3.58 \(\int \frac{\text{PolyLog}(3,a x^q)}{x^4} \, dx\)

Optimal. Leaf size=93 \[ -\frac{a q^3 x^{q-3} \text{Hypergeometric2F1}\left (1,-\frac{3-q}{q},2-\frac{3}{q},a x^q\right )}{27 (3-q)}-\frac{q \text{PolyLog}\left (2,a x^q\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^q\right )}{3 x^3}+\frac{q^2 \log \left (1-a x^q\right )}{27 x^3} \]

[Out]

-(a*q^3*x^(-3 + q)*Hypergeometric2F1[1, -((3 - q)/q), 2 - 3/q, a*x^q])/(27*(3 - q)) + (q^2*Log[1 - a*x^q])/(27
*x^3) - (q*PolyLog[2, a*x^q])/(9*x^3) - PolyLog[3, a*x^q]/(3*x^3)

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Rubi [A]  time = 0.0508152, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2455, 364} \[ -\frac{q \text{PolyLog}\left (2,a x^q\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^q\right )}{3 x^3}-\frac{a q^3 x^{q-3} \, _2F_1\left (1,-\frac{3-q}{q};2-\frac{3}{q};a x^q\right )}{27 (3-q)}+\frac{q^2 \log \left (1-a x^q\right )}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^q]/x^4,x]

[Out]

-(a*q^3*x^(-3 + q)*Hypergeometric2F1[1, -((3 - q)/q), 2 - 3/q, a*x^q])/(27*(3 - q)) + (q^2*Log[1 - a*x^q])/(27
*x^3) - (q*PolyLog[2, a*x^q])/(9*x^3) - PolyLog[3, a*x^q]/(3*x^3)

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 \frac{\text{Li}_3\left (a x^q\right )}{x^4} \, dx &=-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}+\frac{1}{3} q \int \frac{\text{Li}_2\left (a x^q\right )}{x^4} \, dx\\ &=-\frac{q \text{Li}_2\left (a x^q\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}-\frac{1}{9} q^2 \int \frac{\log \left (1-a x^q\right )}{x^4} \, dx\\ &=\frac{q^2 \log \left (1-a x^q\right )}{27 x^3}-\frac{q \text{Li}_2\left (a x^q\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}+\frac{1}{27} \left (a q^3\right ) \int \frac{x^{-4+q}}{1-a x^q} \, dx\\ &=-\frac{a q^3 x^{-3+q} \, _2F_1\left (1,-\frac{3-q}{q};2-\frac{3}{q};a x^q\right )}{27 (3-q)}+\frac{q^2 \log \left (1-a x^q\right )}{27 x^3}-\frac{q \text{Li}_2\left (a x^q\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}\\ \end{align*}

Mathematica [C]  time = 0.0104082, size = 41, normalized size = 0.44 \[ -\frac{G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,\frac{q+3}{q} \\ 1,0,0,0,\frac{3}{q} \\\end{array}\right )}{q x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[PolyLog[3, a*x^q]/x^4,x]

[Out]

-(MeijerG[{{1, 1, 1, 1, (3 + q)/q}, {}}, {{1}, {0, 0, 0, 3/q}}, -(a*x^q)]/(q*x^3))

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Maple [C]  time = 0.34, size = 132, normalized size = 1.4 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{3\,{q}^{-1}} \left ( -{\frac{{q}^{3}\ln \left ( 1-a{x}^{q} \right ) }{27\,{x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}}}+{\frac{{q}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{9\,{x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}}}-{\frac{q{\it polylog} \left ( 3,a{x}^{q} \right ) }{ \left ( -3+q \right ){x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}} \left ( 1-{\frac{q}{3}} \right ) }-{\frac{{q}^{3}{x}^{-3+q}a}{27} \left ( -a \right ) ^{-3\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{-3+q}{q}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x^q)/x^4,x)

[Out]

-(-a)^(3/q)/q*(-1/27*q^3/x^3*(-a)^(-3/q)*ln(1-a*x^q)+1/9*q^2/x^3*(-a)^(-3/q)*polylog(2,a*x^q)-q/(-3+q)/x^3*(-a
)^(-3/q)*(1-1/3*q)*polylog(3,a*x^q)-1/27*q^3*x^(-3+q)*a*(-a)^(-3/q)*LerchPhi(a*x^q,1,(-3+q)/q))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -q^{3} \int \frac{1}{27 \,{\left (a x^{4} x^{q} - x^{4}\right )}}\,{d x} + \frac{q^{3} + 3 \, q^{2} \log \left (-a x^{q} + 1\right ) - 9 \, q{\rm Li}_2\left (a x^{q}\right ) - 27 \,{\rm Li}_{3}(a x^{q})}{81 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^q)/x^4,x, algorithm="maxima")

[Out]

-q^3*integrate(1/27/(a*x^4*x^q - x^4), x) + 1/81*(q^3 + 3*q^2*log(-a*x^q + 1) - 9*q*dilog(a*x^q) - 27*polylog(
3, a*x^q))/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm polylog}\left (3, a x^{q}\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^q)/x^4,x, algorithm="fricas")

[Out]

integral(polylog(3, a*x^q)/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (a x^{q}\right )}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x**q)/x**4,x)

[Out]

Integral(polylog(3, a*x**q)/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x^{q})}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^q)/x^4,x, algorithm="giac")

[Out]

integrate(polylog(3, a*x^q)/x^4, x)