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

Optimal. Leaf size=71 \[ \frac{a q^2 x^{q+2} \text{Hypergeometric2F1}\left (1,\frac{q+2}{q},2 \left (\frac{1}{q}+1\right ),a x^q\right )}{4 (q+2)}+\frac{1}{2} x^2 \text{PolyLog}\left (2,a x^q\right )+\frac{1}{4} q x^2 \log \left (1-a x^q\right ) \]

[Out]

(a*q^2*x^(2 + q)*Hypergeometric2F1[1, (2 + q)/q, 2*(1 + q^(-1)), a*x^q])/(4*(2 + q)) + (q*x^2*Log[1 - a*x^q])/
4 + (x^2*PolyLog[2, a*x^q])/2

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Rubi [A]  time = 0.0317548, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6591, 2455, 364} \[ \frac{1}{2} x^2 \text{PolyLog}\left (2,a x^q\right )+\frac{a q^2 x^{q+2} \, _2F_1\left (1,\frac{q+2}{q};2 \left (1+\frac{1}{q}\right );a x^q\right )}{4 (q+2)}+\frac{1}{4} q x^2 \log \left (1-a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*PolyLog[2, a*x^q],x]

[Out]

(a*q^2*x^(2 + q)*Hypergeometric2F1[1, (2 + q)/q, 2*(1 + q^(-1)), a*x^q])/(4*(2 + q)) + (q*x^2*Log[1 - a*x^q])/
4 + (x^2*PolyLog[2, a*x^q])/2

Rule 6591

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

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

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 x \text{Li}_2\left (a x^q\right ) \, dx &=\frac{1}{2} x^2 \text{Li}_2\left (a x^q\right )+\frac{1}{2} q \int x \log \left (1-a x^q\right ) \, dx\\ &=\frac{1}{4} q x^2 \log \left (1-a x^q\right )+\frac{1}{2} x^2 \text{Li}_2\left (a x^q\right )+\frac{1}{4} \left (a q^2\right ) \int \frac{x^{1+q}}{1-a x^q} \, dx\\ &=\frac{a q^2 x^{2+q} \, _2F_1\left (1,\frac{2+q}{q};2 \left (1+\frac{1}{q}\right );a x^q\right )}{4 (2+q)}+\frac{1}{4} q x^2 \log \left (1-a x^q\right )+\frac{1}{2} x^2 \text{Li}_2\left (a x^q\right )\\ \end{align*}

Mathematica [A]  time = 0.036441, size = 69, normalized size = 0.97 \[ \frac{q x^2 \left (a q x^q \text{Hypergeometric2F1}\left (1,\frac{q+2}{q},\frac{2}{q}+2,a x^q\right )+(q+2) \log \left (1-a x^q\right )\right )}{4 (q+2)}+\frac{1}{2} x^2 \text{PolyLog}\left (2,a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*PolyLog[2, a*x^q],x]

[Out]

(q*x^2*(a*q*x^q*Hypergeometric2F1[1, (2 + q)/q, 2 + 2/q, a*x^q] + (2 + q)*Log[1 - a*x^q]))/(4*(2 + q)) + (x^2*
PolyLog[2, a*x^q])/2

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Maple [C]  time = 0.207, size = 108, normalized size = 1.5 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{-2\,{q}^{-1}} \left ( -{\frac{{q}^{2}{x}^{2}\ln \left ( 1-a{x}^{q} \right ) }{4} \left ( -a \right ) ^{2\,{q}^{-1}}}-{\frac{q{x}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{2+q} \left ( -a \right ) ^{2\,{q}^{-1}} \left ( 1+{\frac{q}{2}} \right ) }-{\frac{{q}^{2}{x}^{2+q}a}{4} \left ( -a \right ) ^{2\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{2+q}{q}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-a)^(-2/q)/q*(-1/4*q^2*x^2*(-a)^(2/q)*ln(1-a*x^q)-q/(2+q)*x^2*(-a)^(2/q)*(1+1/2*q)*polylog(2,a*x^q)-1/4*q^2*
x^(2+q)*a*(-a)^(2/q)*LerchPhi(a*x^q,1,(2+q)/q))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, q^{2} x^{2} + \frac{1}{4} \, q x^{2} \log \left (-a x^{q} + 1\right ) + \frac{1}{2} \, x^{2}{\rm Li}_2\left (a x^{q}\right ) - q^{2} \int \frac{x}{4 \,{\left (a x^{q} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*q^2*x^2 + 1/4*q*x^2*log(-a*x^q + 1) + 1/2*x^2*dilog(a*x^q) - q^2*integrate(1/4*x/(a*x^q - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*dilog(a*x^q), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*polylog(2, a*x**q), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*dilog(a*x^q), x)