3.1.6 \(\int \frac {\text {PolyLog}(2,a x)}{x} \, dx\) [6]

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

[Out]

polylog(3,a*x)

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Rubi [A]
time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, 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 = {6724} \begin {gather*} \text {Li}_3(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/x,x]

[Out]

PolyLog[3, a*x]

Rule 6724

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*}

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Mathematica [A]
time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} \text {PolyLog}(3,a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x]/x,x]

[Out]

PolyLog[3, a*x]

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Maple [A]
time = 0.09, size = 6, normalized size = 1.20

method	result	size
derivativedivides	\(\polylog \left (3, a x \right )\)	\(6\)
default	\(\polylog \left (3, a x \right )\)	\(6\)
meijerg	\(\polylog \left (3, a x \right )\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x)/x,x,method=_RETURNVERBOSE)

[Out]

polylog(3,a*x)

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Maxima [A]
time = 0.28, size = 5, normalized size = 1.00 \begin {gather*} {\rm Li}_{3}(a x) \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [A]
time = 0.40, size = 3, normalized size = 0.60 \begin {gather*} \operatorname {Li}_{3}\left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

polylog(3, a*x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.18, size = 5, normalized size = 1.00 \begin {gather*} \mathrm {polylog}\left (3,a\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x)/x,x)

[Out]

polylog(3, a*x)

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