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

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

[Out]

x*PolyLog[-1/2, a*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0079878, antiderivative size = 9, normalized size of antiderivative = 1., 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*}

Mathematica [F]  time = 0.0069411, size = 0, normalized size = 0. \[ \int \left (\text{PolyLog}\left (-\frac{3}{2},a x\right )+\text{PolyLog}\left (-\frac{1}{2},a x\right )\right ) \, dx \]

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]

________________________________________________________________________________________

Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\it polylog} \left ( -{\frac{3}{2}},ax \right ) +{\it polylog} \left ( -{\frac{1}{2}},ax \right ) \, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{-\frac{1}{2}}(a x) +{\rm Li}_{-\frac{3}{2}}(a x)\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm polylog}\left (-\frac{1}{2}, a x\right ) +{\rm polylog}\left (-\frac{3}{2}, a x\right ), x\right ) \end{align*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{-\frac{1}{2}}(a x) +{\rm Li}_{-\frac{3}{2}}(a x)\,{d x} \end{align*}

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)