3.1.22 \(\int \frac {\text {PolyLog}(2,a x^2)}{x} \, dx\) [22]

Optimal. Leaf size=11 \[ \frac {1}{2} \text {PolyLog}\left (3,a x^2\right ) \]

[Out]

1/2*polylog(3,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 = {6724} \begin {gather*} \frac {\text {Li}_3\left (a x^2\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^2]/x,x]

[Out]

PolyLog[3, a*x^2]/2

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\left (a x^2\right )}{x} \, dx &=\frac {\text {Li}_3\left (a x^2\right )}{2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {1}{2} \text {PolyLog}\left (3,a x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^2]/x,x]

[Out]

PolyLog[3, a*x^2]/2

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Maple [A]
time = 0.82, size = 10, normalized size = 0.91

method	result	size
default	\(\frac {\polylog \left (3, a \,x^{2}\right )}{2}\)	\(10\)
meijerg	\(\frac {\polylog \left (3, a \,x^{2}\right )}{2}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*polylog(3,a*x^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integral(dilog(a*x^2)/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (a x^{2}\right )}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(dilog(a*x^2)/x, x)

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Mupad [B]
time = 0.17, size = 9, normalized size = 0.82 \begin {gather*} \frac {\mathrm {polylog}\left (3,a\,x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x^2)/x,x)

[Out]

polylog(3, a*x^2)/2

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