x, x)

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maple [A]  time = 0.13, size = 10, normalized size = 0.91 \[ \frac {\polylog \left (4, a \,x^{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x^2)/x,x)

[Out]

1/2*polylog(4,a*x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{3}(a x^{2})}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x,x, algorithm="maxima")

[Out]

integrate(polylog(3, a*x^2)/x, x)

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mupad [B]  time = 0.20, size = 9, normalized size = 0.82 \[ \frac {\mathrm {polylog}\left (4,a\,x^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, a*x^2)/x,x)

[Out]

polylog(4, a*x^2)/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x**2)/x,x)

[Out]

Integral(polylog(3, a*x**2)/x, x)

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