3.2.1 \(\int (\text {PolyLog}(-\frac {3}{2},a x)+\text {PolyLog}(-\frac {1}{2},a x)) \, dx\) [101]

Optimal. Leaf size=9 \[ x \text {PolyLog}\left (-\frac {1}{2},a x\right ) \]

[Out]

x*polylog(-1/2,a*x)

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6722} \begin {gather*} x \text {Li}_{-\frac {1}{2}}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[-3/2, a*x] + PolyLog[-1/2, a*x],x]

[Out]

x*PolyLog[-1/2, a*x]

Rule 6722

Int[PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[x*(PolyLog[n + 1, a*(b*x^p)^q]/(p*q)), x] -
Dist[1/(p*q), Int[PolyLog[n + 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \left (\text {Li}_{-\frac {3}{2}}(a x)+\text {Li}_{-\frac {1}{2}}(a x)\right ) \, dx &=\int \text {Li}_{-\frac {3}{2}}(a x) \, dx+\int \text {Li}_{-\frac {1}{2}}(a x) \, dx\\ &=x \text {Li}_{-\frac {1}{2}}(a x)\\ \end {align*}

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Mathematica [F]
time = 0.01, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\text {PolyLog}\left (-\frac {3}{2},a x\right )+\text {PolyLog}\left (-\frac {1}{2},a x\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[PolyLog[-3/2, a*x] + PolyLog[-1/2, a*x],x]

[Out]

Integrate[PolyLog[-3/2, a*x] + PolyLog[-1/2, a*x], x]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \polylog \left (-\frac {3}{2}, a x \right )+\polylog \left (-\frac {1}{2}, a x \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(-3/2,a*x)+polylog(-1/2,a*x),x)

[Out]

int(polylog(-3/2,a*x)+polylog(-1/2,a*x),x)

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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(-3/2,a*x)+polylog(-1/2,a*x),x, algorithm="maxima")

[Out]

integrate(polylog(-1/2, a*x) + polylog(-3/2, a*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(-3/2,a*x)+polylog(-1/2,a*x),x, algorithm="fricas")

[Out]

integral(polylog(-1/2, a*x) + polylog(-3/2, a*x), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\operatorname {Li}_{- \frac {3}{2}}\left (a x\right ) + \operatorname {Li}_{- \frac {1}{2}}\left (a x\right )\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(-3/2,a*x)+polylog(-1/2,a*x),x)

[Out]

Integral(polylog(-3/2, a*x) + polylog(-1/2, a*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(-3/2,a*x)+polylog(-1/2,a*x),x, algorithm="giac")

[Out]

integrate(polylog(-1/2, a*x) + polylog(-3/2, a*x), x)

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Mupad [B]
time = 0.37, size = 7, normalized size = 0.78 \begin {gather*} x\,\mathrm {polylog}\left (-\frac {1}{2},a\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(-1/2, a*x) + polylog(-3/2, a*x),x)

[Out]

x*polylog(-1/2, a*x)

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