3.101 \(\int (\text {Li}_{-\frac {3}{2}}(a x)+\text {Li}_{-\frac {1}{2}}(a x)) \, dx\)

Optimal. Leaf size=9 \[ x \text {Li}_{-\frac {1}{2}}(a x) \]

[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 = {6587} \[ x \text {PolyLog}\left (-\frac {1}{2},a x\right ) \]

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 6587

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 \[ \int \left (\text {Li}_{-\frac {3}{2}}(a x)+\text {Li}_{-\frac {1}{2}}(a x)\right ) \, dx \]

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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm polylog}\left (-\frac {1}{2}, a x\right ) + {\rm polylog}\left (-\frac {3}{2}, a x\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Li}_{-\frac {1}{2}}(a x) + {\rm Li}_{-\frac {3}{2}}(a x)\,{d x} \]

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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maple [F]  time = 0.01, 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 \[ \int {\rm Li}_{-\frac {1}{2}}(a x) + {\rm Li}_{-\frac {3}{2}}(a x)\,{d x} \]

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

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

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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