3.47 \(\int \text{PolyLog}(2,a x^q) \, dx\)

Optimal. Leaf size=54 \[ \frac{a q^2 x^{q+1} \text{Hypergeometric2F1}\left (1,\frac{1}{q}+1,\frac{1}{q}+2,a x^q\right )}{q+1}+x \text{PolyLog}\left (2,a x^q\right )+q x \log \left (1-a x^q\right ) \]

[Out]

(a*q^2*x^(1 + q)*Hypergeometric2F1[1, 1 + q^(-1), 2 + q^(-1), a*x^q])/(1 + q) + q*x*Log[1 - a*x^q] + x*PolyLog
[2, a*x^q]

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Rubi [A]  time = 0.023219, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6586, 2448, 364} \[ x \text{PolyLog}\left (2,a x^q\right )+\frac{a q^2 x^{q+1} \, _2F_1\left (1,1+\frac{1}{q};2+\frac{1}{q};a x^q\right )}{q+1}+q x \log \left (1-a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q],x]

[Out]

(a*q^2*x^(1 + q)*Hypergeometric2F1[1, 1 + q^(-1), 2 + q^(-1), a*x^q])/(1 + q) + q*x*Log[1 - a*x^q] + x*PolyLog
[2, a*x^q]

Rule 6586

Int[PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[x*PolyLog[n, a*(b*x^p)^q], x] - Dist[p*q, I
nt[PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \text{Li}_2\left (a x^q\right ) \, dx &=x \text{Li}_2\left (a x^q\right )+q \int \log \left (1-a x^q\right ) \, dx\\ &=q x \log \left (1-a x^q\right )+x \text{Li}_2\left (a x^q\right )+\left (a q^2\right ) \int \frac{x^q}{1-a x^q} \, dx\\ &=\frac{a q^2 x^{1+q} \, _2F_1\left (1,1+\frac{1}{q};2+\frac{1}{q};a x^q\right )}{1+q}+q x \log \left (1-a x^q\right )+x \text{Li}_2\left (a x^q\right )\\ \end{align*}

Mathematica [A]  time = 0.0450111, size = 51, normalized size = 0.94 \[ q x \left (\frac{a q x^q \text{Hypergeometric2F1}\left (1,\frac{1}{q}+1,\frac{1}{q}+2,a x^q\right )}{q+1}+\log \left (1-a x^q\right )\right )+x \text{PolyLog}\left (2,a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^q],x]

[Out]

q*x*((a*q*x^q*Hypergeometric2F1[1, 1 + q^(-1), 2 + q^(-1), a*x^q])/(1 + q) + Log[1 - a*x^q]) + x*PolyLog[2, a*
x^q]

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Maple [C]  time = 0.207, size = 88, normalized size = 1.6 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{-{q}^{-1}} \left ( -{q}^{2}x\sqrt [q]{-a}\ln \left ( 1-a{x}^{q} \right ) -qx\sqrt [q]{-a}{\it polylog} \left ( 2,a{x}^{q} \right ) -{q}^{2}{x}^{1+q}a\sqrt [q]{-a}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{1+q}{q}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^q),x)

[Out]

-1/q*(-a)^(-1/q)*(-q^2*x*(-a)^(1/q)*ln(1-a*x^q)-q*x*(-a)^(1/q)*polylog(2,a*x^q)-q^2*x^(1+q)*a*(-a)^(1/q)*Lerch
Phi(a*x^q,1,(1+q)/q))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -q^{2} x - q^{2} \int \frac{1}{a x^{q} - 1}\,{d x} + q x \log \left (-a x^{q} + 1\right ) + x{\rm Li}_2\left (a x^{q}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-q^2*x - q^2*integrate(1/(a*x^q - 1), x) + q*x*log(-a*x^q + 1) + x*dilog(a*x^q)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm Li}_2\left (a x^{q}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q),x, algorithm="fricas")

[Out]

integral(dilog(a*x^q), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(polylog(2, a*x**q), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_2\left (a x^{q}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q),x, algorithm="giac")

[Out]

integrate(dilog(a*x^q), x)